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  1. ✷ Ï ❒ ➱ Ï ➮ ❒ ❮ ❪ â ➴ ❰ ➮ ➘ ❐ ❒ ❮Ñ ✃ ❭ ➱ ➘ ➱ Û ➱ ➹ ➮ ❐ Ö Ò Ó ➷ ➹ ❩ ➱ Ï ❰ Ï ➘ ❐ ➴ ❒ ➹ Ö ❮ Û ❒ ❮ Ô ß Ö ➷ Ð ❮ ➘ ➮ ❐ ❒ ❮Ñ ✃ ❬ ➱ ➘ ã ➹ Ú ❰ ä ➹ ✃ ➹ Ö ➴ ✃ ❒ å ➮ Ï ❐ Ï ❒ ❒ ❐ ❒ ➹ ß Ö ➴ ➮ ➘ ➹ ➱ Ö Ï ➮ ❒ ❮ ➱ Ö Ï ➮ ❒ ➹ Ô ❒ ❰ ❮Ñ ➱ ➴ ❮ ➮ ❒ ❮ÏÐ ➹ ➴ ❮ ❒ ✃ ➹ ➴ ➷ ➱ ➘ ✸ Ó ➷ ➱ Ñ ❒ ➮ ❰ ➹ ❳ ➒ ➐ ➋ ➐ ➨ ➫ ➭➯ ➵ ➸ ➯ ➽ ❮ ➾ ➚ ➪ ➚ ❪ ❬ ➮ ➹ ➱ ✃ ❐ ➴ ➹ ➱ ➘ Ö Ï ➷ ➹ ❒ ❮ ❰ ➹ ➮ ➴ ➮ ❮ ➮ ❒ ❮ Ï Ð ➹ ➴ ❮ ❰ ❳ ❲ ➷ Ö ➱ Ï ❒ ❮ Ô ➹ ✃ Õ ➮ ❒ ➹ ➘ × ➹ Ò Ø Ï ➱Ù ❮ Ö ➹ ❒ Ï ➬ ❐ ➔ ÿ øù ø ú û ü ý ó þ ø ò � ó ✁ ò ✂ ✄ ò ☎ ✆ � ø � ô ò ✞ ê ➬ ❨ ➷ Ï ➮ ❒ ➹ ➴ Ò ❩ ë ÷ ì í î ñ ò ó ô õ ñ ö ✝ ✟ ➱ ✾❊ ✳ ✯ ✭ ✺ ✿ ❃❄ ❅❆ ✾ ✿ ✼ ❋ ✮ ● ❁ ❈ ❍ ■❏ ◆ ❖ ◆◗ ❘ ❨ ✯ ✯ ✁ ✗ � ☞ ✌ ✍ ✎ ✏ ✑ ✒ ✡ ✖ ✘✙ ✯✰ ✚ ✢ ✣ ✤ ★ ✩ ✪ ✫ ✬ ✭ ✮✯ ❰ ➷ ➷ æ ➹ â ➹ ❒ ➱ ➮ ❒ ➹ ➴ Ò ✃ ➷ ➱ ➷ Õ ç ➱ ß ➹ ❒ ➬ ➷ ➱ Ï ➘ ❒ ❮ ❒ ❮ Õ ➷ ➱ ❰ Ï ➮ ❐ ❮Ñ ➹ ✃ ➴ ➷ ➱ ➘ ➱ ✃ Ö Õ ✃ ➱ ➱ ➱ ❰ ✃ Ö ❮é ➹ â Õ ❒ ➱ ➮ ➬ Õ ❒ Þ ➹ ä ❮ é ❒ ❮Ñ ✃ ➱ ➘ ❮ è ➮ Ï ➴ ➱ ➱ Ñ ❒ è ❰ Õ Ð Ö ➹ Ô ➴ ❒ ➹ ➷ ➱ ß Ö ➹ ➴ ❒ ❮ ➙ ➼ ➏ ➣ ➐ ➒ → ➔ ➣ ➒ ➛ ➜ ➐ ➉ ➐ ➋ ➝ ➞ ➉ ➣ ➏ ➔ ➌ ↔ ➉ ➋ ❤ ➔ ➎ ➍ ➏ ➐➑ ➐ ➒ ➓ ➏ ➔ ➒ → ➏ ➒ ➔ ➣ ➏ ➔ ➌↔ ➉ ↕ ➔ ➏ ➔➙ ↕ ➏ ➌➍ ➐ ➛ ↔ → ➌ ➉ ↕ ➏ ➉ ➍ ➥ ➐ → ➉ ➞ ➉ ↕ ➏ ➔ ❏ ➦ ➍ ➤ ➔ ➔ ➛ ➙ ➒ ➐ ➋ ➐ ➋ ➐ ➌ ➉ ➟ ➏ ➏ ➐ ➣ ➉ ➐ ➍ ➉ ↕ ➍ ➐ ➌ ➎ ❦ ➉ ❖ ❏ ❀ ❑ ▲ ❀▼ ◆ ■ ❆ ■ P ❍ ◗ ❏ ■ ❘ ❙❚ ❯ ❱ ❲ ❳ ❨ ■ ❀ ❬ ❅ ✹ ✺ ✻ ✿ ❀ ❁ ❂ ❃ ✿ ❄ ❀ ❆ ❁ ❂ ❆❇ ❆ ❈ ❉ ❊ ❋ ❁ ● ❩ ❭ ➇ ❻ ❥ ❸ ⑩ ♥♦ ❻❼ ❽ ❾❼ ❿➀ ❧♠ ➀ ❥ ➃ ➂ ❽ ➄ ➁ ❾ ➅➆ ➂ ➂ ❿ ⑩ ♣ ❵ ♣ ❛ ❴ ❜ ❵❡ ❢❣ ❤ ✐ ❥ ❢❦ ♥ r ⑤ q ♣✈ ✇ ✈ ① ② ♣q ③ q ④ ➐ ❢ ➟ ➠ ➐ ➥ → ➉ ➞ ➉ ↕ ➏ ➔ ➡ ➉ ➦ ➍ ➐ ➥ → ➉ ➞ ➉ ↕ ➋ ➍ ➏ ➌ ➍ ➛ ➏ ➐ ➣ ➉ ➐ ➍ ➉ ↕ ➐ ↕ ➌ ➏ ➔ ❵ ➤ ➛ ↔ → ➌ ➉ ➔ ➔ ➉ ➓ ➟ ➉ ➏ ➐ ➣ ➏ ➔ ➧ ➍ ➜ ➐ ➐ ➏ ➐ ➣ ➏ ➔ ➌ ↔ ➉ ↕ ➉ ➍ ➣ ➜ ➏ ➔ ➌↔ ➉ ↕ ➔ ➤ ➉➩ ➐ ➧ ➦ ➏ ➉ → ➔ ➐ ➒ ➓ ➏ ➔ ➠ ➟ ➉ ➌ ➒ ❡ ❞ P ▲ ➌➍ ➎ ➍ ➏ ➐➑ ➐ ➓ ➐ ➏ ➔ ➒ → ➎ ➏ ➔ ➣ ➏ ➔ ➋ ▲ ➉ ➏ ➏ ➐ ➣ ➏ ➔ ➧ ➍ ➓ ➜ ➐ ➐ ➒ ➣ ➏ ➔ ➌↔ ➉ ↕ ➐ ➏ ➔ ➙ ➌↔ ➐ ↕ ➔ ➒ ➛ ➜ ➣ ➉ ➋ ➝ ➞ ➉ ➣ ➏ ➌ ➔ ↔ ➉ ↕ ➔ ➏ ➔ ➐ ➋ ➐ ➙ ➒ ▲ ➣ ➐ ➐ ❛ ➔ ➐ ➋ ➒ ➔➙ ➋ ➏ ➒ → integer): 4 number /* while (j < n) { */ ✶✷✹✸ 3 ❑✕P n: ✴✕✵ C /* j = j +1 */ - ❫✕❴ /* j = 0 */ ❑▼▲ number, ✱☛✲ double power_1(double x, int n) { power 1 (x: ✐✕❥ while (j > 0 ) { ❵☛❣ number y = 1; j = j -1; double y = 1; integer j = 0; ✥✧✦ ❇❉❈ y = y*x; while (j < n) j = j+1; y = y*x; j = n; return y; return y; ✛✕✜ int j; ❜❂❝ ❀❂❁ } ï✾ð ✽☛✾ ❱☛❲ - ✻☛✼ ✠☛✡ ✓✕✔ } Y ❙▼❚❯ 3 ÚÜÛ ➴✣➷ ➱✍Ò ↕✣➍ ↕✣➍ ➠➢➡ 3 àá➹ ➐✍➨ äá➹ àá➹ ♦✣♣ ↕✣➍ ↕✣➍ ↕✣➍ ↕✣➍ s✉t ➴✣➷ ♦✣♣q ➁✉➂ ❧✔♠ ❐✉❒ ➠➢➡ ➠➢➡ ❹✣❺ ➱✍Ò ➺➊➻ ❝✣❞ ➝✉➒ ❶✉❷ ➴✣➷ ➴✣➷➬ àáÏ ✼✾✽ ⑧✔⑨ ➲✣➳ - Ý✣Þ ➴✣➷ ➶➊➹➘ ⑥✔⑦ àáÏ ❪✣❫❴ ➈➊➉➋ ➈➊➉➋

  2. q ❞ ❪ ❛ s ❝ ❜ ❞ ❜ ❢ ① ❞ ❴ ⑥ ❞ ❡ ❪ ❵ ❢ q ❭ ❵ ❪ ❵ ❣ t ❧ ❴ ❵ ❵ ❜ ❡ ❪ ❦ ❴ ❦ ❞ ❴ ❪ ❥ ❤ ❥ ❛ r ❭ ✐ ❛ ❦ ② ❵ ❞ ❜❝ ❛ ❪ ❵ ❞ ❜❝ ❦ ❝ ① ❫ s ❛ ❜ ❭ ❝ ❜ ❣ ❵ ④ ④ ✐ ❜⑦ ❣ ❤ ⑤ ❤ ❵ s ❛ ❜ ❭ ❝ ❜❣ ❵ ④ ④ ❵ ❜ ❣ ❪ ❭ ❪ ❫ ❭ ❴ ❣ ❭ ❤ ❥ ✐ ❥ ① ❪❜ ❞ ❜ ❤ ❥ t ❴ ② ❵ ① ❦ ❥ ❡ ❪ t ❵ ❜⑦ ✐ ❪ ❫ ❭ ❴❵ ❛ ❪❜❝ ❞ ❡❢ ❜ ❛ ❜❣ ❤ ❥ ❬ ❦ ❤ ❧ ♠ ❜ ❦ ❴ ❞ ❥ ❥ ❤ ❥ ❭ ❯ ❥ ❆ ❉ ❍ ❆ ■ ❑ ❳ ❀ ❋ r ❚ ❨ ❍ ■ ❩ ❑ ❳ ❀ ❋ ❇ ❚ ▲ ❋ ❉ ▲ ❚ ❊ ❛ ❦ ❝ ❵ ❵ t ❧ ❦ ❴ ❭ ❪ ❫ ❭ ❴ ❣ ❪ ✈ ❥ ✇ ✐ ❞ ❴ ❞ ❜ ❜ ❤ q ❞ ❴ s ❞ ❤ ❴ q ❭ ❜ ❛ ❪ ❵ ❝ ❤ ♣ ❣ ❴ ❤ ✐ ❦ ❫ r ❜ ❦ ❴ ❦ ❵ ❥ ❤ ❥ ❝ ❜ s ✐ ❫ ❆ ❪ ❪❜ ❝ ❜ ❤ ❥ t ❴ ② ❵ ❡ ❭ ❧ ❴ ❵ ❭ ❝ ❜ s ❝ ❵ ① t ❡ ❪ ❦ ❴ ❛ ❜ ❪ ❞ ❞ ① ❤ ❥ ❡ ③ ⑤ ⑩ ❝ ❜ ❭ ❵ ❜ ❛ r ♠ ❴ ❞ ❫ ❦ ❞ ❴ ❪ ❥ ❤ ❥ ❭ ❫ ❥ ❾➀ ❵ ❛ ② ❞ ❴ ❣ ③ ⑤ ❹ ❺ ❸ ❺ ❾ ❫ ❹ ❸ ➂ ❼ ➃ ➄ ➅ ➆ ➉ ➊ ➋ ➉➌ s ✉ ❭ ❭ ❴ ❣ ❥ ❪ ❥ ❭ ✇ ❶ ❵ ❞ ❜ ❝ ❵ ❴ ❞ ❜ r ♠ ❴ ❞ ❫ s ❵ ❛ ② ❞ ❡ ❝ ❣ ❦ ❡ ❪ ❵ ❢ q ❭ ❵ ❪ ❵ ❣ t ❧ ❵ ⑤ ❝ ❣ ❭ ❪ ❫ ❭ ❴ ❣ ❭ ❤ ❥ ✐ ⑥ ❡ ① ❪ ❭ ❤ ❥ ✐ ❵ ❛ ❪❜ ❭ ❴ ❪ ❵ ❛ ❜ ❥ ❝ ❞ ❥ ❡ ❛ s ❝ ❥ ❡ ❞ ① ❤ ❥ ❪ s ❥ ❴ ♠ ❜ s ❫ ❧ ❪ ❜ ❦ ① ❪❥ ♠ ❦ ✉ ❞ ❥ ❝ ❢ ❵ ❛ ♣ ❪ ❞ ❥ ❡ ❛ s ❜ ❵ ② ❤ ❛ ❦ ❴ ❣ ❪ ❜ ❜ ❤ ❥ t ❴ ❵ ② ① ❦ ❥ ❡ ❪ ❧ ❪ ❜ ❤ ❛ ❪ ❜ ❘ ❇ ❈ Ð ❮ ❒ Ø Ù ➮ Ò Ö Ù Ó Ú ➮ à Þ Û ✃ Ö Ð Ù❮ Ñ ➮ á Ù Ð Ó ß ➱ Ù ✃ ➮ Ñ ➮ ❮ Ù ➮ Ú ➮ ❮ Ö Ù ❐ Õ Ù ➱ × Ü Ð Ù ➱ Û ❐ Ý Û â Ò Õ Ú æ Ó ❐ ➮ ❚ Õ Ù Ú ❮ Ù ➮ Ù Ò ❒ Ù Ö ❮ Ñ ã ➱ ➮ ã Ð ❮ Þ Ù ❐ ã Ö Õ ❐ ➮ Ú Ù ❒ Ù Ö ❮ Ñ Ö Û Õ Ö ➮ ✃❐ ã ➱ ➮ ã Ð ❮ á Ö Ö Û ❼ ❾ ❿ ③➀ ➁ ❼ ⑨ ❼ ➂ ➃ ➄ ❽ ➅ ❽ ➆ ➇➈ ➉ ➆ ➇ ➊ ➇ ➒ ➍ ➏ ➎ ③ ❼ ➓ ③ s t ✉ ② ③ ④ ⑤ ⑥ ② ⑦ ⑧ ④ ❻ ⑤ ⑨⑩ ⑨ ❶ ❷ ❸ ❹ ④ ❺ ⑨ ③ ➆ ➔ Ø ➴ ➦ ➦ ➛ ➞ ➤ ➼ ➦ ➥ ➝ ➠ ➪➶ ➘ ➭ ➷ ✃❐ ➮❰ ✃ ❮ ÑÒ Ð ➱ Õ Ö ➮× ➛ ➥➻ → ➜ ➣↔ ↕ ➟ ➞ ➛ ➠ ➛➤ ➞ ➥ ➧➨ ➥➩ ➥➩ ➭ ➦ ➨ ➯ ➦ ➟ ➩ ➠ ➥ ➸ ➺ ➛ ➢ á ❮ Ù ✯ ✫ ✕ ✬ ✭ ★ ✛ ★ ✮ ✰ ✕ ✩ ★ ❄ ❙ ❅ ■ ▲ ❖ ✪ ✩ ❆ ✢ ❆ ✖ ✗ ✛ ✜ ✛ ❂ ▲ ✣ ★ ✤ ✥ ✖ ✦ ✛ ✕ ❃ ✧ ✵ ✶ ✙ ❃ ❃ ❂ ❃ ● ❃ ❂ ❍ ■ ❀ ❈ ❋ ❑ ❃ ❀ ▲ ❋❖ ■ ■ ❃❋ ❇ ➱ ✽ ❖ ✷ ✲ ❈ ❇ ✻ ❖ ✼ ✻✾ ■ ❅ ◗ ❂ ❃ ❄ ■ P ❃❅❆ ❇❈ ✚ ✕ ✔ ▼ ❖ ❆ ▲ óô ❆ ■ ◆ õö ù ❖ ú ùû ü ý ❚❊ þ ÿ ✁ é ◗ ✘ ã å ❮ ➮ Ö ▲ Ù Ú Ð ç ❆ ä Õ Û ➮ Û ➮ ➱ è ❳ ÿ ✂ ✄ ✍ ▲ ▲ ❉ ❋ ▲ ☞ ✌ ✎ ❳ ❘ ✏ ❃ ✔ ✕ ✖ ✗ ❄ ❙ ❲ ■ ❋ ✝ ✞ ✟ ✠ ✡ ☎ ❉ ☛ ▲ ❚ ❳ ▼ ▼ ❇ 6 3 ⑧⑨❜ ▼❁◆ ❚❊❱❯ ❣♦♥ ❜❱⑤ ❄❏■ ➇✆➈ ▼❁❋ ❻❏➁ ❉❁❊ ✸✳✹✺ ❉❁❘ ❻❏❿ ③❏④ ❽✆❾ ✉❏❜ ✑✓✒ ❻❏❼ - ✱✳✲✴ ✿❁❀ ❷❁❸ 5 else 3 ✁✆☎ ä✧å return x*y*y; y = power 2 (x,  n/2  ) return y*y; if (n == 1) return x Ó✕Ô ÷ìø ➉➑➐ ➫☛➭ Power 2 ➍❂➎➏ if n is even Power 1 Ï☛Ð Number y ➋❂➌ ➝❂➦ ❒➑❮ ➹❂➘ Ø❂Û ðòñ ➝❂➞ ✈①✇ - í✧îï ➡❂➢ ➙▼➛➜ ➲▼➳➵ ➽▼➾➚ ä✧å êìë 1. 2. 3. 4. 5. 6. ➬▼➮➱

  3. ➍ Ñ ➮ ➱ ✃ ❐ ➬❒ ❮ ✃ ❰ ÑÒ Ó ➘➷ Ô Ú Ö Ú ÖÝ Ø Þ Ü Öß ➬ ➴ Ù ➳ ➏ ↕ ➤ ➏ ➒ ➓ ➙ ➛ ➤ ➵➸ ➘ ➺ ➾ ➚ ➪ ➵ ➶ ➼ ➹ ➸ ➼ à ß ↕ Ø × ß á ä Ö Ö Ü á Ú ç à á ä Ù ã Ý ï ð ôõ ö ÷ ì ß Ù ã ã ä å â Ù ä è à Öß â Ú ê Ø ß Ú ß Ö ë é Ö Ú ➫ ➨ ö ➌ ❿ ➎ ➃ ❽ ➅ ➆ ➇ ❾ ➈ ➍ ➀ ➐ ➓ ➔ → ➣ ➣ ➓ ➔ ➐ ➂ ➁ ➓ ⑦⑧ t ✉ ♥ ♠ ①② ① ③ ⑦⑧ ⑨ ❺ ➀ ⑦ ③ ④ ⑤ ❻ ⑤ ❼ ❽ ❾ ❿ ➒ ➙ ➐ ➜ → ↕ ➞ ➨ ➔ ➩ → ➫ ➏ ➭ ➟ ➣ ↕ ➐ ➣ ↕ ➞ →➯ ➓ ➛ ➐ ➓ ➞ ➛ ↕ ➐ ➜ ➣ ➝ ➏➞ ➟ ➐ ➐ ➏ →➠ ➟ ➓ → ➡ ➤ ➧ ➐ ➣ ➝ ➏ ➞ ÷ ú ♠ ❃ ✻✼ ✽ ✽ ✸ ✾ ❁ ❂ ❃ ❂❄ ❁ ✶ ❂ ❀ ❅ ✺ ❆ ❁ ❂ ❃ ❂ ✺ ✱✲ ❑ þ ✝☛ þ ☞ ý ✁ ✄ ☛ ✞ ✠ ÿ ✮✯✰ ✛ ✠ ✜ ý ✜ ✣ ✥ ✩ ★ ✭ ❍■❏ ❖ ✟ P P ❙ ▼ ❍ ❲ ❍ ◆ ● ❋ ◗ ▼ ❍ ❖ ❘ ❑ ▲ ◆ ◗ ❙ ● ❍■ ❍ ❱ P ❑ ◗ ❍ ❖❘ ❑ ▲ ❍ ◗ ❍ ■ ▲ ❯ ▼ ◆ ❖ ● ❑ ● ❑ ❚ ◆ P ✆ þ ö ÿ þ ÿ ✍ ✆ ☞ ✎ ✂ ✟ þ � ✞ ✁ ✄ ✆ � ✠ þ ✏ þ ✁ ✄ ✠ ☛ ✆ þ ú û ÿ � ✁ ✆ ✝ ✞ ✄ ✟ ✆ þ ✆ ✠ ✡ ✝☛ þ ☞ ý ✁ ✌ ✆ ✠ ✄ û ☛ ✞ ✠ þ ÿ ✛ ý ✜ ✣ ✠ ✌ ✁ ✠ ✁ ✏ ✆ ✝ � ✆ ✝ ✞ ✆ ✁ ✆ ✓ ✄ ☛ ✒ � ✁ ☛ þ ✁ ✏ ☛ ý ✔ � ✆✕ ✖ ÷ ✗ û ✝☛ þ ☞ s ➃➄ ❡ ➭ õ ÿ ÷ ☎ ➯ ✆ ÿ ÷ õ ✁ ø ÿ ù ö ✁ ♠ ☞ ✌ ÷ ☎ ☎ ✆ õ ➲ ☞ ✌ ö ✁ ✂ õ ✄☎ ù ✓ ÿ ü ☎ ÿ ✄ ý ☎ õ ✒ ➥ ➜ ÷ ✱ ✮ ✮ ✯ ★ ✬ ✭ ✰ ➧ ✫ ✭ ✲ ✱ ✰ Ó → ✶ ★ ✶ ✧ ✬ ✔ ✖✗ ÷ ý ✄ ú ✌ ➥ ➫ ✕ ✘ ✤ ✙ ✚✛ ✜ ✢ →➩ ✦✧ ★ ✩ ➨ ý ➦ ✷ ò ç ✃ à æë ➴ ➽➘ ï ðñ ➹ å ó ➶ ö ÷øù ➚ ø ➾ ú æ ä ü Ú ❰ ❮ ❒ Ö Ð Ñ Ñ Ù Ò âã ❒ Û Ó ÔÜ ❒ ❰ Ý ❮ á û õ ✠ ✞ ✝ ✂ ✞ ✟ ☎ ✟ ✠ ✂ ÷ ÷ ➥ ó ✡ ö ☛ ÿ ✂ ✞ õ õ ✁ ÷ø ➻ ý ÷þ ➼➽ � ù ÿ ö ✁ ✂ ➺ ☎ õ ✄☎ ✆ ➸ ù ÿ ø ✆ ÿ ✫ ✳✴ ✬✧✸ ● ▲ ● ❊ ❇ ● ❇❈ P ❘ ❖ P◗ ❙ ❋ ❖ ❙ ➜➝ ▲ ↔ ➣ ❍ ▲ ❲ ● ✰ ★ ✩❀ ❄ ❅❆ ★ ❇❈ ❉❊❋ ❍ ❍ ❉■ ❏ ❑ ➜ ●▲ ▼◆ ❉ ❖ ● ❱ → ✰ ➑ ✐ ❥ ❵ ❦ → ❡ ➔ ❴ ❧ ❣ ♠ ♥ ➐ ➏ ❴ ♦ ♣ ❴q r ❤ ➣ ❳ ❵ ➛ ❨ ➙ ➔ ❩ ↕ ❫ ❴ ❛ ↔ ❜ ❫ ❝ ❞ ❴ ❵ ❛ ❡❢ ❡ ✮ ➞ ★ ✤ ✩❀ ✵ ✧ ✦ ★ ✩ ✩ ✩ ✬ ✿ ✩ ✸ ✿ ✫ ✦ ✧ ➡ ✿ ✧✾ ✫ → ➦ ➥ ➤ ✢ ✣ ✫ ✽ ✸ ★ ✩ ➜ ➢ ➣ ✰ ✮ ✰ ✮ ★ ✵ ✬ ✶ ✤ ★ ✰ ✷ ✩ ✱ ✲ ✳ ✻ ✩❀ ✾ ✻ ❃ ✦ ➟ ✫ ✽ ✮ ÓÔ ★ ❁ ✰ ✮ ❂ ✮ ★ ✲ ➠ ✰ key mathematical key, key key, S 8 v = u. ➏➦➥ ➢✍➏ 3 ➏❯➲ Û✥ç v < u ã➦æ ❙✑● ❐ÐÏ í⑥î classes of mathematical objects) ✌☎✤ u < v u,v) ✞✑� ✿☎❀ á✥â S øóù ❸✥❹ ▲☎▼◆ false ➉✑➊➋ ✠✢✜ key u key u ∈ S? ⑩❷❶ ✳✙✴✵ ↔✥↕ true ❋✑● ñóò ✂☎✄ Û✍Ü operations ✂☎✄ ∀ u,v ④⑥⑤ ❇❈❊❉ S) «u ∈ S áÐé ×⑥Ù ✩✫✪✬ ➑✥➒ key. ×⑥Ø ➻➽➼ ❬❪❭ - ø ✙✘✚ ➎✍➏ ➎✍➏ ü✍ýþ ✷✹✸ Õ➽Ö ✦✙✧★ ✈✥✇ 7 ❉❯❚ 3 ✯✍✵ ➷➮➬➱ ×➮Ø ú❁ÿ ✎✑✏ è✳éê Ï❏Õ ☞✍✌ ➾➪➚ ✪✍✫ ✺✼✻ Ï✆Ò ✹✍✫ ✣✥✤ ➒✓➓ Ï❏Ð - Þ✳ßà ➳❏➵ ô⑨õ ìîí ❐❁❒

  4. ❳ ➁ ❺ ❷ ❸ ➓ ➣ ➂ ❺ ➒ ❸ ❽ ➌ ➐ ➑ ➝ ➇ ❺ ➍ ❷ ❿ ❾ ❹ ➈ ➀ ❿ ❹ ❺ ❻ ↔ ❿ ❽ ➄ ❹ ❺ ❻ ❾ ❿ ➀ ❿ ❽ ➃ ➂ ➃ ➔ ❼ ➈ ➆ ➍ ➆ ➑ ➡ ❸ ❷ ❸ ❿ ➣ ❻ ➆ ➍ ➆ ➑ ➠ ➙ ❿ ❸ ❼ ➍ ➆ ➑ ➍ ❸ → ➀ → ➑ ➠ ➟ ❷❸ → ➇ ❸ ➅ ❼ ➑ ➄ ❽ ➈ ➣ ➂ ❺ ❸ ➃ ➣ ❸ ➂ → ➇ ❿ ❷ ❨ ❻ ❺ ➁➌ ➛ ❸ ❿ ❺ ➉ ➂ ➌ ↕ ➣ ❻ ❽ ❿ ❺ ➈ ➆ ❻ ➍ ❼ ➌ ➛ ➓ ❸ ❷ ➑ ❸ ❸ → ➌ ➂ ❷ ➎ ❸ ➍ ❷ ➓ ❾ ➌ ❽ ➊ ❷ ➈ ❺ ➁ ➇ → ➄ ❽ ➜ ➆ ➣ ➏ ➇ ❷ ➒ ❻ ❺ → ❷❸ ➓ ➒ ➍ ❽ ➍ ➁ ❿ ➏ ➇ ➈ ➆ ❿ ❷ ❷ ➶ ➶ ➾ × ➪ Ø ➬ Õ ➬ Ô Ô ➴ × ➪ Ø ➬ ❒ ➶ Õ Ö Ô ➴Ó ➴Ð ➴ ➚ Ï ➮ Ñ Ò ➮ Ô Ð ➴ ➹ Ð ❰ Õ ➶ ➴ ➮ ➬ Ð Ï ù ú ø ù ý ú þ ø ÷ ú ø ✂ ô ☎ ✝ ú ✞ ✟ ø ✠ ✡ ù ò ❰ ß ➶ ➴ ➮ ➹ ❐ Ð Ù ➾ Ûß ðñ â ã ä å èé ê ï î ➮ ➾ → ❛ ➣ ➌ ❳ ❨ ❩ ❬ ❭ ❵ ❜ ➍ ❝ ❞ ❵ ❡ ❢ ❛ ❜ ❝ ❻ ❷ ❣ ↕ ➇ ❸ ➅ ❼ ➑ ➢ ➁ → ↔ ➇ ❿ ❽ ➁ ➍ ➆ ➑ ↕ ➀ ❣❤ ✐ ➶ ➵ ③ ➥ ➩ ➫ ➭ ➯ ➲ ➳ ➾ ♣ ➪ ➹ ➪➶ ➶ ➴ ➮ ➹ ❐❒ ♦ ✇ ❥ ♣ ❦ ❧ ❜ ♠ ❣ ❛ ♥ ♦ ❛ ✈ q r ❛s t ♦ ❣ ♦ ✉ ❽ ➒ ➇ ❽ ➃ ❺ ➒ ➆➓ ❽ ➁ ❻ ➀ ❺ ❿ ❣ ❻ ❾ ❿ ➀ ❿ ❸ ➂ ❸ ❽ ❷ ➂ ❸ ❻ ➎ ➏ ❿ ❷➐ ❺ ❥ ➑ ✐ ❸ ➍ ➉ ❿ ➈ ❺ ❽ ➔ ➉ ❿ ❜ ❛ ❽ ➃ ❽ ❽ ➈ ➇ → ❸ ➍ ➄ ❢ ❡ ❺ ➍ ❣❤ ➇ ➃ ❺ ➄ ❸ ❻ ➃ ❺ ❺ q ➉ ❝ ❿ ❿ ❸ ➁ ➃ ➇ ➃ ❸ ➁ ❼ ♥ ❛ ❻ ❣ ➀ ❿ ➂ ♦ ❺ ➃ ❺ ❷ ❽ ➄ ❾ ❻ ⑩⑧ ❸ ✇ ❛s t ♦ ❣ ♦ ✉ ✈ ♣ ③ ♦ ❛ ③ ④ ♣ ⑥ ② ❿ ❿ ❻ ❽ ❽ ❹ ❺ ❻ ❦ ➀ ❿ ➃ ❼ ❺ ❻ ❾ ❿ ➀➌ ➍ ➁ ❹ ❻ ❸ ➉ ➅➆ ➇ ➆ ❻ ♠ ❺ ➈ ➃ ❧ ❺ ❹ ❺ ❻ ❾ ❿ ➀ ❜ ❵ ❺ ❸ ➈ ❺ ❛ ➍ ➁ ❿ ❸ ➇ ❼ ➍ ➍ ❽ ➀ ❵ ➣ ➃ ❭ ❻ ❷ ➁ ❜ ➍ ➆ ➌ ➑ ❸ ➍ ➉ ❿ ➇ ❺ ➃ ❺ ❸ ➃ ➣ ❺ ❺ ➆ ❽ ➁ ❻ ❷❸ ➂ ❽ ❷ ➈ ❺ ➇ ❩ → ➄ ❸ ➍ ➈ ❺ ➃ ➄ ❽ ➓ ➇ ➏ ➒ ❽ ➍ ➁ ❿ ➏ ➈ ❬ ➆ ➍ ➆ ❽ ➓ ➈ ❽ ➂ ➁ ➂ ➓ ❿ ➁ ➍ ↕ ❽ ➈ ➄ ➙ ❺ ➈ ➃ ❸ ❷ ❻ ❽ ❼ ➆ ➍ ➁ → ❺ ❿ ❽ ➂ ➣ ❞ ❸ ➍ ➉ ❝ ❿ ❺ ❺ ↔ ➂ ❽ ❷ ➈ ❺ ➁ ➇ → ❷ ➓ ➈ r ❼ ❻ ❷❸ ❼ ❹ ❻ ❺ ❹ ➆ ❺ ❻ ❻ ➑ ❷ ➇ ❸ ❺ ➃ ➒ ➓ 10 3 ➱✑✃ f(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0 ➷✹➬ í✑î Ï☎Ù ë⑨ì – ➻✑➼➽ ÿ ✁� ➱✑✃ ➸✑➺ æ✑ç = lg(lg n) lg k n = (lg n) k = log 2 n = log e n – ➷✹➬  x  ,  x  , [x] û✹ü ➘☎➴ à✙á lg lg n Ý➦Þ lg n ln n ö✑÷ ➧✑➨ ❪❴❫ n! - ➚✹➪➶ ❮☎❰ Ú☎ÛÜ óõô ✄✆☎ ➤➦➥ 9 ❾☎❿ ❾☎❿ 3 ❿✢➛ ❾☎❿ ❹⑨❺ ❹⑨❽ ➟☎❷❸ ➉☎➌ ❾☎❿ ❹⑨❺ ❾☎❿ ❾☎❿ ❹⑨❺ ⑦⑨⑧ ❿➋➊ ❾☎❿ ❹⑨❽ ①✑⑤ ❹⑨❺ ❹⑨❽ ❹⑨❺ ➞✹❷ ❪❴❫ - ❾☎❿ ❾☎❿ ❾☎❿ ❶✹❷❸ ❶✹❷❸ ①✑②

  5. ☛ ➧ ➻ ➥ ➶ ➩➪ ➵ ➚ ➵ ➞ ➻ ➠ ➻ ◆ ☞ ▼ ➤➾ ➦ ➽ ➩ ➯ ➧ ➵ ➼ ➞ ➧ ➫ ❈❖ ❰ Ù ❈ Ö ❉ Õ Ô Ó Ò ❊ ❈ ➷ ❮ ➹ ❒ ➷ ➱ ➷ ✃❐ ❈ ➬ ❏ ❑ ▲▼ ➾ ➧ ❍ Û ➙ ➍ ❈ ➛ ➒ ➇ ➆ ➎ ➍➎➏ ❉ ❊ ❨ ➆ ● ➙ ➌ ➈ ➇ ➍ ➆ ➎ ➆ ↕ ✳ ➏ ❈ ❘ P ➫ ➸ ➞ ➩ ➩➵ ➠ ➳ ➭ ➫ ➩ ▲ ➦ ➩ ➡ ➯ ➥ ➭ ➫ ➩ ❙ ➩ ▲ ➠ ➦ ❱ Ú Ü → ✗ ✮ ✯ ❂ ✮ ✥ ✚ ✤ ✣ ✺ ✘✙✚ ✴ ✻ ✸ ✖ ✕ ✺ ✏ ✑ ✎✏ ✍ ✌ ✌ ☞ ✬ ✯ ✳ ✱ ✳ ✲ ✱ ✰ ✹✿ ✲ ✺ ✼ ✻ ✺ ✲✹ ✲ ❁ ✶ ✴ ✱ ✸ ✷ ✵ ❀ ✱ ✸ ✳ ✲✳✴ ✹ ✡☛ Ý ä ë ê Ú é è ç Ú à æ Û å ã í â ❃ ❄ à àá Ü Û Ú ß Þ Ù Úì è ✆✝ ø ý ☎ ✄ � ✂ ✹ þ ✲ ✿ ò ù ÷ à ✭ ô ✱ ❃ ✹ è é ð ï î è ↕ ➒ ↔ r ❏ ■ ❍ ● q ❞ ❈ ❀ ❇ ❆ ❅ ❂ ❖ ❀❁ ✿ ❵ ❴ ✺ ✽ ✾ ✱✽ ✼ ✻ ✺ ❑ ◆ ✸ ❵❛ ❩ ❝ ❡ ❦ ❧ ❢ ❡❢❣ ❞ ❝ ❴ ❵ ❢ P ❘❪ ❯ ❥ ❩ ❨ ❱ ❳ ♦ ❚ ♣ ✐ ✹ ✷ ♦ ✙ ✣ ✢ ✜ ✉ ❥ ✈ ✚ ✚✛ ✖ ✕ ✔ ✘ ✕ ✓ ✗ ✖ ✕ ✔ ✓ ✉ ✎ ❛ ✍ ✌ ✤ ✥ ✱ ✧ ✶ ❦ t ❛ ❴ ❛ ✧ ★ ✯ ✮ ✭ ✚ ✚ ✧ ✬ ✔✫ ✪ ✩ ✔ ★ ✧ ✦ ❡ ✔ ♥ ❝❦ ❦ ⑦ ❫ ✈ ❣ ❞ ❝ ♥ ♠ ❦ ⑧ q ❝ ❡ ❭❪ t q ❝ ② ✇ ⑥ ❴ ❡ ❤✐ s ✐ ❣ ⑨ ❷❸ ② ❉ ➣ → ❈ ➒ ➌ ➑ ➐ ➇ ➆ ➎ ➍➎➏ ❘ ❹ ❈ ❉ ➇ ➅➆ ❊ ❈ ❿ ▼ ❈ ▲ ❻ ❺❻❼ ♣ ♣ ③ ✉ ❡② ♦ ❡ ❥ ❝ ♥ s✈ ❥ ❤ ♥ ❥ ❥ st ❥ ❛ ❝ ♥ ❝ ♥ ♠ ❦ r q s ♣ ✴ ❤ ❡ ❥ ✐ ❦ ❴ t ✈ s ❥ ❤ ✈ ① ✇ ❡ t s ❥ ✐ ❴ ✈ ❤ y n-k+1 = -1): y n- 12 ✞✠✟ 3 ❍✔■ ✹✾✽ ❋★● y n-k+1 = x n-k . ❋★● y n-k+2 = y n-k+1 *x. ❡✾✇ ❅❇❆ ÿ ✁� j, j = n, …, 0. y n-j+1 =x n-j ❍★❬ k, n ≥ k ü✵ý y n-n+1 = y 1 = 1 = x 0 = x n-n ú✵û o y n-k+2 = x n-k+1 ✩✔✪✫ ✵★✶ ♠★♥ ❍✔■ ◗✔❘ ✦★✧ ❢✠❣ ✒✔✓ k+2 = x n-k+1 . Power_1 ❵✢❞ ❲❯❳❨ ✰✢✱ ❜✠❝ õ✵ö j = n: ✛✢✜ ×✒Ø ❚❯● - x n-k . ❵★❛ ❵★❥ ✭✔✮ ❛sr ñ✲òó A k+1): 11 y j = 1, …, n+1 k, 1 ≤ k < n +1, y k = x k-1 . 3 y 1 = 1 = x 0 = x 1-1 = x j-1 . y k = x k-1 . y j = x j-1 . ➧✁➲ ❒ÑÐ Power 1 ✈⑤④ y k+1 = y k *x. ▲▼✲◆ Ï✵❮ ➽❙➠ o j. y j ❉❋❊ ❚❴❫ ➓❙➔ – ➮✵➱ ➧✵➨ ❽✁❾ ➐✵➜ y k+1 = x k ➈➊➉ ❬❲❭ ❤✵✐❥ j- y k+1 = x k . ➢❙➤➥ ➃✲➄ ❃✆❄ ✏✒✑ ➟❙➠ ✳✵✴ - ❯❲❱ ➀➂➁ ⑩✵❶ ✰✲✱ ❜✆❝ ◗❙❘ ➝✁➞ ➺✵➻ A ➘✲➴ ➷⑤➬ j ❧✆♠ ➋✲➌ ➋✲➌

  6. ① ñ ñ ✞ ò ó ý ñ ❑ ❑ ë ÿ ñ é ù ❘ ü ò í ù ü ✞ í ☎ ð é ✆ ð ì ñ ✆ ï ❑ ❚ ✞ ✞ ñ ù ì í þ ì ñ ï ñ ò ò é ✞ ï í ✝ ÿ ñ é ò í ï ð ❈ ñ ✠ ù ï û ò ì ✠ ✞ ✠ ☛ ò ✠ ☛ ð ❈ ù ☎ í ✞ ✠ ☛ � é ✞ ë þ ð ò ñ í ☛ ü é ✞ í ù ì í þ ì ñ ï ï ñ ò é ò ì ó ☛ ö ❯ ð ✁ ì ✠ ò é ÿ ü ☛ ò þ ò ï ð ❘ ✞ ÿ í ❑ í ✆ ñ ù ì í ◗ ❑ ú P ð ❈ ✠ ✞ î ï ù ð ì é ❘ í ì ✝ ✡ þ ❍■ ò ✠ ☛ ð ù ❈ ☎ ② ✞ ✠ ☛ ● ❏ ❏ ✌ ñ ù ❑ ü ò ✠ ò ñ ◆❖ ● ❍ ■ ë é ý ò ð ì þ ñ ✞ ❈ ñ ☛ ð ê ñ ì ë ð ò ñ é ñ ù ü ò í î ❑ é ÿ ü ê ù ñ ☎ ë ❑ ❑ ñ ✞ ✝ ✆ í ❑ ñ ☎ ð í ð ï û ✆ ✁ ✆ ö ✡ ❙ ✟ ì ü ✆ í ☛ é é ✠ ò ð û ✆ ñ ù ì ü ◗ ❑ ✠ ï ñ ñ ✡ ê é ✆ ü ï ✠ ✞ ✠ ☛ ð ✆ ü ❜ ö ù ð ñ ù ì û ù ð é ✆ ñ î ❝✁ ú ❈ ï í ☎ ð ☎ í ï û ✆ í ☛ ñ ì é ú ☛ ❈ í ✟ é í ñ ì é ú ï ü ☛ ✞ ò íé ð ✆ ❈ ✁ ✆ ò í î ù ❈ ✆ ñ ÿ ñ ö ñ ❈ ✆ ☛ ñ ÿ ñ ý ò í ï û þ ð ú í ò ð í î ù ì í ◗ ❑ ó ï ñ ò í ☛ ❈ ù ñ ö î ù ì í ◗ ❑ ó ï ñ ò í ☛ ❲ ò ❳❨ ❳ ❵ ❳ ❛ ❳ ✡ ❜ ò í ù ì ü í ☛ ❑ ð ï ð ò ñ ◗ ❑ ✠ ò ó ù í î ÿ í ❘ ì ë ❱ ð é ò í ï û þ ð ú ◗ ✠ í é í ◗ ❑ ó ï ñ ò í ☛ ð ❈ ✆ ñ ò ù í ✌ ✠ ☎ ✝ ✆ ñ ï ✠ ✞ ò ✠ ✆ ì î ï ☎ ñ ñ ✆ î ❝ ❚ ✞ ð ✁ ☛ ✞ ð ✝ í ✆ ñ ï ✠ ò í ï û þ ð ú í ☛ ò ☛ í ✞ ➘ Ý ➺ ➵ ➳ ➭ ➳ ➷ ➫ Û ➯ Ï ➭ ➘ Ü ➷ Þ ➩ ➽ ➾ ➯ á á ➺ ❐ ➵ ➯ ❐ ❮ ➯ ➱ ➫ ➨ ➴ ➭ ➲ ➯ ❒ ➯➲ Ñ Ü Ò Ó × Ú ➫ ➫ Û ➯ ➘ Ï ➘ ❒ ➲ ➲❒ í íî ï ð ï éñ ò é ï ó ò î ➳ ô í ú û ò íî ï ð ü ò é ê ✠ ß ➫ ➽ ➾ ❐ Û ➴ ➭ ➫ ➲ ➵ â ã å ➫ Û ➯ ➘ Ï ➘ ➲❒ Ü æ Û ➴ ç ➺ ➭ þ ➄ ➅ ➆ ⑩➇ ➈ ➃ ❻ ➃ ➉ ➊ ➋ ➃ ➄ ➎ ➏ ➒ ➓ ➔ ➍ → ↕ ➙ ➎ ➛ ⑩ ➃ ➝ ⑩ ③ ④ ⑤ ⑨ ⑩ ❶ ❷ ❸ ⑨ ❹ ❺ ❶ ➂ ❷ ❻❼ ❻ ❽ ❾ ❿ ➀ ❶ ➁ ❻ ⑩ ➜ ➍ ➫ ➾ ➩ ➭ ➳ ➵ ➳ ➼ ➯ ➩ ➚ ➪ ➶ ➯ ➵ ➩ ➹ ➸➘ ➬ ➮ ➶ ➾ ➱ ➲ ➯✃❐ ➻ ➸ ➯➲➳ ➓➞ ➦ ➓ ➛ ➟ ➒ ➓ ➙ ➝➠ ➒ ➍ ➡ ➦ ➍ ➭ ➝ ➦ ➍ ↔ ➒ ➍ ↔ ➔ ➢ ➤ ➫ ý ➵ éñ ✣ ì ñ ✌ ✔ ✕ ✖ ✗ ✘ ✜ ✢ ✤ ✍ ✥ ✜ ✦ ✧ ✢ ✣ ✤ ★✩ ★ ✪ ✫ ✑ ✠ ✭ � ì ñ ✡ ò é í ☛✌ ✒ ✓ ✏ ö ù ✞ ñ þ ✁ þ é ÿ ó î ù ü ú ÿ ✬ ✣ ö é ❅ ❆ ❇ ✺ � ï ù ð é ì ÿ ❃ ü ☛ ò ì ü ù í ☛ ï û ò ì ❄ ❁❂ ✮ ★ ★ ✢ ✯ ✰ ✱ ✢ ✲ ✳ ✢✴ ✵ ✰ ✰ ✽ ✶ ✷ ✸ ✱ ✰ ✼ ✽ ✾ ✿ ❀ ✹ ✑ ð ✏ þ ✆ é ✞ ✟ î ì é ✞ ï ü é ò ✎ ò ✠ ✆ ò é ï ó ✡ ù ð í ð é ✂ ë ú ð � ù ñ þ ✁ þ é ÿ ü ó ï ï ñ ✄ ù í ☎ ð é ÿ ✆ ✝ íî ì ñ ñ ð þ ù ✁ þ é ÿ ó î ù ò ü ú ò ✍ ✠ ✞ 14 ü ☞☛ 3 ü ☞☛ n n ✹✻✺ : ❪❴❫ ❩❭❬ ✙✛✚ - ❉❋❊ ▲❋▼ y: : z = y/2 < y): m(x,y) = (x+x)*y/2 = x*y. 13 ≠ 0 3 ➴★➷ z = y-1 < y): m(x,y-1) = x*(y- m(x,y) = x + x*(y-1) = x + x*y –x = x*y x,y m(x,y) = x + m(x,y-1). E x*y y ø✢ù z, 0 ≤ z < y. y = 0 m(x,y) = m(x+x, y Ð✢➻ õ÷ö x + m(x, y- m(x+x, y m(x,y) = x*y, ∀ y ➣✔↔ ➌✢➥ – ➽❇➾ m(x,y) = x*y. ➢❯➤ y. { ➶àß y = 0, m(x,y m(x*y) = x*y. ➺✢➻ = ➐✔➑ m(x,z) = x*z ü ☞☛✌ ➌✢➍ Ø★Ù ❰❯Ï m(x,y) ⑥⑧⑦ ➲✾ä ë✢ì - ÔÖÕ ë✢ì y y èÖé ➧➨★➩

  7. ❞ ü õ ✂ ✄ ö � ☎ ✆ ✟ þ ✠ ✡ ☛ ☞ ✏ ✑ ✒ ✓ û � ✏ ÷ ñ é ò õö ÷ ø ÷ ö ù ö ú ø ÷ û ü ý þ õÿ ✔ ✕ ì ✤ ✩ ✤ ✗ ✤ ✪ ✫ ✬ ✥ ✰ ✑ ✷ ✰ ✻ ✼ � û ✽ ✾ ★ ✧ ✖ ✛ ✑ ✒ ✓ ✗ ✘ ✗ ✙ ✚ ✜ ✦ ✒ ✢ ✗ ✑ ✣ ✤ ✥ ✑ í ë õ ➟ ↕ ➑ → ➌ ➏ ➔ ➐ ➛ ➝ ➑ ➌ ➉ ➐ ➔ ❡ ➆ ➣ ➌ ➔ → ❐ ➐ ➛ ➈ ➴ ➷ ➠ ➛ ➐ ➔ ➛ ➩ ➩ ➇ ➐ ➔ ➊ ➋ ➷ ➭ ➔➬ ❒ ê Ö ➝ ➌ ➇ ➩ ➋ ➥ Ð ÔÕ ×ØÙ → Ú Ý Ü Þ ßàá â ã é ➐ ➑ ➼ ➈ Ï ➵ ➱ ➶ ➑ → ➫ ➭ ➍ ➋ ➊ ➞ ➝ ➌ → ➔ ➛ ➩ ô ö → ♣ ❉ ❆ ❇ ❍ ❘ ❏ ❳ ❙ ❳ ♥ ◆ ■ ❲ ❘ ❳ ❖ ▼ ❈ P ❘ ❥ ❴ ❘ ❳❵ ❛ ❜❝ ❞ ❤ ✐ ❜ ❩ ❦ ● ● ❏ ◆ ❧ ❘ ♠ ❴ ❳ ❖ ❏ ■ ▲ ❳ ❖ ❲ ❩ ❚ ● ◆ ❖ ❧ ❘ ❳❨ ❍ ❴ ❘ P ● ▼ ❳ ❩ ❑ P ❳ ❭ ❊ ❱ ❳ ● ❨ ❭ ◆ ❊ ● ❱ ❳ ● ❉ ❭ ❊ ▼ ❳ ✾ ❆ þ ❁ û ❄ ❅ ü ✽ ❁ ❇ ö ❈ ❉ ❆ ❇ ❈ ●❍ ■ ❏ û õ ◆ û ù ✆ ÷ � ✿ ÿ õ ❀ ú ÿ � ü ö ❁ ✝ ❂ ✾ ü ❃ ▼ ■ ● ❳ ▲ ● ❳ ◆ ❚ ❙ P ❖ ❫ ❨ ❖P ❘ ❱ ❳ ● ❨ ❑ ❍ ❙ ● ❍ P ❖P ● ▲ ❏ ▲ ●❱ ❲ ● ❳❨ ❳ ❩ ❚ ❊ ❬ ❭ ❈ ❨ ● ➔ ➑ ➐ ➋ ➞ ➉ ➟ ➇ → ➙ ➑ ➙ ➠ ➋ ➍ ➊ ➞ ➌ ➛➡ ➇ ➟ ➢ ➌ ➑ ➐ → ➌ ➔ ↔ ↕ ➉ → ➐ ➔ ↔ → ➔ ➙ ➜ ➊ ➓ ➝ ➌ ➔ ➐ ↔ ➔ ➙ ➛ → ➊ ➤ ➞ ➍ ➤ ➊ ➛ ➈ ➙ ➙➉ → ➐ ➆ ↕ ➇ ➊ ➧ ➐ ➋ ↔ → ➆ ➤ ➇ ➊ ➉ ➔ ➛ ➓ ➍ ➊ ➞ ➓ ➍ ➤ ➊ ➛ ➈ → ➙ → ➐ → ➌ ➉ ⑧ ❧ ② ③ ④ ❧ ⑤ ⑥ ❧⑦ ③ ① r ③ ⑨ ⑩ ❶ ④ ③ ❹ ❺ r ♠ ❼ ❦ ❢ ❣ ❤ ❦ ❧ ♠ ♥ ♦ ♣ ✇ q ❧ ♠ ♥ rs r t ✉ ✈ ❻ ❽ ➋ ➋ ➊ ➐ ➍ ➐➔ → ➊ ➌ ➉➊ ➆ ➋ ↕ ➇ ➊ ➋ ↔ ➉ ➓ ↔ ➙ ➓ ➐ ❷ ➆ ❺ ❾❿ ➀ ➁ ➂ ➃ ➄ ❸ ➅ ➇ ➒ ➋➌ ➍ ➌ ➎ ➌ ➇ ➏ ➐ ➑ ➌ ➍ ➉ ➞ ➋ ➙ → ➑ ➒ ➙ ➌ ➎ ➑ ➟ → → ➙ ➐ ➊ ➓ ➨ ➉ ➐ ➐ ➤ ➙ ➨ ➝ ➇ ➌ ➙ ➑ ➙ ➞➉ ➆ ➛ ➠ ➠ ➋ ➫ → ➑ ➶ ➅ ➨ ➘ ➪ ➐ ➔ ➫ ➠ ➵ ➇ ➭ ➠ ➐ ➵ ➳ ➨ ➉ ➲ ➞ ➍ ➣ ➥ ➟ ➊ → ➇ ➇ ➹ ➉ ➌ ➟ ➏ ➔ ➙ ➋ → ➋ ➐ ➋ ➙ ➥ ➇ ➛ ➞ → ➠ ➌ ➛ ➔ ➌ ➇ ➏ ➔ ➐ ➩ ➐➡ ➔ ➔ ➌ ➟ ➟ ➉ ➞ ➑ ➋ ➩ ➙ → ➋ ➓ → ➋ → ➞ ➐ ➇ ➓ ➎ ➑ ➞ ↔ ➑ ➓ ➍ ➌ ➊ ➇ ➔ ➡ ➇ ➍ n ≥ 16 ◗✺❘ ◗✺❘ ❑✯▲ f(n 3 c ❈❋❊ an ≥ g(n 0 ≤ cg(n) ≤ f(n) n). c = b & n 0 g(n)), f(n c ∈ R + (c ❭❪❊ c = b f(n g(n an 2 + (b-c)n ≥ 0 ⇒ ≥ an 2 + bn ≥ cn ⇒ n 0 ❳✞q ✸✺✹ an + b-c ≥ 0. ♥✺♦ f(n ✴✶✵ ❙❯❚ ✱✳✲ n ≥ n 0 ❡❣❢ ✌✎✍ - ❑✯▲ ◗✺❘ ✭✯✮ A 15 n 0 ≥ b n ≥ n 0 f(n 3 n ≥ n 0 c ➣✻↔ ➋➯➭ g(n)), f(n) = O(g(n 0 ≤ f(n) ≤ cg(n) äæå a,b ö ✞✝ an 2 + bn ≤ cn 2 ó☞ô ➒☞➫ c ∈ R + (c ➞➦➥ ➋➯➭ Û➯Ü c-a)n - b ≥ c = a c & n 0 f(n) = an 2 +bn 0 ≤ n [(c-a)n –b] ⇒ 0 ≤ (c-a)n 2 – bn ⇒ g(n n 0 ≥ ? f(n) = O(n 2 ). (c-a)n - b ≥ 0 ➾❴➚ î❋ïð f(n ➻✻➼➽ ➈☞➉➊ ❮❭❰ ✐✛❥ ✁ ✻û ➸❴➺ - Ñ☞ÒÓ f(n ❷✻❸ ➮✃➱ ç☞è

  8. r ✭ ✶ ✧ ✪ ✴ ✧ ✦ ✳ ✦ ✲ ✭✱ ✮ ✬ ✯ ✮ ✬ ✵ ✫ ✱ ✵ ✵ ✪ ✧ ✭✴ ✮ ✛✜ ✧ ✦ ✳ ✦ ✲ ✶ ✱ ✮ ✯ ✯✺ ✥ ✸ ✹ ✸ ✷ ✧ ✦ ✳ ✦ ✲ ✭✱ ✮ ✬ ✮ ✫ ✭ ✬ ✭ ✮ ✯ ✬ ✮ ✱ ✭ ✲ ✦ ✳ ✦ ✧ ✬ ✭✱ ✬ ★ ì ô î æ î ó àò ñ ð à s î í à æ á ö à á â ã ß ä å á ë â æç æ è é ê õ ï ✭ ✡ ✪ ✧ ✥✦ ✤ ✣ ✛✜✢ ✎ ✚ ✗ ✙ ✌ ✡ ✒ ✕ ✑ î ✂ ù ú û ü ý þ ✄☎ ✌ ✆✝ ✞ ü þ ✡☛ ☞ ✫ ✳✻ ß ❅ ❑ ❉ ❄ ❅ ❊ ■ ❆ ❅ ❉ ❍ ❄● ❊ ❉ ❋ ❆ ❖P ❉ ❊ ❈ ❉ ❄ ❉ ❆ ❅ ❄ ❃ ✛✜ ✧ ✦ ✳ ▲ ◆ ✲ ❘❙ ❳ ❱ ❲ ❱ ❯ ❙ ▲ ▼ ❑ ◆ ▼ ▲ ❑ ❚ ❱ ◗ ▲ ❘❙ ❚ ❑ ▲ ❑ ▼ ❙ ❨ ❲ ❱ ❳ ❨ ❙ ❩ ✦ ✱ ✬ ✾ ✺ ✵ ✿ ✧ ✦ ✺ ✾ ✵ ✪ ✲ ✴ ✮ ✧ ✺ ✥✦ ✶ ✦ ✭ ✬ ✮ ✭ ✱ ✲ ✳ ✵ ✦ ✧ ✼ ✽ ✮ ✦ ✲ ✳ ✭ ✭ ✮ ✬ ✭ ✱ ✵ ✼ ✯ ✱ ✧ ✦ ✳ ✦ ✲ ✱ ✮ ✥ ✪ ✳ ✻ ✹ ✦ ✩ ✦ ✧ ✬ ✴ ✾ ✺ ✵ ✱ ✭ à ï ➹ ❽ ➔ ❻ ➡ ➓ ➶ ➠ Ü ➜ ❼ ➝ → ↔ ➠ ➲ ↕ ➛ ③ ❾ ➵ ➚ ➾ ➽ ❿ ➺ ➵➸ ➳ ➭ ➫ ➁ ❹ ➓ ➢ ➒ ➯ ❸ ➣ ↔ ➠➬ ➹ ➜ ➛ ➝ ➴ ③ ④ ❺ ➫ ➩ ⑨ ➠ ➒ ➘ ➶ ➛ ➯ ➔ ③➀ ➫ ❷ ➒ ➒ ➣↔ ➝ ➥ ➠ ➔ ❽ ➤ ❼ ➅ ➑ ➟ ➞ ➡ ➣ ➜➝ ➑ ➆ ↕ ➔ ➇ ➣↔ ➇ ➓ ➏ ➑➒ ➦ ➒ ➲ ➂ ➭ ➔ ❼ ↔ ➣ → ➔ ➔ ⑨ ❼ ➝ ➣ ➔ ➧ ➃ ➫ ➩ ➭ ➔ ➄ ➫ ➩ ➡ ➓ ➠ ➨ ➢ ➮ ❶ ➞ ↔ ➛ ➯ ➮ ➭ ② ③ ➣ ↔ ➬ ➣ ➧ ➷ ➯ ➮ ④ ➠ × ➱ ➧ → ➣ ↔ ➬ ➯ ➑ ➜ ⑤ ➠ Û Ú Ù Ø t ✉ ✈ ➢ ↔ ➣ ➒ ➙ ➞ ➝ → ➠↔ ↕ ➛ ➶ ➣ ➡ ➘ ➹ ➑ ↔ ➮ ➐ ⑥ ➶ ② ➠ ➹ ➜ ➛ ➨ Ó Ò ↔ ➠ Õ ➝ ⑧ ③ ④ ⑤ ⑨⑩ ⑨ ❰ ❮ ✃❐❒ Ô ⑦ Ö ➑ ➫ ↔ ➒ ➘ ➠ ➙ ➷ ↔ ↔ ➮ k: 18 ★✰✶ 3 n log n ■✁❏ ✖✘✗ ÿ ✁� ✒✔✓ ❯✔❱ ✮✰✯ ✍✏✎ ✫✔✬ ✮✰✯ ✮✰✯ ▼✰◆ ★✰✶ ÷✯ø ❇✔❈ ✫✔✬ ✫✔✬ ✟✁✠ O(n logn): ★✏✩ logn): O(n 2 ): O(n 3 ): O(2 n ): O(n): ❀❂❁ Ý✎Þ n k - ✮✰✯ ▼✰◆ c-a)n-b = c = a/2 & f(n) = f(n) = 17 f(n 3 n ≥ n 0 ➔❋➯ g(n ➠↔➡✞➢ n 2 ÏÑÐ g(n)). n ≥ c > 0 & n 0 ≥ g(n)), f(n f(n an 2 + bn ≥ cn 2 ➙❯➛ c = a 0 ≥ n [(c-a)n –b] ⇒ 0 ≥ (c-a)n 2 – bn ⇒ g(n f(n ➍✺➎ (c-a)n -b ≤ 0. ➙❯➛ ➔✶→ f(n -an/2-b ≤ ➊✶➋➌ n 2 ). O(g(N ➈✳➉ ➻❣➼ ✇✎① - ➧❯➷ n 0 ➪✳➶ A

  9. ❬ ➦➻ ➧➫ ➭ ➯ ➨➲ ➧ ➨ ➒ ➍ ➳ ➵ → → ➼ ➐ ➟ ➢ ➤ ➥ ➥ ➔ → ➦ ↕ ➙ ➛ ➜ ➝ ➨➩ ➍ ➞ ➥ ➑ ➍ ➑ ➍ ➒ ➍ ➒ ➍ ➒ ❭ ➢ ➤ ➥ ➔ ➒ → ➦ ↕ ➙ ➛ ➜ ➝ ↕ ➞ → ➟ → ➣ ↕ → ➑ Ø Ô ❰ ➧ Õ ➧ ❰ Ö ➨× ➫ Ï ➫ ➯ Ó Ó Ó Ð ÛÜ Ý Þ ß Þ à á âã ä å æ çè ➫ Ð ➟ ➾ → ➣ ➐ ➨➩ ➧➫ ➭ ➯ ➨➲ ➧ ➨ ➑ ➍ ➽ ➚ Ñ ➪ ➴➷ ➹ ➷ ➬➮ ➾ ➱ ✃ ➹❐ ❒ ➱ ➮ ➲ ➍ ➍ ➍ ❻ ❥ r ① ② ③ s r ④ ⑨ ⑧ ⑩❶ ❺ ❸ ❼ ✇ ❽❾ ➀ ❾ ➁➂➃ ❿ ➄ ➅ ➆ ➃ ➇ ➂ ➈ ➂ ❾ r ❞✈ ➍ ❢ ❪ ❫ ❴ ❝ ❞ ❡ ❢ ❣ ❝ ❤ ✐ ❞ ❡ ❥❦ ✉ ❥ ❧ ♠ ♥ ♦ ❡ ♣ ❥ ❞ q r s ❞ t ➌ ❿ ➎ ➝ → ➣ ↔ ➣ →↕ ➙ ➛ ➜ ↕ ➓ → ➟ → ➣ ➡ ➍ ➡ ➍ ➡ ➔ ➞ ➐ ➒ ➐ ➑ 19 h(n)) h(n)) h(n)) f(n)) f(n)) 3 ➐➋➠ ➐➋➠ ➐➋➠ h(n)) ⇒ f(n h(n)) ⇒ f(n h(n)) ⇒ f(n ÔÚÙ ➐➋➠ ➐➋➠ g(n g(n ❷❹❸ ➐➋➠ ➐➋➠ ➐➋➠ g(n)) & g(n g(n)) & g(n g(n)) & g(n ⑦✘⑧ ⑤✰⑥ (f(n)) (f(n)) ➐✰➧ ➐✰➧ g(n f(n) = O(g(n ➍✰➏ Ò✰Ó ➶➘➹ f(n) = f(n) = ❵❜❛ Ï✏Ð - ➸✔➺ ➉➋➊ ➐➋➠ ➐➋➠ ➐➋➠ ➐➋➠ f(n f(n f(n f(n f(n ❮✔❰

  10. é á Ù ❐❒ ➴ ❮ß ➘➴ ❐ Ï ➱ Ù ❮ × ➴ × ➱ ➷ ã ä ë ì ë ä è ã ìï à ❮ ò ➹ ❒ Þ ❒ ➴ ❮ß ❮ ➶ Ð ➴ ❐ Ï Ö ➴ ➷ Ð ➴ ➱ ➷ ❐ Ö ❮ ➹ Ö ❮ ðñ ❒ Ö ❐ ❐ Ï ❐❒ ❐ ó ➹ Û ➘ ➴ ❮ ➴ ß Ñ ➶ Ó ❒ ❮ Ö ❮ ß ➴ ❐Ô ❐ ➶ ß ➹ ❮ß ➶ ó ➹ ó ➹ Û ➘ ➴ ❮ ➴ ❐ ß Ó ❮ ❒ ❮ Ö ❮ ß ❐ ➷ Ó Ü ➱ ➴ ➹ ➱ Ó Õ ➴ Ø × ➱ ➹ ➴ ❐Ô ➷ Ù Ó ❮ ➚ ➱ Ú ➷ Ò ➹ Ö ❐ ➶ ❐ ➹ ➱ Ó Ó × ➱ ➴ ✃ ➴ ❐❒ ❮ ➷ ➹ Ð ➴ Ñ ❒ ➘ ❐ ➱ ➹Ò ➱ Ó ê ❒ ➴ ❐Ô Õ ➱ Ö ➴ ➘ Ö Ý ❒ ➴ Ø ➘ Ø ➘ ➱ Ö Û ❰ ➱ ➬ ❮ Û ➘ ➴ ❐ ➹Ò ➱ Ó ➴ ❐ Ô Õ ✃ ➷ ❒ Û Õ ❰ ➱ ➘ ➴ ❐ ➹Ò ➱ Ó ❐ ➴ ❐Ô ➪ Ô ➮ ❮ ➷ Ï Ð Ü ➱ ➹ ➴ Ø ❒ ➘ ❐ ❐Ô ➷ ù ÿ ú ù ✟ ✂ ÷ û �✠ ✞ � ✡ ✞ ✝ ☛ ✞ ý ÷ û ✂ ✠ ☞ ù ÷ ù ✞ ù û ü ❮ Ö ➱ ß â Ú ❒ õ ö ÷ û ù ú ú ý û þ ùÿ �✁ ú ú ✂ ÷ ✄ ý þ Ó ✄ ✕ û ú ✠ ✞ � ÿ ✕ ✂ ÷ ✂ û ✂ ✁ ü ù ✞ ý ÷ ø ✂ ✠ ✔ ✖ ú ú �✠ ÷ ú ✠ ✞ ÿ ✝ ú ✂ ú ✠ ✌ ✠ ✎ ✝ ✒ ✓ ÷ ú û ✔ ✠ û ✁ ✟ ✠ ✞ ÿ ❒ ➶ ➴ ➘ ❮ Ö ➱ ß â ò ❒ ❮ß ó ➹ Û ➴ Ó ❮ ➴ ❐ ß ➶ Ó ❒ ❮ Ö ❮ß ❐ ❒ ➶ Ó ❮ ❮ ➘ ➴ ❐ ➹Ò ➱ Ó ❮ ➱ Ó ❒ ➹ ➹ Ï ❐❒ ❐ ó ➹ Û ➘ ➴ ❮ ➴ ❐ ➷ Ü ➹ ❮ ❐ Ó ❐Ô ➴ ❮ ➘ ➴ ❐ ➹Ò ➱ Ó ➱ ❐ Ó ❒ ❮ ➹ ô ó ➹ ➴ ❮ ➴ ❐ ➶ ❐Ô ➱ Û ➴ ❮ ➹ Ñ ß ❐ Ï ❐❒ ❐ ó ➹ ➘ ➴ ➴ ❮ ➴ ❐ ß ➶ Ó ❒ ❮ Ö ❮ ß ➹ Ñ ➘➴ ❫ ◆ P ❚ ◗ ❑ ❲ ❚ ❲ ❳ ❭❪ ❴ ❖ ❩ ❭❵ ❭❪ ❛ ❴ ❩ ❭❜ ❝ ❜ ❭ ❞ ❱ ◆❯ ❩ ❖ ✾ ✿❀ ❁ ❀ ❅❆ ❅ ❇ ▲ ▼ ◆ ◆ ◗ ● ❑ ❉ ■ ❘ ◗ ❉ ❙ ❉ ❑ ❚ ❡ ❤ ✱ ❦ q ❛ ❤ ❭ ❜ ❞ ❧ ❩ ❦ ♣ ❥ ♣ ❛ ❜ ❵ ❤ r ❩ ♥ ❵ ❭❜ ❫ ❛ ♠ ❡ ✐ ❞ ❞ ✐ ❝ ❛ ❫ ❛ ❥ ❦ ❣ ❜ ❝ ❫ ❦ ❧♠ ❩ ❭ ❥ ❜ ❢ ❞ ❤ ✐ ❝ ✺ ✸✹ ❤ ✝ ñ ✁ ✂ ñ ✄ ☎ ÿ ÷ ÿ ✆ ✞ ÿ � ÿ ✟ ☞✌ ✍✎ ✓ ✔ ✕ ✑ ✓ ✙ � þ ✚ ö ë ì í ð ñ ò ó ô ð õ ñ ñ ò ó ÷ø ÷ ù ú û ü ò ý ÷ ✘ ✑✛ ✱ ✭ ✜ ✬ ✦ ✢ ✓ ✣ ✓ ✜ ✜ ✘ ✦ ✤ ✤ ✖ ★ ✓ ✖ ✘ ✲✳ ✲✴ ✵ ✶ ✷ ✫ ✦ ✜ ✧ ✚ ✑ ✤ ✖ ✓ ✦ ✜ ✜ ✤ ✑ ✑ ✑✪ ✛ ✢ ★ ✜ ✜ ✚ ✩ ✕ ✦ ✜ ✕ ❥ ❦ ❜ ➅ ❝ ❼ ❽ ❾ ❿ ➀ ➁ ➄ ➆ ⑨ ➇ ➈ ➄ ➉ ➊ ➅ ➆ ➇ ➋➌ ⑧ ❺ ➍ ④ ❛ ❜ ❵ ❤ r ❩ ❤ ❦ ❡ ③ ❹ ⑧ ⑨ ⑨ ⑩ ⑧ ❶ ⑩ ⑧ ⑨❷❸ ➋ ➎ ❦ ➧ ➤➥ ➦ ➞➧ ➨ ➞ ➩ ➭➯ ➞ ➲ ➠ ➳ ➵ ➸➺ ➻ ➲ ➵ ➾ ➚ ➹ ➡➢ ➟ ➏ → ➐ ➆ ➑ ➋ ➅ ➒ ➓ ➔ ➅ ➣ ➓ ➅↔ ↕ ➓ ➋ ➓ ➙ ➛ ➜ ➔ ❥ ⑨❻ ❩ ❤ ❤ ❭ ❞ ❤ ❦ ❜✈ ❛ ❭ ❦ ❭ ❡ ❫ ❴ ❩ ❭ ❵ ❭ ③ ❫ ✉ ❛❩ ❩ ❦ r ❩ ♠ ❩ ❭ ❞ ❧ ❩ ❭ ❝ ❫ t ❭ ❩ ❤ ❜ ❵ ❛ ❴ ❴ ❴ ❞ ✐ ③ ❪ ❪ ⑤ ❥ ❵ ❤ ❤ ❧ ❛ ❦ ❭ ❭❵ ❴ ❞ 22 d- ø ✆☎ ❐✯â j] = a ✌✆✍ 3 f ❰✽Ï ø✒ù d j 1 ,…,j d a ➦☛➫ j] u ➬✒➮➱ ➼✒➽ ✎✑✏ í☛î ø✒ùú Initialize(X,a): j 1 , … , j d ➪❄➶ Û❏➘ o Assign(X,j,a): ø✒ù Access(X,j): ➝❏➞ Iterate(X,f): d-1)- Length(X): d- é❬ê ø✒ù ç✽è ➂♦➃ - å❏æ o o o o o o o o 21 ❴❬❦ ♣s❦ 3 ✇②① ✜✯✮ ❢✽❣ ✥✒✖✦ ✢☛✣ ❭♦♥ ♣s❦ ✗✒✘ ✠☛✡ P✒◗ ❴❬❦ l…u ✻✽✼ l ≤ j ≤ u. ✗✒✘ j- ❂❄❃ ❍❏■❑ ✕☛✖ ❋☛● ⑥☛⑦ X[j î❜ï - l,u ④✒⑤ ❨❬❩ ❈❊❉ j j ✏✒✑ ✰✒✱

  11. ✗ ✦ ✤ ✥ ✦ ✧ ★ ✤ ✩ ✪ ✥ ✧ ✛ ✫✬ ✫ ❬ ❭ ✭ ✮ ✯ ✰ ✦ ✱ ❨ ② ✥ ❞ ♠ ❤ ❱ ❩ ❨ ✈ ❩ ❱ ❝ ❛ ✚ ➂ ❚ ❨ ➆ ➇ ➈ ❬ ✗ ✘ ✙ ✫ ✲ ❱ ➏ ▼ ❽ ❇ ❈ ❉ ❊ ❇ ❛ ❢ ➏ ➋ ❺ ⑧ ➐ ④ ❹ ⑩ ❷ ❶ ❜ ⑨ ▲ ✾➊ ✳ ✺ ✴ ✥ ✵ ✶ ✥✷ ✸ ✳ ✫ ✳ ✹ ✻ ❖ ✴ ✳ ❾ ➙ ➉ ❆ ❏ ❆ ❉ ❿ ♦ ❚ ❻ ❛ ✐ ❭ ❾ ❿ ❭ ❴ ❴ ❴ ❭ ❽ ❛ ✐ ❭ ➦ ❛ ➅ ❭ ❾❿ ❭ ❴ ❴ ❽ ❭ ❭ ❭ ♣ ➠ ❛ ❽❾ ❭ ❾❿ ❭ ❛ ❽❾ ✐ ❿ ❿ ❭ ❴ ❴ ❴ ❭ ❛ ❽❾ ❭ ➧ ❴ ❛ ❤ ❽ ❤ ❚ ❲ ❢ ❵ ❜ ❞ ❲ ❛ ❤ ❚ ❿ q r ❞ ❤ ❡ ❜ ❚ ❬ ❢ ❲ ❬ ❽ ❲ ➤➥ ➞ ➢ ➠ ➠➡ ➞➟ ➜➝ ➛ ✐ ❦ ❫ r ❳ ❭ ↕ ✘ ❬ ❵ ❢ ❲ ❚ ➁ ❹❺ s ❩ ❢ ❢ ❝ ❩ ❨ ✈ ❩ ❱ ❝ ♥ ❬ ❤ ❚ ❱ ❨ ❬ ❫ ❲ ♦ ✈ ❡ ❜ ❜ ♠ ❤ ❬❢ ❞ ❬ ❡ ❬❢ ♣ r ❬ ❬ ❫ ❚ ♦ ❲ ❢ ❚ ❬❢ ② ✐ ➂ r ➓ ➒ ❦ ❱ ❵ ❜ ❢ ❬ ❚ ❬ ❤ ❚ ❲ ❢ ❵ ❞ ❚ ❬ ❜ ❨ ❳ ❝ ❫ ♣ ❨ ✇ ② ↔ ❻ ❚ ➔ ❲ ❦ ❫ ❞ ❨ ❬❡ ❬ ❴ ❚ ↕ ➐ ❺ ④ →➣ ❸ ④ ❹ ⑩ ❷ ❶ ✇ ⑨❹ ❜ ❢ ❙ ❩ ❡ ❤ ❫ ❞ ❨ ❬ ❡ ❬ ❝ ❜ ❨ ❤ ❚ ❲ ❢ ❵ ❜ ❞ ❬ ❛ ❽ ❢ ✇ ❭ ① ❦ ❩ ❩ ❜ ❚ r ❢ ❡ ❳ ❝ ❵ ♥ ❜ ❚ ❬ ❢ ❲ ❚ ❜ ❚ r ❬ ❢ ♥ ❲ ❢ ➂ ➒ ➂ ❴ ❴ ❴ ➂ ♥ ❣ ❬ ❨ ✉ ❲ r ❜ ❚ ❢ ❡ ♣ ❤ ❚ ✐ ❦ ❢ ❫ ❬ ❡ ♣ ♦ ❜ ❾ ❬ ❬❡ ❨ ❞ s ❲ ❲ ❡ ❢ ❨ ✇ ➑ ❫ ♣ r ❵ ❩ ➟ ❬ ❨ ❜ ❞ ❲ ❚ ❲❦ ❴ ➥ ➇ ❧ ➟ ❢ ❜ ❞ ❨ ❬❢ ❚ ❲ ❩ ➦➝ ❜ ♦ ❵ ❲ ❝ ❣ ❞ ❨ ❬❡ ❬ ❜ ❞ ❨ ❬❢ ❡ ❬❢ ❜ ❚ ➅➳ ❚ ❱ ❝ ❤ ❚ ❲ ✐ ❥ ❤ ❲ ❡ ➤ ❜ ❲ ❤ ❚ ❳ ❤ ❚ ❲ ❢ ❵ ❞ ❲ ❲ ❚ ❲ ❦ ❫ ❞ ❨ ❬❡ ❬ ➥ ➄ ❚ ♣ ❦ ♦ ❜ ❤ ❚ ❲ ❢ ❵ ❜ ❞ ❲ ❚ ❜ ❲ ❦ ❫ ❞ ❨ ❬❡ ➠ ❚ ❳ ❚ ❫ ❨ ❤ ✶ ✱ ✫ ✥ ➥ ✲ ✳ ✴ ✥ ✵ ✥✷ ✰ ✸ ✳ ✫ ✳ ✹ ✺ ✻ ✴ ✳ ➞ ✦ ✯ ➧ ✤ ✙ ✚ ➈ ✛ ➝ ✤ ✥ ✦ ✧ ★ ✩ ✮ ✪ ✥ ✦ ✧ ✫✬ ✫ ➺ ➞ ➨ ✭ ➺ ✼ ❜ ➡ ❖ ❇ ➯ ❂ ▼ ➤ ➞ ➝ ➇ ❱ ❚ ❆ ❲ ➜ ❲ ❩ ➵ ➝ ❵ ❴ ❛ ❭ ▼ ❋ ➸ ❄ ❀ ✾ ➆ ❄ ❂ ➅ ❇ ➟ ❊ ❇ ❋ ❄ ➩ ■ ❏ ✾ ❑ ❆ ❉ ➸ ◆ ▲ ❞ ❡ r ❚ ❫ ❞ ❨ ❬ ❡ ❬ ② ❬ ❦ ❳ ❲ ❫ ❴ ❵ ❴ ♥ ❞ ❨ ❜ ❚ ❬ ❦ ❚ ❤ ❲ ❲ ❫ ❲ ❞ ❲ ❤ ❱ ❩ ❬ ❚ ❣ ❝ ❲ ❚ ❜ ❞ ❚ ❲ ❚ ♠ ① ❲ ❢ ❚ ❜ ❡ ❞ ❨ ❬❡ ❜ ❝ ❬ ❫ ❲ ♦ ❱ ❜ ❢ s ❲❨ ❚ ❬ ❢ ❡ ❬ ❚ ♣ ♥ ❫ ❲ ❱ ➞ ➄ ❩ ❜ ❚ ❬ ➡ ➆ ➤ ⑤ ④ ⑨ ➠ ⑧ ⑩ ④ ❶ ➦ ⑨❹❺ ❻ ➠ ➄ ❱ ❲ ❚ ➭ ❲❦ ♠ ❝ ❚ ❲ ❩ ✈ ❡ ❲❝ ❚ ❫ ❲ ❞ ❨ ❬ ❡ ❬ ❣ ❜ ❨ ❜ ❞ r ✉ r ♦ ❚ ❬❢ ❤ ❚ ❱ ❣ ❢ ❜ s ❦ ❜❝ ❨ ❤ ❱ ➲ ➄➅➯ t ♣ ❫ ❲ ❢ ❨ ❬❢ ♥ ❤ ❲ ❢ ❜ ❞ ❚ ❬ ❢ ♠ ❨ ❬ ❚ ❲ ❲ ❢ ❵ ❜ ❞ ❲ ✉ r ❲ ❦ ❫ ❩ ❨✇ ❢ ❤ ❚ ❳ ❡ ❬❢ ❬ ❨ ❚ ❞ ❡ ❬ ❦ ❚ ❲ s ❵ r ❱ ❤ 24 ≤ j,k ≤ n. 3 k < j ❳✆❝ 86 23 43 34 31 87 2 31 87 45 0 0 18 0 2 ❅✿❆ ➨➫➩ 0 43 0 1 1 34 6 n)*n n k,j] = X[j,k] ≤ k,j ≤ n. 0 7 0 0 5 k,j ➧➫➺ 21 34 ▲❃▼ 1 5 ❷❯❸ ♠✆♥ ❷❯❸ s✆❨ ✜✣✢ - k > j ➌➎➍ m], ... , 23 C ♠✆❝❼ 3 X + L*(j-1). ▲❃▼ … [0,m] … [1,m] … [n,m] … … m ❳✆❝ ... ●❍❆ [0,1] [1,1] [n,1] m P❘◗ … ♠✆♥ j- 1 ❫❪❴ k,j ❳✆❝ ❬❪❭ ❈❃❉ n,m]. [0,0] [1,0] [n,0] ➃✆➄➅ ... … L*(m*k + j). ❅✿❆ 0 ❳✆❨ strings). ❷❯❸ n 0 1 ❬❪❭ C L ❁❃❂ ❞✆❡ C ❙❯❚ q❃r ⑥✿⑦⑧ ✜✣✢ - ✽✿✾ o o o ❽✆➀ ❙❯❚ ③✆④

  12. ➻ ø ë ê ø ñ è ñ ù ÷ ò ✛ ➼ è ☞ ✆ ç ✏✑ ✒ ✍ ✏ ✓✔ ✕ ✟✖ ✗ ç ì ✑ ✝ ñ ý ç ñ ò ö ✥ ✣✤ ✢ é ✆ ✞ è ✟ ✠ ✡ ☛ ✄ î ö ò ñ ð ö ✘ ø øü ô ✜ ê ë ç è ì è ✢ ö è ç î ê ì è ø ç ð ö ✜ ø ✚ ü ø í ø ò ✙ è ê ø ✚ þ ì ð ü ù ý ñ øü ç ✛ ñ ë ù ñ ø ò û øü ✛ ü è ç û èì ✃ ✲ Ñ Ò Ó ➘ Ô Õ ➘Ö × Ò Ò ✃ Ø Ù Ú Ó Ò ✰✱ ✯ ✮ ✭ Þ ß ➘ Ð á ➱ ➽ ➾ ➚ ✰✱ ➹ ➘ ➴ ➷ ➬ ➹ ➮ ➘ ➴ ➴ ➷ ✃❐ ✃ ✳ ✱ ✮ ❒ ❮ ❰ Ï à ✭ î ñ ï ÷ ✧ ✦ ê ø ù ø ù ñ ô ú Ý ç ù ñ ø òû ð ë ú ç ñ øü ð ö è ç ãäå ✬ ✪✫ èé ê ë ç èì èé í ✩ ð ñ èò è ó èò ô ★ ✦ è ç î ø 25 í❍ø -3 3 0 0 0 0 1 0 0 þÿ ✁� (7 1 2 -3 4 12) ô✿ñ 0 0 0 0 0 0 0 0 é❪õ à❃â 0 0 4 0 0 0 1 0 Û✿ÜÝ î✆ï 7 2 0 0 1 1 0 0 ➪✣➶ - ✌✎✍ 12 ✂☎✄ 0 1 0 0 1 0 1 æ✆ç

  13. ✴ ➐ ➣ ➄ ➁ ➊ ➃ ➔ ➓ ➈ ➁ ➞ ➏ ➏ ➅ ↔ ➞ ➈ ➏ ➒ ➅ ➈ ➓ ➐ ➧ ➊ ➈ ➧ ➜ ➞ ➐ ➙ ➑ ➇ ➈➉ ➊ ➀ ➄ ➒ ➃ ➁ ➒ á ➄ ➒ á ➔ ➇ ➈ ➃➄ ➐ ➉ ➊ ➏ ➏ ➏ ➈➉ à ➉ ➊ ➉ ➒ ➃ ➞ ➅ ➑ ➇ ➏ ➅ ➅ ➁ ➒ ➈➉ ➄ ➅ ➊ ➔ ➒ ↔ ➁ ➦ ➏ ➁ ➊ ➞ ➐ ➏ ➅ ➜ ➁ ↕ ➅ ➁ ➏ ➉ ä ➉ ã ➉ ➄ ➓ ➅ ➁ ➏ ➧ ➏ ➇ ➃ ➁ ➧ ➅ ❰ ➮ ➱ ➸ ✃ ❐ ➸❒ ❮ ➮ ➚ ➮ Ï ➸ Ð ➱ ➮ Ñ Ò Ø ØÙ Ú Ý ÙÞ ß ➬ ➚ ➏➒ ➵ ➦ ➨ ➩ ➫ ➭ ➯ ➵ ➸ ➺ ✵ ➼ ➽ ➷ ➾ ➸ ➺ ➻ ➚➪ ➚ ➶ ➹ ➘ ➴ ➺ à ➔ ➏ ➒ ➅ ↔ ➞ ➈ ➏ ➒ ➅ ➃ ã ➔ ➊ ➐ ➅ ➁ ➐ ➂ ➁ ➙ ➅ à ➓ ➐ ➧ ➏ ➏ ➒ ➆ ➊ ➃ ➂ ➑ ➇ ➈➉ ➊ ➃ ➊ ➉ ➊ ➊ ➁ ➁ á ➠ ➐ ➏➒ ➈ ➓ ➐ ➏ â ➜ ➜ ➈➉ ➃ ➙ ➆ ➇ ➈➉ ➊ ➅ ➃ ➊ ➉ ➠ ➉ ➐ ➙ ➏ ➃ ➉ ➞ ➆ ➏ ➈ ➙ ➉ ➄ ➓ ➅ ➉ ➔ ➈ ➏ ➉ ➊ ➒ ➙ ➏ ➣ ➞ ➃ ➉ ➊ ➙ ➙ ➂ ➙ ➈ ➓ ➔ ➔ ➇ ➃ ➐ ↕ ➈ ➣ ➄ ➞ ➦ ➞ ➐ ➏ ➅ ➆ ➇ ➈➉ ➏ ➅ å ➌ ➏ à ➉ ➄ ➇ á ➊ ➌ ç ➓ ➔ ➠ ➁ ➧ → ➄ ➔ ➠ ➁ ➁ ➊ ➉ ➄ ➅ ➋ ➓ ➆ ➣ ➏ ➄ ➇ ➙ ➈ ➙ ➁ ➏ ➈ ➓ ➐ ➧ ➣ ➔ ➄ ➔ ➣ ➄ ➇ ➙ ➈ ➙ ➁ ➏ ➈ ➏➒ ➒ ➆ ➊ ➃ ➞ ➅ ➆ ➇ ➈➉ ➏ ➅ å ➦ ➄ ➓ ➓ ➏ ➒ â ➏ ➇ ↕ ➐ ➏ ➇ ➐ ➊ ➒ ä ➋ ➊ ➔ â ➂ ➅ → ➐ ↕ ➈➉ ➜ ➅ æ ➐ ➉ ➐ ➒ â ➏ ➇ ↕ ➐ ➧ ➏ ➐ ➏ ➇ ➊ ➈➉ ➒ → ➐ ↕ ➈➉ ➜ ➅ ➦ ➝ ➣ ➒ ➄ ➅ ➊ ➄ ➁ ➒ ➃ ➞ ➅ ➑ ➇ ➈➉ ➏ ➅ ➀ ➁ ➜ ➞ → ➒ ➈➉ ➄ ➅ ➊ ➔ ➒ ↔ ➁ ➠ ➻ ➒ ④ ❥ ❧ ② ⑤ ✉ r ❥ ❧ r ⑨ ■ qr ④ ❧ ❷ ✉ ♣ ④ ⑨ ❍ ✽ ♠ ⑨ ● r ♣ ④ ✽❏ ⑤ ❤ ④ ❹❺ ✉ ♣ ❥ ④ ❧ r ④ ⑨ qr ④ ❧ ❷ ✉ ♣ ❻ ❼ ❷ ➇ ❈ ➌ ➍➎ ➀ ➃➏ ➐ ➂ ➆ ❇ ➈➉ ➋ ➊ ❆ ❅ ❃ ➑ ➏➒ ➉ ➄ ➓ ❲ ❿ ➅ ♠ ⑨ ✉ ❽ ❥ ♦ ♥ s ❋ ❻ ❊ ④ ❤ ➊ ❾ ✽ ❻ ❃ ➉ ➁ ❉ ➅ ➆ ✾ ✉ ❧ ➇ ♠ ✇④ ♦ ④ ❥ ⑤ ⑥ ▲ ❋ ⑦ ⑥ q ❦ ⑧ ④ ⑨⑩ ❃ ❶ ⑥ ⑤ ④ ⑨ ③ ❥ ④ ❧♠ ❨ ❜ ❝❞ ❯ ❙❚ ❙ ❥ ❦ ❦ ♥ ✇ ♦ ❋ ❥ s ● ◆ ▼ ❥ ❧ ② r q ④ ❦ ④ ⑤ ❤ ❸ r ④ ⑩ ♠ ⑥ ⑧ s ④ ⑩ t ❶ ✉ ✇ ❸ r ④ qr ♥ ④ r r ④ ❧❷ ✉ ♣ ④ ❋ ✉ ❑ ❥ ❧ ④ ♣ ⑤ ❤ ❸ r ④ qr ④ ❧ ❷ ✉ → ➔ ➏ ➉ ➄ ➉ ➞ ➆ ➏ ➈ ➁ ➊ ➙ ➊ ➅ ➈ ➓ ➐ ➞ ➐ ↕ ➈ ➙ ➣ ➄ ➉ ➏ ❀ ➏ ➣ ➒ ➈➉ ➔ ➅ ➟ ➒ ➉ ➁ ➒ ➊ ➆ ➇ ➈ ➉ ➊ ➀ ➅ ↕ ✼ ✷ ➌ ➏ ➣ ➒ ➈➉ ➔ ✸ ➏➒ ➊ ➎ ➐ ✶ ➧ ➒ ➊ ➟ ➄ ➔ ➏ ➦ ➍ ➐ ✿ ➢ ➆ ➒ ➈➉ ➤ ➐ ➈ ➏ ✾ ➒ ➌ ✽ ➏ ➣ ➒ ➈➉ ➔ ✼ ➏ ➌ ➞ ❿ ✾ ➜ ➆ ➇ ➈➉ ➏ ➅ ➀ ✿ ➌ ➝ ➊ ➣ ➒ ➈➉ ➔ ➒ ➏ ➒ ❂ ➂ ➣ ➒ ➐ ➓ ➣ ➄ ➓ ➈ ➓ ➐ ➂ ❃❄ ➅ ➣ ➊ ➄ ➈➉ ➙ ➔ ➇ ➛ ➄ ➐ ➉ ➞ ✽ ➊ ❿ ➏ ➇ ➐ ➒ ➍ ➡ ➂ ➎ ➌ ➊ ❁ ➝ ➣ ➒ ➈➉ ➔ ➞ ➟ ➏ ➐ ➊ ❫ ➠ ➂ ➅ ➓ ➁ ➁ ➆ ➊ ↔ ➓ ➐ ➄ 28 3 Û❳Ü Û❳Ü ➞①→ : o Õ◗Ö× : Ó✎Ô ➲✻➳ - L 0 L = < >, false 27 ➥✁➦ L k+1 3 L k-1 , 0 < k < n-1. true ❩❭❬❪ ♣①✇ L 0 , L 1 , ..., L n-1 L|. L j L| = n) t✈✉ ➞①➟ linear list ❱❳❲ ➞①➟ ↔❳↕ L|-1. ➞①➟ head) MakeEmptyList(): (tail) IsEmptyList(L): ❖◗P❘ Concat(L 1 ,L 2 ): L k Access(L,j): ➇✎➈➉ Length(L): ♣✎qr ➂①➃➄ ❡◗❢ L n-1 : j ✹✻✺ - L 0 ❿✈➀ |L n ≥ ❴❛❵ ❣✐❤ ♣①✇

  14. è ❒ ì Û ☎ ➴➷ ➴ ➬ è ❐ ✃ ➺ ❐ Ï ✃ ➬ ➮ ➱ Ñ➱ ➷ ➸ ➶ Ý ➺ ➯ ➲ ➢ ➳ ➨ ➡ ➵ ➸ ➡ ➪ ➻ ➼ ➡➽ ➾ ➸ ➨ ➸ ➚ Ô Ø ➫ ß â à Û Ý ß Ø ã à á á × Ùäå ä æ × Ø Û áç Ý Ù Ù à Ú ÛÜ Ý Ú Û Þ ß Ö Ù Ø Ø Þ Û Ú Û Ú Ûá ß Ö ➭ Û Ö ⑨ ❪ ❡ ❪ ✐ ❽ ❶ ⑨ ⑧ ❣ ⑧ ❤ ✐ ❦ ⑩ ❴ ♥ ➀ ❶ ⑨ ❤ ❡ ❢ ❽ ❶ ❾ ✐ ❪ ✐ ♥ ì ❡ ♥ ❿ ❡ ❛ ❵ ❴ ❣ ❦ ❴ ❵ ❤ ➨ ➢ ↕ ➙ ➛ ß ➜ ë ➠ ➡ ➤ Ú ➥ ➠ ➦ ➧ ➡ ➢ ➤ ➨➩ ↔ ß ❴ ➌ ❫ ➁ é ➅ ➃➆ ➇ ➈ Ù ä ➣ ➈ Þ ß ➑ ➇ ➊ ➒ ✆ → ß å × á ÿ ê Ù á Ø Û ß Ú × ã Ø Þ è í ß Ø Ù á ë Ø ï þ Û í æ Ø Û Ü Ú Þ Ø ÛÜ Û Ü × Ø Û áç ß Ö � Ø Ø × ✁ ÿ ï Ù á ß Ú á Ø ß Þ è í ß á Ù à ✂ á í ß í Þ ß Ú á × Ø Þ è ß è á Û í ß Ú á × Ø Þ Þ × à Ø Ø Ûá ç ß Ö å à × Û æ é î î ✄ ð ñ òó ôõö × Ùí ø Ø × Ø Û è à Ù Þ Û ãë Þ ê Ù á Ø ã ì á Ùä ÷ ñù á á Ö × Ø Ûáç Û ê á ü Ù à ä Þ Ù í æ ý ß Ú Ø Ù ï × é ú á × Ù äå ä æ Ø û Û Ø è î Û ë Ø ã ë ❼ ❡ Ö ✳ ✦ ✮ ✵ ✮ ✦ ✱ ✫ ✦✶ ✤ ✳ ✥ ★ ✮✷ ✬ ✣ ✲ ✭ ✤ ✴ ✱ ★ ✭ ★✮ ✬ ✣ ✥ ✦ ✮ ✪ ✯ ★ ✲ ✪ ✬ ✥ ✰ ✭ ✬ ✮ ✤ ✦✱ ✥ ✸ ★ ★ ✫ ✸ ✶ ✬ ✮ ✥ ★ ✴ ✫ ✤ ✩ ✶ ✦ ❻ ★ ✤ ✥ ★✮✷ ✥ ✮ ✭ ✞ ✦ ✥ ✰✹ ✺ ✴ ✫ ✟ ö ✽ ✧ ✾ ✿❀ ❁❂ ❃❄ ❅ ✾❆ ✝ á ✬ ✧ ✧ ✣ ð ñ ü ö ð ô ý þ ÿ � ú ✁ ð ✂ ✄ þ ö þ ☎ û ù ✝ ó ê ë ì ➣ ï ð ñ ò ï ø ô õ ð ñ ò ö÷ ö → ✍ ✆ ÿ ★ ✪ ✛ ✜ ô ó ✥ ✦ ✧ ★✩ ✧ ✏ ★ ✫ ✬ ✣ ✭ ✦ ✥ ✫ ✚ ✒ þ ✏✑ ó ✟ ö ✞ ✡☛ ✡ ☞ ✌ ✒ ✙ ✏✑ ✠ ✖ ✏ ☞ ✌ ✍ ✘ ✍ ✬ ✣ ★ ✐ ❥ ❣ ❤ ✐ ❫❧ ❡ ❦ ❥ ❤ Û ❣ ❤ ♠⑨ ❞ ✐ ⑨ ❛ ✐ ❝ ❜ ✥ å à s t✉✈ ✇ è ③ ♠ Ý ❵ ⑥ ⑤⑦ ❭ ❡ ❫ ❣ ❴ ❵⑧ ❥ ❝ ❜ à ❡ ❫ ❤ ♠⑨ ❷❸ ❹ ❫ å ❶ Ø ❡ ❢ ❫ ❣ ❤ ❛ ❶ ❜ ❛ ❴ ⑩ ❤ ♠ ♥ ❣ ❤ ✐ ë ❦ ❴ ❤ ♥ ì ß ì ❡ ❢ ❫ ❣ à ❺ ÛÜ é ◆ ❖P◗ ❘ ë ❚ ❱ ❲ ❩ è ❨❬ ❭ Þ ❵ ❛ ❴ ❜ ❝ î ä ❞ ✤ ✰ ✹ Ù ❇ ✦ ✭ ✥ ✣ ✥ ß ★✮✷ ★ Ø ❞ ❉❋ ● ● ❍ ❫ ❴ ❡ ❣ ❪ Ù ❥ ❜ ❴ ❦ ✐ ❥ ❤ ❤ ✐ ❫❧ ❡ ❦ ✐ ❥ ❤ ♠♥ ❛ ❫❴ ✐ ❞ ❡ ❛ ❫ ❡ ❢ ❫ ❣ ❤ ✐ ❛ ✐ ❜ ❜ ❝ ❤ ✐ ❥ Q Q 0 Q| = 0, false First In 30 3 Ò✠Ó ✎✑✏ Q (Q = < Q 0 , …, Q |Q|-1 true é❊ê (Queue) Concat(Q, <x>)). Û✼ï x Ï✠Ð Q 0 . – ✡☞☛ ❮✕❰ – First Out, FIFO). MakeEmptyQueue(): IsEmptyQueue(Q): ➮✎➱✃ Enqueue(x,Q): Dequeue(Q): Front(Q): ➹✠➘ ➝➟➞ - ÕÖ✠× S S| = 0, Last 29 3 S ➓❊➔ ❪✎❫❴ true ➏➄➐ (Stack) S, |S|-1)). – ❳▲❨ ④▲⑤ ✖✠✗ ➍✠➎ In – First Out, LIFO). ①❯② ✓✕✔ ❙❯❚ MakeEmptyStack(): ➉➋➊ false. IsEmptyStack(S): ✌✎✍ ❈❊❉ ♦♣▲qr x ■❏▲❑▼ Push(x,S): ➂➄➃ Top(S): Pop(S): ✞✠✟ í✻î - ★✼✻ ✢✣✠✤

  15. ✒ ⑥ ④ ➎ ➆ ④ ➇ ❺ ➂ ❶ ⑩ ➏ ➅ ⑥ ➈ ➅ ❸ ⑨ ②③ ③ ⑧ ④ ⑧ ⑨ ③ ④ ⑥ ② ③ ③ ② ➅ ➎ ③ ⑦ ➅ ❼ ⑦ ③ ④ ➆ ③ ② ➄ ❻ ⑤ ❸ ⑨ ➃ ⑧ ➂ ➋ ⑥ ❻ ➃ ⑥ ④ ➐ ③ ⑥ ➅ ❸ ➐ ④ ③ ➈ ⑤ ➉ ⑤ ③ ➅ ③ ❸ ➂ ❾ ❿ ➑ ⑤ ➃ ⑦ ② ⑦ ⑤ ❷❸ ③ ❻ ⑤ ⑥ ✓ ❽ ③ ❶ ❸ ➉ ⑤ ➃ ❻ ⑤ ⑥ ❼ ④ ➂ ③ ⑦ ⑥❺ ❻ ⑤ ⑥ ④ ➆ ❸ ⑤ ➉ ⑥➈ ➅ ❼ ⑦ ③ ② ③ ⑩ ❽ ➊ ③ ④ ⑦ ③ ➈ ➋ ⑥ ➅ ⑥ ➃ ❸ ❹ ❸ ❼ ③ ➅ ➎ ⑧ ⑦ ❼ ⑨ ④ ❸ ⑥ ③ ➅ ➋ ➉ ⑩ ❻ ➃ ⑥ ❻ ⑤ ⑥ ❼ ❽ ③ ❶ ➅ ➈ ⑧ ➁ è ì ë í î î î ï â á á ç ã ä ñ é í ä ç ë ÙÚÛ ÓÔ Ð × Ñ ➱ ❐ Ø Ü é ✃ ➮ Ý ã ä å æ á ã á Ñ ✁ ✁ ý ✂ ø ÿ ü ✄ ✞ ÿ ý ✂ ✄✟ ù ✠ ✡ ø ☛ ☞ � ÿ ô ð â á ð Þ õ ß á ö ý â å ó ì ð ú ý þ ➱Ò Ð ❺ ➛ ➓ ➔ →➣ ↔ →↕ ➣ ➙ ➙ ➎ ➛ ↕ ➜ ➝ ➞ ➟ ➠ ❻ ⑤ ➥ ⑤ ❶ ❸ ⑤ ③ ❽ ➋ ➃ ⑧ ❻ ❺ ❸ ❼ ⑦ ③ ④ ❿ ➒ ➡ ➦ ❒ ➽ ➦ ➚ ➪ ➦➶ ➹ ➽ ➯ ➘ ➽ ➴ ➷ ➾ ➽ ➬ ✃ ➱ ❮ ➾ ➼ ➧ ➨ ➨ ➩ ➥ ➫ ➭ ➦ ➧ ➯➲ ➦ ➯ ➳ ➵ ➸ ➺ ➧ ➻ ➯ ⑤ ❼ ◗ ❱ ❭ ❛ ❡ ❳ ❪ ◗ ❩ ◗ ❙ ◗ ❚ ✼✽ ❳ ❲ ❘❙ ❙ ✜ ❱ ❬ ❲ ❛ ◗ ❚ ◗ ♦ ✻ ❁ ❤ ♠♥ ✛ ❤ P ❱ ✐❥ ✺ ❘❙ ❝ ❫ ❙ ❱ ❲ ❭ ❴ P ❱ ❘❙ ✮ ❱ ❲ ✻ ❚ ✺ ❱ ❪ ❪ ❡ ❙ P ❤ ◗ ✥ ✤ ❚ ❱ ❵ ❨q ✼ ❂ ❲ ❩ ❯ ❵ ❫❴ ❙ ❭ ❪ P ◗ ❙ ❲ P❭ ❱ ❬ ❱ ❨ ❫ ❙ ❙ ✢ P❚ ❯ ❏ ❘ ❑ ◗ ❱ ✦✧ ❍ ❉● ❲ ❳ ❘ ❭ ◗ ✿ ❳ ❲ ❱ ❘❙ ❂ ❫ ✺ ❤ ✽ ❃ ✸ ❚ ◗ ❛ ❲ ❛ ❞ ❋ ✦ ❪ ❱ ❛ ❱ P ❬ ❡ ❘❙ ◗ ❙ ❱ ❨ ❱ ❘❙ ❳ ⑤ ❹ ✛ ✜ ✛ ⑥ ✭ ④ ③ ✮ ② ③ ⑥⑦ ✯ ✢ ②③④ ✬ ✛ ✰ ◗ ❚ ◗ ❛ ❲ ❬ ✱ ✛✲ ❛ ❱ ❙ ◗ ✦ ⑥❺ ❞ ❼ ✔ ③ ❽ ❼ ⑥ ⑤ ❻ ❸ ★ ❻ ❺ ➃ ⑤ ② ❸ ❻ ❺ ⑤ ⑥ ❼ ❽ ✜ ③ ✫ ➂ ✚ ✪ ✖ ✩ ✕ ➁ ❲ ◗ ❭ ❝ ✵ ❭ ❲ ❱ ❙ ❫ ❘❙ P ❱ ✴ ❲ ❳ ◗ ❝ ❴ ❱ ❬ ❳ ❝ ✯ ❤ ✚ ♥r ✶ ◗ ❘❙ s t ❙ ❭ ❵ ❱ ✮ ✣ ✦ ❪ ✉ ❝ ❪ P◗ ❞ ❱ ❯ ◗ ✈ ✮ ❘❙ ❫ ✳ ❪ ❭ P ◗ ❨ ❱ ❲ P❭ ❱ ❬ ♦ ❱ ❘❙ ❩ ❙ ◗ ❳ ❲ for (i=0; i < N; i++) Infos[i] = ∅ ; /* initialize */ 32 3 else (return(Infos[Length – 1])); if (IsEmptyStack()) then error; Õ❀Ö /* return (Length == 0) */ if (Length == 0) return 1; void MakeEmptyStack(void) boolean IsEmptyStack(void) (N) ☎✝✆ ê☞ë ò☞ó (1) ❰☞Ï MakeEmptyStack(): else return 0; IsEmptyStack(): ç✹è Length = 0; info Top(void) ❐❀❒ ê✹ð (1) ➮☞➱ ➢✙➤ û✹ü Þàßáâ Top(): - ÷✑øù ❜✑❝ C) Length Infos 31 A[0 … n-1]. ❼✑③ & A[0 … N- Infos(S)). 3 struct int Length; info Infos[0..N-1]; ⑧✹⑨ ❄❆❅❇ ⑧✑⑦ ✾❀✿ fos ✇❊① length ⑥❧➌ ◗❣❢ S = <S 0 , … , S n-1 > ✾❀✿ ❼✑➇ ❾➀❿ ❾➍➌ S->Length == N: ▲▼✹◆ n ≤ N S->Length == 0: C ❈❊❉ ✷✹✸ ❝❧❦ global j] = S j ■✹❏ ✐❥✑♣ ➄☞➅ ❥✑♣ ✗✙✘ A[n- ⑧✑❷❸ - ❖❊P◗ ❈❊❉ A info ⑩❆❶ S

  16. ✌ ❇ ✾ ✴ ✿❀ ❁ ✽ ❂ ❃ ❄ ✾ ✴ ✷ ❅ ❆ ❬ ✸ ✽ ✴ ❭ ❪ ✺ ❫ ✴ ✳ ❆ ❋ ●❍ ■ ❏ ❑ ✽ ✺ ▼ ✪ ✗ ✘ ✜✢ ✜ ✣ ✤ ✥ ✦ ✗ ✧ ✜ ✖ ★ ✩ ✖ ✴ ✫ ✍ ✖✭ ✮ ✩ ✜ ✩ ✯ ✰ ✱ ✪ ✩ ✲ ✶ ▲ ❍ ✛ ♣ ● ❍ ❵ ❢ ■ ▲ ❝ ❚ ❵ ❦ ♥ ♦ ♥ ♣ q ❋ r ♥ s ❤ ♣ ♠ t ♥ s t① ❥ ❤ ❳ ④ ❡ ❵ ◗ ❨ ◆ ❍ ❑ ❚ ❙ ❯ ❱ ❲ ❳ ❨ ❱ ❩ ❲ ❨ ❴ ❍ ❖ ■ ❍ ◆ ❏ ❑ ❛ ◗ ❯ ❏ ❙ ❍ ❞ ■ ✖ ✬ ✚ ✸ ✩ ✜ ✩ ✯ ✰ ✱ ✪ ✩ ✲ ✶ ✴ ✺ ✽ ✖✭ ✾ ✴ ✿❀ ❁ ✽ ❄ ✾ ✴ ✷ ❅ ❆❇❈ ❉ ✶ ✮ ✬ ❊ ✘ ✎ ✏ ✑ ✕ ✖ ✗ ✘ ✙ ✕ ✚ ✛ ✖ ✗ ✜✢ ✫ ✜ ✣ ✤ ✥ ✦ ✗ ✧ ✜ ✖ ★ ✩ ✪ ✖ ✕ ✳ ❍ ▼ ✏ ✎ ✍ ✌ ❋ ✕ ✖ ✗ ✘ ●❍■ ❏ ❑ ❑ ▲ ❍ ✙ ◆ ◗ ❚ ❙ ❯ ❨ ✑ ❱ ❩ ❲ ❨ - 3 - 3 ✒✔✓ ✒✔✓ ✳✵✴ ✷✹✸ ✻✵✼ ❂✹❃ ✳✵✴ ✷✹✸ ✻✵✼ pointer MakeEmptyStack(void) pointer S; /* temporary pointer */ S = newcell(Stack); /* malloc() */ S->Length = 0; return S; info Pop(void) boolean IsEmptyStack(pointer S) if (Length == 0) return error /* return (S->Length == 0) */ else if (S->Length == 0) return 1; x = Top(); else return 0; Length = Length –1; return x; info Top(pointer S) void Push(info x) if (IsEmptyStack(S)) then error; else (return(S->Infos[S->Length – 1])); if (Length == N) then error else Length = Length + 1; ◆P❖ ❘✵❙ Infos[Length-1] = x; MakeEmptyStack IsEmptyStack(): (1) Top(): (1) ◆P❖ ❘✵❙ Pop ❱❲✝❳ ❘P❵ ❜✵❝ Push(): (1) ❣✐❤❥ ❧P♠ ✉✝✈ r✵✇ ②✐③ 33 34

  17. ⑤ ë û ú ù ì ø ÷ ð ô ôõ î í ì ê ù ➶ é Ý × è ❒ ✃ ➱ Õ å× ❒ Ü ë üýþ Ô ÷ ù ✆ õ ò ô ñ ò ✂ ð ì í ✁ � ô ó ð ì ò ë ë ÿ ì ÷ ð õ ð ô ð ✃ ➷ ê Ò × Ò Ï Ñ ➶æ å ❒ ✃ ä ❮ ➷ Ó Ñ ❰ ❒ ⑥ Ñ Ö â ➚ ➶ ❰ Ð ➱á ➷ à ❰ç ❮ ç ❰ Û Ù Ñ Ï ❰ × Ü ❒ ❰ ➱ á ➱ ❒ Ï Ò ➷ Ñ➱ Ï Ñ Ð ➱ ✃ Ü ➮ Ø Ô Ñ ✝ ñ ➮ ✕ ✒ ☞ ì ð ë î ò ë ô ð ì ò î ð � ✓✔ ð ï î í ì ë ✒ ✑ ÷ ð ☞ ì ✑ ✤ ✭ ★✬ ✤ ✗ ✫ ✙ ✜ ✪ ✦ ✜ ✥ ★✩ ✧ ð ✢ ✥ ✙ ✦ ✜ ✥ ✜ ✤ ✗ ✣ ✢ ✘ ✟ ✑ í òó ó ì ò ë ë ÿ ì ÷ ð õ ð ô ð þ ô ✞ ù ë û ú ù ì ø ÷ ð ô õ ô ð ÷ ÷ ✂ ✏ ✁ ò ï î í ì ✎ ð ☞ ô ☛ ò ö � � ð ✡ � ò ì ë ô � ð ð ì í ✁ ß ã ❰ ⑦ ❹ ➂ ❸ ❷ ➃ ⑨ ❹ ⑧ ⑥ ❻ ⑤ ➴ ➾ ➶ ➽ ➪ ➄ ❾ ➾ ❺ ❷ ➽ ➁ ❾❿ ➆ ➅ ❸ ❾ ➄ ❹ ➃ ➂ ➀ ❼ ❾ ➀ ➁ ❾ ❾❿ ❺ ❹ ❸ ❽ ➶ ❸ ❸ → ➆ ➌ ➍ ➎ ➇ ➆ ➏ ➒ ➑ ➔ ➆ ↕ ➙ ➑➛ ➜➝ ↕ ➠ ➙ ➑ ➢➤ ❾ ➋ ➪ ➲ ➅ ➆ ➇ ➼ ➺ ➻ ❸ ➫ ➧ ➵ ❸➊ ➈ ➧ ➯ ➭ ➫ ➩ ➦➧➨ ➥ ➉ ➓ ➇ ➈ Ñ ✃ ➷ ✃ ➱ ❰ ❒ Ñ Ñ➱ Ï Õ ❸ Ô ÑÔ Ï ➷❮ Ó Ò Ñ ❒ ✃ Ô × ➉ ➱ ⑦ ⑧ ⑨ Ü Ð Û × Ð ❮ ➷ ➷Ø ✃ Ú Ù ➷❮ × ➷ ❷ Ð ❐ ❰ ➱ ❹ ➎ ➐ ➐ ➑ ➹ ❼ ❽ ➆ ➇ ❸ ➒ ❹ ➍ ❺ ➌ ➆ ❾ ➆ ➋ ❸➊ Ï ➢ ➐ ➘ ❰ ➷❮ ➮ ➷ ❺ ✃ ✃ ➱ ➮ ❻ ❷ ➢ ➡ ➓ ➑ ➙ ➠ ➡ 36 Ö✐× ö✐ò 3 Stack n ❮ÞÝ ö ☎✄ Stack[0…n- ö✐ò ❐✵Ð Stack[n- n ñ✹òó ✝✍✌ ï✝ð ö ✠✟ ❐✵❒ ⑩✔❶ - ✙✛✚✜ ➴➬➷ ✖✍✗ 35 3 S->Length = S->Length + 1; S->Length = S->Length –1; S->Infos[S->Length-1] = x; if (S->Length == 0) return error if (S->Length == N) then error ➞✹➟ void Push(info x, pointer S) ➣✵↔ ➸✵➺ x = Top(S); info Pop(pointer S) return x; ➓✹➔ ➲P➳ ➐✵➑ else else ⑩✔❶ ➽➾✝➚ - Push ➽➾✝➚ Pop

  18. ✮ ❅ ❄ ❅ ❆ ✷ ❇ ❈ ✷❉ ❊ ❅ ✽ ❋ ✽ ● ❍ ❆ ❅ ❏ ◗ ✉ ❚✈ ✇ ❘ ✷ ❃ P ✶ ✮ ✯ ✰ ✱ ✲ ✶ ✷ ✸ ✹ ✺ ✻ ✸ ✼ ✷ ✸ ✯ ✽✾ ✽ ✿ ❀ ❁ ❂ ❬ ❲ qr ➎ ❷❹❺ ❻❼ ❿ ❷➀ ➁ ➂➃ ➇➈ ➉ ➊ ➋ ➏ ❷ ➎ ➁ ➂➃ ➄ ➅➆ ➇➈ ➉ ➊ ➋ ➎ ❸ ❶ ◗ ❬ ❲ ❏ ① ⑤ ◗ ❏ ✉ ✉ ❲ ① ◆ ⑩ ❚ ❲ ❪ ② ③ ⑥ ❢ ❛ ❦ ❤ ♦ ❛ ✹ ❦ ❅ ❅ ❆ ✷ ❇ ❈ ✷❉ ❊ ❅ ✽ ❋ ✷ ❜ ❍ ❆ ❅ ▲ P◗ P❘ ❙ ❱ ❚ ❄ ✽ P ✻ ✰ ✱ ✲ ✶ ✷ ✸ ✹ ✺ ✶ ✼ ❃ ✷ ✸ ✹ ✽✾ ✽ ✿ ❀ ❁ ❂ ✸ ❲ ● ❲ ❘ ❚ ❭ ❪ ❏ ❘ ❵❛ ❜ ❝❞ ❞❢ ❳ ❝ ❢ ✐ ♠ ♠ ❛ ♥ ♠♦ ♣ ❘ ❡ ▲ ❚ ▲ ▲ ❏ ❩ ❱ - 3 - 3 ✳✵✴ ✳✵✴ s❑t ②④③ ②④③ ■❑❏ ▼❖◆ ▲❯❚ ❳❖❨ n- ❬❯◆ ⑦⑨⑧ ❽✛❾ n! ❫✍❴ ❣❯❤ ❥❧❦ ➄⑨➅➆ ➌❯➍ integer factorial(integer n) if (n == 0) then return 1; else return (n * factorial(n-1)); ➌❯➍ integer factorial(integer n) integer j, product; j = n; product = 1; while (j > 0) product = j * product; j = j-1; return product; 37 38

  19. ➐ ➙ ➫ ↕ ➩ ➨ ➧ ↕ ➟ ➦ ➥ ↕➯ ➤ ➢ ➡ ➟ ➟➠ ➛ ➙ ↕ ➞ ➭ ➲ ↔ ÷ ý ➽ ➹ú ➶ ó ➻ ➽ ö ó ➨ ô ñ ➨ ➩ ➸ ➵ ➳ ➨ ➟ ➝ ➜ ➾ ❐ Ö Ñ ❰ Ø ❰ Ñ ❐ Ó Ø Ú Ú Ï Ù Ø Ú ❰ Ù Ý Ü Ò ➑ ➛ Ï ➙ ↕ ↔ ➔ ➓ ➒ ➑ ➐ Ò Ò Ó Ö Ý ❒ à ❰ × Þ Ü ❐ ➶ ò ð ✴ ✣ ✸ ✮ ✪ ✫ ✷ ✶ ✵ ✤✳ ✣ ✰ ✙ ✣✯ ✧ ✥✮ ✩ ✥ ✭ ✬ ✹ ✮ ✤ ✤ ✾ ✳ ✤ ✫ ✸ ✽ ✜ ✸ ✣ ✣ ✙ ✼ ✻ ✤ ✣ ✜ ✥ ✸ ✺ ✦ ✫ ➽ � í ì ✂ ì ✁ ç ➴ è ➷ ✞ è ➴ ÿ þ ➘ ➹ ➶ ➪ ➾ ✄ ✟ ✪ ✡ ✥ ✩ ★ ✧ ✥✦ ✣✤ ✗ ✔ ✓ ✒ ✠ ☛ ✒ ☛ ✑ ✎ ✏ ✌ ☛☞ ✡ Ô Ò ❒ ❰ ❰ ÒØ Ò× Ó Ô Õ ✃ Ó Ï Ð Ï ❰ ❒ ❮ ❒ ➱ ➮ Ù ❐Ú ➘ Ð Ï ➓ ➔ Ù ❐à Ï ß Ó ➛ Þ Ù Ý Ü Ô Õ ❰ Û ➴➷➬ ➹ Ú ➢ ↕ ➙ ➟ ➦ ➙ ➥ ➤ ➡ ➨ ➟ ➟➠ ➛ ➙ ↕ ➞ ➝ ↔ ➧ ➩ ➚ ➸ ➾ ➻➽ ➼ ↔ ↕ ➨ ➩ ➵ ↕ ➳ ➨ ➟ ➨ ➲ ↕➯ ➭ ➫ ❐ ➜ Û ➷ç é ì í ì ë ê ç ➒ æ ê ❰ Û ❰ ❐ Ï Ù Ø â ç î Ü ❰ Õ Ø ï Ð à ❐ Ò Ñ Û ➷ Ú ❐ Ï Ù Ô Õ ➮ ➬ ❒ Ú Ú Ö Ø â × à ❐ Ú Ù Ð á Ô à Õ ❐ Ñ Ò - 3 - 3 →✵➣ →✵➣ ➺❯➻ ➪❯➶ ò❯ó õ④ö ø❯ù û④ü ➚④þ Q: ✃④❐ Ñ✍Ò Ñ✍ÒÔ A=Q->Infos Ñ✍Ö Ñ✍Ò F=Q->Front pointer MakeEmptyQueue(void) n=Q->Length pointer Q; /* temporary pointer */ Ñ✍ã Öåä Q = NewCell(Queue); /* malloc() */ Q->Front = 0; Q->Length = 0; return Q; è❯é boolean IsEmptyQueue(pointer Q) return (Q->Length == 0); info Front(pointer Q) If IsEmptyQueue(Q) then error; else return (Q->Infos[Q->Front]); x 0 , …, x n-1 ✕✝✖ ☎✝✆ ✍✝✎ A[F], A[(F+1) mod N], A[(F+2) mod N] , … , ✛✢✜ ✱✢✲ ✘✚✙ A[(F+n-1) mod N ✘✚✙ x 0 , …, x n-1 . 39 40

  20. ✿ ➃ ❶ ⑩ ⑨ ⑧ ⑦ ⑥ ③ ③ ❷ ➁ ⑥ ④ ⑤ ❻ ⑤ ❸ ⑧ ⑤ ⑧ ⑤ ❷ ⑧ ❶ ⑨⑩ ⑧ ❷ ⑥ ➂➅ ❻ ❷ ⑧ ❶ ❷ ❾ ➄ ❸ ❷ ➁➂ ⑤ ➆ ✇ ❞ s ❦ ❀ r ♦ ② ① ❞ ❴ ❭ ❢ r ♦ ❞ ❢ ❜ s t ➀ ⑤ ❿ ❸ ❷ ❷❾ ❷❽ ❹ ❸ ❶ ❵ ⑨⑩ ⑧ ③ ✇ ❢ ❜ ❴ ❝ ⑥ ➁ ❝ ⑩ ❶ ⑤ ➇ ➄ ⑧ ⑦ ⑤ ❷ ⑩ ❾ ➟ ➈ ➞ ➣ ➙➛ ➍↕ ⑨ ⑦ ➣ ➠ ➢ ❺ ➡ ③ ⑩ ➁ ⑥ ➈ ❿ ❽ ❷ ⑧ ➂ ⑤ ➇ ⑤ ➊ ↔ ➑ ⑩ ➈ ⑨ ➊ ❷ ❻ ➁ ➁ ⑥ ❷ ⑤ ➉ ❷ ➈ ⑨ ➇ ❻ ③ ➃ ⑦ ❽ ➒ ➂ ➍ ➏ ➋ ❷ ⑥ ⑤ ➅ ❷ ❸ ⑧ ❶ ❷ ⑧ ❶ ⑨ ❾ ➄ ❴ ❣ ♦ ❵ ❡ ✐ ❣❤ ❭ ❢ ❡ ❜ ❣♠ ❭ ❫ ❩ ◗ ❘ ❨ ❧ ❬ ❲ ❀ ❉ ❈ ❇ ❃ ❂ ❁ ✿ ❡ ✉ ♦ q ❤ ❣ ♣ ♠ ❳ ◗ ❋ ❋ ❊ ❉ ❈ ❍ ● ❇ ❊ ■ ❉ ❈ ❇ ❃ ❂ ❁ ■❏ ❑ ■ ❘ ◗ ❱ ❈❯ ❚ ❙ ❈ ◗ ▲ P ❈ ■ ❖ ❉ ◆ ▼ ❊ ➤ ❇ ❚ ❳ ❲ ◗ ■ ◗ ● ❈❯ ❙ ❘ ❈ ❘ ◗ P ❈ ■ ❖ ❨ ◗ ◆ ♦ ♠ ❡ t t ✇ ❤ ✈ ❞ r t ❞ s ❞ s ❦ ❣ ❉ ❱ ❍ ❉ ▲ ▼ ❑ ■ ❊ ■❏ ❈ - 3 - 3 ❄❆❅ ❄❆❅ ❬❪❭ ❴❛❵ ❝❪❞ ❥❛❦ ♥❛♦ ✉❛❦ ✉❛❦ (Q->Front + Q->Length –1) mod N S Node) ④❛⑤ ⑥✢⑦ ④❛❷ ❺❼❻ next ④❛⑤ data ➌➎➍ ➐➎➑ ➓→➔ ➜➎➝ info Dequeue(pointer L) pointer MakeEmptyStack() if IsEmptyQueue(Q) then error; return NULL; else x = Q->Infos[Q->Front]; boolean IsEmptyStack(pointer S) Q->Front = (Q->Front+1) mod N; return (S == NULL); Q->Length = Q->Length –1; return x; info Top(pointer S) if IsEmptyStack(S) then error; procedure Enqueue(info x, pointer Q) else return S->data; if (Q->Length == N) then error; else Q->Length = Q->Length+1 Q->Infos[(Q->Front + Q->Length –1) mod N] = x 41 42

  21. ➥ ➷ ➲ ➬ ➮ ➲➱ ✃ ➴ ➼ ➴ ❐ ❒ ❮ ➴ ➴ ❰ Ñ Ð Ô Ó Õ Ö × Ø Ð ÙÚ Û ➷ ➘ Ü ➺ ➥ ➦ ➧ ➨ ➩ ➯ ➲ ➳ ➵ ➸ ➯ ➦ ➲ ➲ ➳ ➵ ➼➽ ➼ ➾ ➚ ➪ ➶ ➳ ➹ ➼ × Ý ö ü ✡ ☛ ÿ ✁ ☞ ÿ ✌ û ú ✏ � � ✟ ✂ ☛ ✠ ✝ ù ý ✔ ✕ ✙ ✚✛✜ ✢ ✥ ✦ ✠ ý ✂✞ Þ Þ Ù Ý ß Ö ß Ö à á Ý Ó â Ø ✁ Ð Ò ã ä â úû ü ý � ✁ ý ✂ ✝ ÷ ➻ õ ❰ ➮ ➲➱ ✃ ➴ ➼ ➴ ❐ ❒ ❮ ➷ ➴ ô ➲ Ð Ô Ó × Ø Ð ÙÚ Û × Þ Ù ➬ ➷ ß ➲ ➧ ➨ ➩ ➯ ➲ ➳ ➵ ➸ ➯ ➺ ➻ ➳ ➴ ➵ ➼➽ ➼ ➾ ➚ ➪ ➶ ➳ ➹ ➼ ➲ ➘ Ý Ñ â ñ ò î ê êï íî ä ã ï ê ó Ò é Ð Ø Þ Ö å æ çè ß ð à Ö Ó â - 3 - 3 ➫❆➭ ➫❆➭ Ï❪Ð Ò❛Ó Õ❪Ö Ü❛Ý Ï❪Ð Ò❛Ó á❛Ý P->next S->next S=S->Next info Pop(pointer S) void Push(info x, pointer S) if (IsEmptyStack(S)) then error; else pointer P; /* temporary pointer */ P = NewCell(Node); /* malloc() */ x = Top(S); S = S->Next; P->data = x; P->next = S; return x; S = P; ø❪ù þ❪ÿ ✄✆☎ Extra ✠✎✍ ✑✓✒ : n n ✖✘✗ ✣✓✤ ë✝ì 43 44

  22. ✧ ❛ ♥ r ❞ ❥ ❜ ❭ ❥ ❦ t ❭ ❛ ❞ ♥ ④ ❦ t ❤ ❥ ⑨ ❺ ❷ ⑨ ❶ ⑩ ❶ ③⑩ ⑦ ❛ ③⑧ ⑥⑦ ⑤ ❢ ❣❤ ♠ ❜ ✇ ❣ ❼ ❣ ❝ t ② ❦ ❥ ❣ ❜ ❬ ❭ ❦ ❦① ❭ t ✇ ❥ ❛ ♦ ♠ ❤ t ❞ ③ ❦ ❜ ❛ ❧ ❦ ✉ ❞ ♥ ❦ ❞ ★ t ♠ ❾ ➁ s ✿ ❖ ◗ ➐ ❳ ➏ ✾ ❆ ❖ ❅ ❄ ✾ ✶ ✾ ❃ ✰❂ ❘ P ❀ ❇ ❋ ❨ ❳ ❲ ❱ ❊ ❈ ❏ ▲ ➓ ➑ ❊➒ ❈ ➐ ❚ ➑ ❁ ✰ ➀ ✧ ✱ ✰ ✯ ✫ ✪ ✩ ★ ➈ ✳ ➇ ➌ ➅ ➈➉ ❼➇ ➆ ➅ ✲ ✯ ✿ ✺ ✾ ✽ ✰ ✶ ✼ ✱ ✻ ✹ ✴ ✸ ✶ ✶✷ ✲ ✱ ✰ ✵ ❫ ♥ ✻ ❀ ✶ ❩ ❬ ❭ ✰ ✾ ❴ ✯ ❃ ✰❂ ❁ ❞ ✫ ✱ ❛❡ ❢ ✰ ❝ ❣❤ ✿ ✾ ❦ ❬ ✽ ❦ ❬ ♠ ✾ ❄ ❛ ❯❱ ❊ ❋ ●❍ ■ ❈ ❏ ▼ ◆ ❏❖ P ✾ ✿ ❚ ❆ ❋ ❨ ❅ ✶ ✶✷ ✲ ✱ ✰ ✵ ✴ ❡ ✯ ✳ ✲ ♥ ✪ ✩ ❦ ❡ ♥ ❞ ❦ ❛ ❥ ❦ ♦ ✉ ♠ ❦ ♥ ❞ ❤ ❞ ❧ ❛ ❜ ❦ ❞ t ❡ ♥ ❞ ❦ ❜ ✈ ❣ ✺ ❵ t ✹ ❴ ❦ ♦ ♣ ❢ ♦ ❛ q ♠ ❦ ♦ ✰ ✐ ❣ ❥ ✼ ❛ ❵ ✸ ❫ s ✱ ❫ ♠ ✶ ❵ ❛ - 3 - 3 ✬✮✭ ✬✮✭ ◗❙❘ ❇❉❈ ❑❉▲ ❑❉▲ ❲✎❳ ❨❉➐ ❋✘❍ Back->next Front Back ❪✓❫ ❵✘❛ ❜✓❝ ✐❙❥ ❵✘❛ ❜✓❧♠ Node ❜✓r next ❜✓❝ data Front, Back global ✐❹❸ ❻❽❼ ❿❽➀ ➂➄➃ ➊❽➋ ➍❽➎ void MakeEmptyQueue(void) Front = Back = NULL; boolean IsEmptyQueue(void) void Enqueue(info x) return (Front == NULL); pointer P; /* temporary pointer */ P = NewCell(Node); info Front(void) P->data = x; P->next = NULL; if (IsEmptyQueue()) then error; if (IsEmptyQueue()) then Front = P; else return (Front->data); else Back->next = P; Back = P; 45 46

  23. ➔ í ã å ä ã â ê ÷ ê ã è ê â ìô è ó ê Þ è à ê ê â á ù á è â ç ã ï î í ì ë à ä ò ð ð Ý ã å → á à ß Þ Ù ãè ➚ ➪ Ü ➴ Û Ö ➪ Þ ã Þ ë ã ç î ìí ð è ñ íð ä ê â ï î ìí ë à õ ë Ú ó ë ÷ ã ë ÷ ú è ä í Þ ê ð ú ò ã ç ã ö ì ☛ ✌ ☞ ✑ ✡ ✍ ☞✌ ✁ ✡ ê ✆ ✝ ✁ ✄ ÿ ã õ ì õ ú ê ö í ã ë ÷ ã è ä õ û ç ê ã ë ÷ ô ê ã ß ù þã ã ß ú ý ç á ô ë ì ü õ ã ë ÷ ô ú ➮ ä Ò ➣ Ù ➤ ➫ ➞ ➩ ➨ ➧ ➦ ➤ ➤➥ ➟ ➔ → ↔ Ð ➞ ↕ ➜ ➝ ➝ ➞ ➟ ➠ ➜ ➳ ➢ ➢ ➝ ➞ ➝ ➚ ➤➥ ➭ ➝➸ ➺ ➯ ➤ ➯ ➻ ➼ ➽ ➲ ➯ ➝ ➲ ➯ ➾ ➶ ➶ ➪ ➴ ➘ ➮ ➱ ➪ ✃❐ ❒ ➮ Ï ✃ ❐ Ó ➟ ➡ ➤ ✃ ➲ ➟ ➯ ➞ ➝ ➜ Ï ✃❐ Ð Ô ↕ Ñ Õ ↔ ➼ ❰ Ö ➬ Ö ➣ ➬ × Ø ❰ ➘ ➷ ➹ ➴ ➱ ➽ ➵ ➻ ➞ ➝ ➭ ➤ ➯ ➯ ➫ ➲ ➝ ➳ ➜ ➵ ➩ ➝➸ ➨ ➧ ➠ ➡ ➦ ➺ ➯ ➤ - 3 - 3 ➙✮➛ ➙✮➛ ➚❉➪ ➹✘➘ ➷❉➬ ❮✘❰ Ð✘ÑÒ Front->next Q Q Queue â✘ã æ❙ç é✘ê (Q->Front Q->Back õ✓ö Node æ❙ç é✘ê õ✓ø next data �✂✁ ☎✂✆ ✞✠✟ ✎✂✏ ✒✂✓ pointer MakeEmptyQueue(void) pointer Q; /* temporary pointer */ Q = NewCell(Queue); /* malloc() */ Q->Front = Q->Back = NULL; info Dequeue(void) return Q; if (IsEmptyQueue()) then error; boolean IsEmptyQueue(pointer Q) else return (Q->Front == NULL); x = Front->data; Front = Front->next; info Front(pointer Q) if (Front == NULL) then Back = NULL; if (IsEmptyQueue(Q)) then error; return x; else return (Q->Front->data); 47 48

  24. ✔ ✵ ✢ ✯ ✰ ✱ ✢ ✲ ✳ ✢✴ ✰ ✮ ★ ✰ ✶ ✷ ✸ ✱ ✰ ✹ ★ ✣ ✻ ✜ ✖ ✗ ✘ ✜ ✢ ✣ ✤ ✥ ✦ ✭ ✧ ✕ ✣ ✤ ★✩ ★ ✪ ✫ ✬ ✽ ❁ ✔ ❫ P ❙ ◗ P❚ ❲ ❱ ❳❨ ❩ ❴ ✻ ❴❵ ❴ ❡ ❭❪ ❢❣ ❤ ❛ ✐ ❖ ▲ ✿ ❊ ❄ ❅ ✻ ❆ ❇ ❈ ❄ ❉ ❋ ❁ ❆ ❇ ● ❍ ■ ❄ ✻ ◆ ❑ ✕ ✢ ★ ✧ ✰ ✶ ✷ ✸ ✱ ✰ ✢ ✦ ✰ ✜ ✥ ✤ ✹ ✽ ❁ ✿ ✣ ✵ ❅ ✮ ★✩ ✤ ✪ ✫ ✬ ✭ ✣ ★ ✢✴ ✢ ✯ ✰ ✱ ✢ ✲ ✳ ❄ ✻ ✻ ❄ ✖ ✗ ✘ ✜ ✢ ✣ ❆❇ ▲ ❁ ❑ ❏ ✻ ✻ ■ ❆❇ ❈ ❄ ❋ ★ - 3 - 3 ✙✛✚ ✙✛✚ ✺✼✻ ✾❀✿ ❂✼❃ ❉❀❊ ●❀❍ ✺✼✻ ✾❀✿ ❂✼❃ ✺✼▼ ✺✼▼ Q->Back->Next Q->Front- >Next Q Q Q Q Q Q info Dequeue(pointer Q) if (IsEmptyQueue(Q)) then error; void Enqueue(info x, pointer Q) else pointer P; /* temporary pointer */ x = Q->Front->data; P = NewCell(Node); Q->Front = Q->Front->next; P->data = x; if (Q->Front == NULL) then P->next = NULL; Q->Back = NULL; if (IsEmptyQueue(Q)) then Q->Front = P; return x; else Q->Back->next = P; Q->Back = P; ◗✂❘ ❯✼❱ ❬❀❭❪ ❛❝❜❞ ❴❥❧❦ 49 50

  25. ♠ ❹ ⑨ ✈ ⑩ ④ ✉ ❶ ❷ ❸ ✉ ❺ ⑦ ✉❻ ❼ ❷ ④ ❷ ❽ ❾ ❿ ❸ ⑧ ⑥ ò ✉ í å ñ ♠ ♥ ♦ ♣ q t ✈ ④ ✇ ① t ② ③ ✉ ✈ ✇ ④⑤ ❷ ó çð ✷ ✫ ✬ ✭ ✮ ✯✰✱ ✲ ✳ ✴ ✵✶ ✸ ★✩ ✷ ✸ ✼✽ ✾ ✻ ✿❀ ❁❂❃ ❄ ❈ ✪ ✦✧ ô ✁ õ ö ÷ ø ù ú û ü ÿ � ✂✄ ✤ ✡ ✍ ✑ ✏ ✖ ✙ ✚✛ ✜ ✢ ✣ ï îï ❊ ➘ ➽ Ñ ➭ ➻ ➮ ❰ ➾ Ò ♥ ➶ ➻ ➘ ➵ ➯➸ ➪ ➾ ➹ Ð Ñ Ò ➸ ➲ Ú ➺ ➲ ➸ ➲ ➻ ➳ ➵ ➯➸ ➺➻ ➹ ➽ ➸ ➺ ➾ ➺ Ï ➭ ➸ ➲ ➯ ➲ ➪ ➫ å â ➫ ➘ ➸ ➽ ➹ Õ ➽ ➮ á ã Ú äå ê ë à â ç ì í ã ➪ ➺ ➪ ➽ ➴ ➫ Ð ➼ ➵ ➾ ➘ ➸ ➺ ➲ ➽ ➘ ➽ ➯ ➺ ➼ ➷ ➼ ➭ ➮ Þ ❉ ❋ ➽ ➧ ➦ ➢ ➩ ➵ ➢ ➫ ➳ ➤ ➩ ➨ ➤ ➻ ➼ ➽ ➼ ➪ ➶➹ ➷ ➽ ➼ ➠ ➳ ➹➬ ➤➥ ➐ ➎ ➒ ➜ ➛ ➒ ➝ ➣ ➢ ➦ ➲ ➥ ➢ ➩ ➫ ➠➭ ➯ ➲ ➦➳ ➯ ➶ ➮➱ ➒ ❰ ❐ ❒ ➱ ➬ ❮ ➶ ❰ ➱ ➶ ➹ ➮ ➚ ➬ ➼ Ï ➹ ➚ ❒ Ñ ➴ ➘ ➶➹➬ ❒ Ð ➱ ➬ ❮ ➶ ❰ ➱Ï ❒ ➱ Ï ❰ ➼ ➹ ➚ ➮ ➚ ❐ ➶ ➷ ➽ ➷ ➽ ↕ ➑ ●❍ q ❭ ❪ ❞ ❣ ❤ ✐ ❤ ✐ ♠♥ r ❜ ♠st ✉ r q r✈ ✇ ✉ ❧ ① ❛❝❞ ❫ ⑦ ❳ ■ ▲ ▼ ◆ ❖ P ❙❚ ❯❱❲ ❘ ❯ ❪ ❚ ❨❩ ❬ ❫ ❴ ❩ ❴❵ ❫ ❛ ♥ ⑧ ➛ ➓ ➅➈ ➉ ➊ ➋ ➎ ➏ ➐ ➑➒ ➍ ➍ ➅ ➓ ➒ ➋➣ ↔ ➒ ↕ ➐ ➎➋➙ → ➇ ➃ ⑨ ❷ ⑩ ⑧ ❸ ❹❺ ⑧❽ ⑦❾❿ ❹ ➀ ⑦➁ ⑨ ❷➂ ❷ ❾ ❽ ❹ ➀ ❿ ⑤ ❻ ❺ ⑦ ➯ ➫ Ö Ò ➺ Ï ➭ ➸ ➲ ➯ ➲ ➹ ➪ ❰ ➴ Ñ ➘ ➯ ➴ ➼ ➹ ➘ ➹ ➾ ❒ ➽ ➬ ➾ ➯ ➴ ➾ ➸ ➘ ➳ ➘ ➯➸ ➱ ✃ ✃ ➨ ❐ ➡❒ ❮ ➛ ➡ ➤ ➛ ➫ Ñ ➸ ➯ ➯➸ ➲ ➳ ➵ ➯ ➸ ➺ ➘ ➽ Õ ➘ ➼ ➘ ➸ ➺➽ ➮ × ➽ ➯ Õ ➽ Ö Ó ➪ ➾ ➲ ➸ ➽ Ñ ➭ ➮ ❰ ➾ ➹ ➫ Ò ➪ ➻ Ô ➘ ➾ ➴ ➯ Õ ➷ ➲ ➘ ❷ ❶ ❷ ❸ ✉ ❹ ❺ ✉❻ ❼ ④ ④ ❷ ❽ ❾ ❿ ❸ ❷ ➂ ➃ ➄ ✉ ⑩ ➆ ② ♦ ♣ q t ✉ ✈ ✇ ① t ③ ✈ ✉ ✈ ✇ ④⑤ ④ ⑥ ⑦ ⑧ ⑨ ➅ ➇ ➯ ➯➸ ➡ ➥➦ ➥ ➦ ➩ ➫ ➲ ➳ ➵ ➺➻ ➟ ➼➽ ➺ ➸ ➲➾ ➺ ➚ ➪ ➶ ➴ ➠ ➝ ➈ ➄ ➁ ➁ ➂ ➈ ➋ ➇ ➐ ➄ ➅ ➅ ➙ ➈ ➑ ➐➒ ➐ ➣ ↔ ➐ ↕ ➙➛ ➼ ➚ ➘ Û ➭ Ñ ➽ ➻ ➮ ➸ ➲ ➯ ➫ ➲ ➸ ➭ ➾ ➺➻ Ï ➺ ➺ Ö ➺ ➻ ➪ Ð ➚ Ñ ➺ Ò ➪ Ú Ð ➫ ➪ ➴ ➫ Õ ➹ ➽ ➾ Ô ➹ ➾ ➪ Ò Ñ ➽ Ð Ñ ➹ ➺ Ð ➯ ➘ ➪ ➺ ➴ ➸ ➾ ➾ ➮ ➘ ➘ ➵ ➹ ➸ ➻ ➲ ➻ Ô ➵ Ñ Ö ➪ ➾ ➫ ➚ Ü ➽ ➶ ➺➽ ➸ ➴ ➹ ➸ Ð Ñ Ò ➪ ➻ ➺ Ø ➺ ➸ ➘ ➳ ➘ ➴ ➻ Ð ➸ ➸ Õ ➽ ➸ ➺➽ Ô Ó ➺➽ ➵ ➮ ➘ ➳ ➘ ➸ ➹ ➾ × ➪ Ò ➹ ➪ ➵ ➯➸ Ý ➺ Ù ➚ ➴ ➯➸ ➳ Ó ➼ ➘ ➘ ➹ ➲ ➳ ➲ ➺ ➪ ➸ ➺ ➵ ➾ ➪ ➻ ➯➸ ➺➽ ➸ ➚ Õ ➘ ➾ ➸ Ð 52 ❡❇❢ ❅❇❆ ➸✝➺ ➔✝→ 3 ÒÔÓ - ☎✝✆ ✗✝✘ ①③② ✒✔✓✕ ✹✝✺✻ ý✂þ ✃✌❐ ➝③➞ - ❭✌❪ ➌✔➍ ✎✝✏ ❻❇❼ ☛✌☞ ☛✌☞ ♦✌♣ ➘❇➴ ✞✠✟ ➧❇➨ ❶✔❷ ❥✝❦❧ 2 - ✣✠✥ ❏✌❑ ➄✔➅➆ ➾✔➚ ◗✝❘ - ④⑥⑤ ➟➡➠ r✛s o o o o - 51 3 ➌❀➍ ➭❝➹➘ ➓➔➆→ a n x n + a n-1 x n-1 + … + a 1 x + a 0 ➉✂➊ Ð❝Ö æèç ➧✼➨ ➢✂➤ Ð✼Ñ ➎✂➏ ➀✂➁ ➾❧➮ æèçé ➜✼➝➞ r✛s - ➭✼➯ Ð✼Ñ Ð✼Ñ ß❝à

  26. Õ ➳ ➘ ➹ ➶ ➚➪ ➾ ➽ ➸ ➟➵ ➡ ➥ ➭ ➜ ➝ ➢ ➞ ➷➬ ➝ ➞ ➩ ➢ ➤ ➲ ➟ ➡ ➠ ➥ ➩➯ ➭ ➠ ➫ ➴ ➮ ➩ í þ úýþ ù õ ü ö û ú õù ö õ ó ðò ðñ ç ➱✃ Ö ❐ ✃ ❰ ÑÒ ÓÔ Õ × æ ÓØ Ù ÝÞ ß à ãäå ➤ ➡ ✂ ➈ ➌ ➋ ➀ ➈ ➇ ❽ ❾ ➊ ➇ ❾ ➈ ➉ ➆ ❾ ➆➇ ➍ ➅ ➄ ➃ ➂ ❿➀ ❷ ❻ ❻❼ ⑧ ⑨ ❺ ❹ ❷❸ ⑧ Ö ➈ ➥ ↕ ➨ ➦ ➧ ➟➦ ➤➥ ➠ ➢ ➝ ➡ ➠ ➞ ➟ ➞ ➙➛ ↔ ➒ → ➓ ➀➔ ➆➇ ➅ ❾ →➣ ➄ ➀➔ ❾ ❿ ➍ ➓ ❾ ➇ �✁ ✄☎ ❶ ❨❩ ❖ P ❖ ❭ ❵ ❯ ◆ ❳ ❩ ❪ ❖ ❭ ❬ ❯ ❳ ❴ ◆ ❲ ◆ ❖P ◆ ❍ ❉ ✷ ✶✼ ❀ ❈ ❆ ❄ ❇ ❨ ❜ ❆ ❯ ❯ ❪ ❖ ❛ ❨ ❯ ❳ ▼ P ❯ ❛ ❢ ❏ ❳ ❳ P ❏ ❳ ❝ ▼ ❳ ❯ ❴ ❯ ◗ ◆❞ ❯ ❖P ❯ ❝ ❴ ❅ ❅ ✆✝ ✖ ✢ ★ ✘ ✧ ✦ ✥ ✤ ✢ ✢✣ ✙ ✘ ✗ ✜ ✛ ✚ ✩ ✙ ✘ ✗ ✖ ✒ ✑ ✏ ✎ ✍ ✌ ☞ ☎ ✡☛ ☎ ✗ ✪ ✽ ✸ ❄ ✴ ❃ ✷ ❂ ✿ ✶ ✽❁ ❀ ✿ ✾ ✴ ✽ ✺ ✹ ✫ ✢ ✗ ✬ ✭ ✗✮ ✯ ✪ ✪ ✷✸ ✰ ✱ ✲ ✫ ✪ ✶ ⑨ ➋ ⑩ ❀ ✺ ✮ ✲ ✾ ✺ ✽ ✿ ✮ ✺ ❄ ✺ ❂ ✾ ✺ ✼ ✾ ✽ ✼ ✻ ✴ ✺ ✵ ✷ ✴ ✶ ✵ ✮ ✾ ❅ ✩ ❆ ▲ ❉ P◗ ❉▼ ❇❖ ● ❇ ❉▼ ❏ ●❍■ ❉ ❈ ❇ ✲❆ ✼ ✱ ✮ ✲ ✼❄ ✻ ✼ ❄ ✲ ✵ ✹ ✶ ✮ ✸ ✪ ★ ❘ èé ö õ ô ó ò ïðñ î í ì ë ê è ç õø æ å ä ã â á à ß Þ Ý Ù Ø × ÷ ù ✧ ✍ ✦ ✢ ✛ ✘ ✜ ✚ ✛ ✚ ✘✙ ✍ ✑ ☛✏ ✎ ✌ úû ☞ ☛ ✟ ✝ ✞ ✆✝ ☎ � ÿ þ ý ü ú ● ✵ ❏ ❚ P◗ ▼ ❡ ❋ ◗ ❛ ● ❯ ❢❣ ❍ ❆ ❈ ❤ ✐ ❥ ♥ ▼ ❇ P◗ ❯ ❱ ▼ ❉ ◆ ❋ ❨ ◗ ❋ ▼ ❱ ❯ P◗ ❇ ▼ ❯ ❍ ❖❳ ❲ ❩ ❬❭ ❴ ❵ ❛ ❉ ◗ P ❉ ❜❝❞ ◗ ❙ ♦♣ ◗ ● ♥ ❉ ❉▲ P◗ ▼ ❋ ④ ⑤ ◗ ③ ✈ ♥ ❯ ❍ ✇ ❚ ♦ ⑥ ❆ ❉ ❍ ✇ ❈ ▲ ❙ q ❏ ♣ ✈ ❘ ● ✉ ❉ ♥ ❉ ❍ ✇ ✉ ❧① ② ❏ ③ ❘ 54 ❣✐❤ ❪❙❛ 3 /* temporary pointers */ while (C != NULL) and (C->Num > K) do if (C != NULL) and (C->Num == K) then /* malloc */ ❫●❴ P void LLInsert(integer K, pointer L) if (P == NULL) then L = ptr ; ✻✵✼ ptr = NewCell(Node); else P->next = ptr; C = C->next; ptr ❬✵❡ pointer C, ptr; ptr->num = K; ptr->next = C; ❚❱❯ return; P = NULL; P = C; ✳✵✴ ◗❙❘ C = L; ✓✕✔ - ▲✵▼ ❊●❋ ■❑❏ 53 ✿❁❄ ➌➏➎ Ú✳ÛÜ 3 ✻❃❂ ✯✄✰ ⑦✄⑧⑨ ✿❁❀ ÷✳ø ✣✥✤ ✯✄✰ Ï✳Ð ó✗ô ✠✄✡ ❑✔◆ ❒✳❮ ✸✄✹ î✗ï ❑❁▲ ✕✗✖ ➁✳➀ ê✗ëì ✒✔✓ ✞✠✟ ✱✳✲✴ ✁✄✂ ✿❁❄ t❁✉ è✗é ✯✄✰ ❊✭❋ ❽✭❾ r❁s ✫✭✬ ❪✔❫ ➜✔➝ ÚÜÛ - num next á✳â ➺✳➻➼ L ❦♠❧ û✥ÿ ➐♠➑

  27. ❥ ❳ ❘ ❙ ❚ ❯ ❙ ❲ ❨ P ◆❩ ❙ ❲ ❘ ◗ ❳ ◗ ▲ ❬ ❅ ✻ ❀ ✻ ❂ ❃ ❄ ❆ ❑ ✻ ✽ ❈ ❀ ❊ ❋ ❏ ❨ ❩ ❂❃ ❢ ❫ ❚ ❙ ❜❝ ❞ ❡ ❣❤ ❲ ✐ ❥ ❣ ♠ ♥ ♦ ❲ ❚ ❭ ❴ ❩ ❪ ❲ ❫ ❨ ◆ ◗ ❙ ❘ ❙ ❚ ❪ ❳ ❨ ❵ ❄ ❁ q û � î ✂ ó þ î ó ú þ ✂ ✕ ✖ ✗ ✘ ò � ✢ þ ✎ ✏ ✑ ✒ ✓ ☛ � þ ✝ ó ü ô ÿ ❦ ✔ ✙ ✣ ✿❀ ✶ ✱ ✲ ✣ ✳ ✴ ✣✵ ✱ ✣ ✩ ✱ ✷ ✸ ✹ ✲ ✱ ✰ ✩ ✤ ✤ ✥ ✦ ✢ ✧ ★ ✣ ✥ ✯ ✩✪ ✩ ✫ ✬ ✭ ✮ ✤ ♣ t ✌ t s ✉ ③ ⑨ ⑤ ✇ ⑦ ❹ ❻ ✉ ③ ✈ ⑤ ➁ ⑩ ➀ q ② ❾ q ⑩ t ❻ ⑦ t ⑩ ✇ ❸ ✇ s ⑥ ❹ ❸ ❻ ➀ ⑨ ⑦ ✉ ⑨ ❽ ✇ ⑦ ♣ ❹ ♣ ④ ① ⑦ ❹ ⑩ ⑦ q s ③ ❿ ❹ ① t ➂ ③ ⑨ ❻ ✇ ⑦ ✇ s ① ✉ ③ ④ ⑩ ③ ✉ ❷ ⑨ ⑤ ✇ t ✉ ❶ ⑩ ✉ ✇ ⑨ ✉ t ③ ❸ ③ ⑤ ❻ t ✈ ✇ s ① ② ③ ✇ ⑦ ④ ① ✇ t⑤ ⑧ ② ✇ ❺ t ✉ ⑤ ✇ ⑦ ⑤ ✉ ③ ⑨ ✇ ⑧ t ✉ ❶ ❷ ✉ ⑨ ❽ ② ⑤ ✇ ⑨ s ① ❼ ② ⑨ ♣ ❽ ❽ ❾ ⑨ ❿ ❻ ⑦ ✇ ⑨ ✍ ô ➶ ➢ ➪ ➚ ⑧ ➼ ➲ ⑨ ➸ ➵ ➲➳ ➯ ⑩ r ❶ ❷ ➡ ➸ ➠ ➟ ➢ ➠ ➝ ➛ ➤ ➢ ↔ ➡ ➣ ➥ r❸ ➔ ➣ ➶ ➹ ➠➡ ÏÐ à ß ÙÞ ⑨ × ⑤ ØÙÚ × Ö ÔÕ ÐÓ Ò ❒ Ñ ⑥ r ⑦ ➲ ➵ ➘ ① ➲ ➘ ➬ s ➼ ➴ ➮ ➲ ➘➱ ✃ ➛ ➟ à ⑨ ➜ ➛ ① ↔ ➣ → ➓➔ ➉➍ ➏ ➒ ➑ ➍ ➐ ➏ ➋ ↔ ➄ ❼ ➁ ❾ ➀ ➂ ➃ ➀ ➊ ➅ ➆➇ ❻ ❺ ➈ ➉ ➣ ➝ ➞ ➨ ➛ ➥ ➡ ➛ → ➩ ➣ ↔ ➦ ➛ → ➥ ↔ ➝ ➨ ➠➡➧ ⑨ ➛ ➠➡ ➛ ➟ ➢ ➠ ➝ ➤ ➣➦ ➠ ➢ ➡ ➣➥ ❹ ➛ á Ý â þ q r ✆ s õ ô ü ✝ ✆ ✂ ü ô ù t ✂ þ þ ✂ ✉ î ô ✁ ð ú � Þ ð � þ ü ÿ þ ü ö ✁ ô õ ✠ � ✟ þ ñ ❧ ♠ ♥ ñ ✝ ú ð î ✂ ï û ð ü î ñ ✂ î ó ✞ ✂ î ö û ý ô ôõ ü î q ü î ✈ ø ✇ õ ô ÷ ò ö r ó ñ s t ì ë ①② è ç æ å ① ③ ④ ô ⑩ ü ò þ îý û ö þ ö ü ÷ ÿ î ô crabapple, 56 ⑧❖⑩ 3 ⑧❖⑩ ❯☎❛ ✽❉❈ e. ❅❇❆ cup). ⑥☎⑦ r☎s ▲❖❱ ⑥☎⑦ ✽✼✾ ✺✼✻ ▼❖◆ ❦❖❧ ⑥☎⑦❹ r☎s ●❉❍■ ✚✜✛ - 1) 2) w L = <canary, cat, chickadee, coelacanth, collie, 55 w. ➞✠➟ 3 collie. strings áäã ➞➭➫ ö ☎✄ ↕●➙ ➴●➷ û✠ü û✠ü Û✠Ü ↕●➙ ù✠ú w ❿●➀ ❽❱❾ w = crabapple ↕●➙ ➽✠➾ corn, cup>. ➺✠➻ ò●ó é✵ê ➌➎➍ ♦✕♣ - ❮●❰ ï➎ð ❐❱❒ ✡☞☛ H í❱î

  28. ➃ ✐ ❩ ❭ ♠ ❤ ❣ ♥ ❩ ❡ ❛ ❜ ❝ ❛ ❝ ❱ ❙ ❳ ❪ ❢ ❛ ❢ ❪ ❜ ❳ ❞ ❦ ❝ ❝ ❜ ❧ ❛ ❝ ❳❭ ❤ ❛ ❚ ❪ ❱ ❱ ❞ ❛ ❭ ❫ ❞ ❝ ❙ ♠ ❜ ❞ ❝ t ② ⑤⑥ ⑦ ❚ q ♦ ❜ ❢ ❩ ❪ ❱ ♣ ❛ ❝ ❝ ❳ ❭ ❝ ❱ ❫ ❨ ❳ ❪ ❫ ❳ ❘ ● ◆ ❆ ❖ P ❑◗ ❙ ▲ ❚ ❲ ❳❨ ❳❭❪ ❫ ❲ ❴ ▼ ❆ ❨ ❀ ✺ ✻ ✼ ✵ ✽ ✾✿ ❁ ➄ ✿ ❂❃ ❄ ❃ ❆ ❂ ✿ ❜ ❝ ❳ ❚ ❤ ❚ ❡ ❱ ❫ ❙ ❫ ❵ ❝ ❱ ❫ ❙ ❜ ❞ ❝ ❪ ❛ ❱ ❢ ❫ ❞ ❝ ❪ ❡ ❛ ❛ ❥ ❡ ❪ ❨ ❣ ❤ ✐ ❪ ❡ ❧ ✸ ❺ ❿ ➈➉ ➄ ➊ ➇ ➋ ➌ ➄ ❽ ➋ ➆ ➊ ➂ ➊ ➃ ➇ ❽➂➃ ➂ ❺ ❜ ❞ ❝ ❷ ❸ ❹ ❻ ➁ ❼ ❽ ❾ ❿ ➀ ❺ ❻ ❿ ➍ ❱ ➨ ➞ ➟ ➠ ➡ ➢ ➧ ➩ ➛ ➫ ➭ ➧ ➨ ➭ ➥ ➯ ➜➝ ➣ ➀ ❻ ❺ ❽ ❼ ➆ ➀ ❼ ➎ → ➎ ➎ ➎ ➏ ➐➑ ➒ ➓➔ ❙ ❝ ⑧ ❡ ♦ ❝ ❣ ❦ ❩ ❪ ❱ ❝ ❨ ❩ q ❝ ❱ ❞ ❱ ❳ ❛ ❛ ❫ ❝ ❱ ❞ ❱ ❝ ❱ ❞ ❤ ❝ ❱ ❙ ❜ ❞ ❝ ❳ ❵ ❭ ❞ ❳ ❵ ❩ ❱ ❞ ❱ ❨ ❝ ♠ q ♠ ❚ ❞ ❛ ❭ ❫ ❪ ❞ ❨ ❝ ❝ ⑨ ❡ ❢ ❩ ❪ ❫ ❳ ❭ ❢ ❜ ❞ ⑩ r ❶ ❢ ✹ ❑ ✷ ➻ ➸ ➷ ❰ ➘ ➹ Ñ ➪ ➴ ➷ ➱ ➪✃ ➴ ✵✶ ➸ ➘ ➹ ➚ ➮ ➱ ➷ ❐ ➴ ✃ ➸ ➘ ➽ Ï Ð ❐ ❰ Þáâ ➛ ➙ ➤➥ ➦ ➩ ➯➲ ➳ ➻ ➼ ➽ ➾ ➹ ➘ ➴ ➴ ß Þ ➷ ÙÚ Ø ➬ ➾ ➼ ➴ ➻ ➴ ➷ ➮ ➻ ➘ ➡ ➪ ➘ Ð Õ ❐ ❒ ➸ ➴ ➷ Ô ➷ ➪ ÒÓ ➱ ❒ ➘ ➴ Ñ ➘ ➱ Ð ❰ ➻ ❰ Ï ✃ ➻ ➴ ➾ ➴ ➹ ➪ ➻ Ð ➾ ➘ ❒ ➶ ➪ ➮ ➪ ➾ ➘ ➽ ➸ ➴ ➷ ➼ ➪ ➾ ➻ ➴ ➶ ➘ ➬ ➘ ➴ ➪ ➴ ➼ ➴ ➢ ➠ ➓ ➐ ✤ ✎ ➋ ➌ ✣ ➍ ✢ ➑➒ ✜ ➑ ✎ ✕ ✎ ➔ ✛ ✏ ✚ ✙ ✘ ✗ → ➣ ✕ ✖ ➌ ✕ ✥ ✦ ✏ ➊ ✴ ➅ ✳ ✬ ➆ ➇ ✲ ✯ ✱ ✰ ✯ ✮ ➋ ✧ ➌ ➍ ➎ ➊ ➏ ✢ ✣ ✪ ✩ ★ ✢ ✕ ✢ ➙ ✑ ✎ ò ý ûü ø ú ø ù ➝ óö ô ò ➋➞ ó î þ ñ ï ð ï ç ➟ ì éêë è ➙ äå ã ➑ ✔ ➻ ó ✞ ✓ ✍ ✒ ↔ ✑ ✏ ✎ ✍ ➑ ✡ ➋ ú ✟ ✠ ✝ ✁ ➙ ➜ ó � ➛ ↕ ✆ ûü ➋ 58 3 ❳sr Zig-Zag ❯❇❱ ➅❏➆ ❳sr ❣❉❤ ❍❏■ n w ❊✭❋● w ➤➦➥ ❁❉❈ ✫✭✬ n ❯❇❱ ❅❇❆ ❯❇❱ ❚❬❩ ❣❉❤ String: ❵❇❛ n ③❉④ Num: Next ☛✌☞ ✇❬① - ↔➙↕ ✉❬✈ w 57 ➻×Ö /*find the last word in L ending with the same letter as w*/ ❮➺❰ while (C != NULL) and (C->string < w) do function FindLast(pointer L, string w): string same letter as w) then P = C; 3 /* return NULL if there is no such word */ If (P == NULL) then return NULL; ÷❖ø if (C->string ends with the ô❖õ ➫❉➭ ➧❇➨ else return P->string; C = C->Next; í❉î P = NULL; Û➺à ✂☎✄ C = L; æ❖ç ➚❇➪➶ ➈✜➉ ÛÝÜ - ô❖ÿ ➵➺➸

  29. ➲ ✧ ✮ ✪ ✩ ✥ ✣ ★ ✥ ✩ ✤ ✣ ✚ ✙ ✘ ☞ ✌ ✥ ✯ ✓ ✻ ✺ ✷ ✾❂ ❁ ✳ ✽✾ ✼ ✺ ✰ ✹ ✰ ✸ ✰✶✷ ✳ ✰ ✱ ✔ ✒ ✼❄ ý ✆ ☎ � ÿ þ ✄ ✂ ✁ ✝ � ÿ þ ý ú ù ø ☎ ✞ ☞ þ ☎ ☞ ✑ ✏ þ ✎ ✍ ✌ ✟ ☞ ☛ þ ☎ ✡ ÿ ✠ ❃ ✶ ➳ ❃ ❃ ✿ ✷ ✰ ✿ ✼ ❆ ✶ ✼ ✰✶ ✸ ✶ ❅ ✻ ✰ ✾ ✰ ❈ ✰ ❋ ✷ ✿ ✳ ✰ ✼ ❊ ✵ ✿ ✹ ✵ ✺ ✽ ✵ ✱ ❉ ✱✶ ✼ ✷ ✱ ✹ ✰ ✶ ✱ ✷ ✿ ❄ ❃ ✵ ✾ ✽ ✵ ✰ ❅ ✿ ✹ ❅ ✷ ❃ ✰ ✹ ✶ ❄ ✷ ✵ ❅ ✹ ✱✶ ❄ ✷ ✲ ❅ ✹ ✳ ✿ ✶ ÷ ö è ➴ ❐ ➚ ✃ ➱ ➮ ➬ Ú ➴➷ ➾ ➪ ➚ ➾ ➘ ➽ ➶ ➪ ➴ à ➾ ❮ ❮ ❰ Ö × Õ Ô Ó ➴ ❒ ❮ Ò ➾Ñ Ð Ï ➾ ❰ ❮ ➚ ➹ ➽ Û ã ➺ ì ô ó ðñò îï æ Ù ç Û Þ à ß ê é Ý â õ á ➸ Ù Û ➵ ßà 60 ✺✗✶ 3 ✫✭✬ Zig-Zag ✸❀✿ ✸❀✿ ✤✦✥ ✛✢✜ ✸❀✿ ✕✗✖ ✴✗✵ ✲✦✳ û✌ü ❁❇❆ -      Next 59 > − Back(P,Q): 3 Q P P      ←      Next NULL    > ä❉å − Q P P L         P ←      Next    Q StartTraversal(L):    > Ü❉ÝÞ − Forward(P,Q): Q Q P      ←      Next Ø❉Ù ➻✌➼ - > − ëíì P Q Q     

  30. ● ➀➁ ➊ ➋ ➈ ➌ ➀ ❾ ❿ ➀ ❿ ➂ ➀ ➃ ➀ ➄ ➅ ➃ ➀➁ ➉ ➁ ➀ ⑩ ❹ ❺ ❽ ❾ ❿ ➀➁ ➂ ❿ ➀ ➃ ➀ ➁ ⑨➏ ➀ ➈ ➅ ➆ ➅➇ ⑨⑩ ➐ ❾ ➃ ➒ ➑ ❿ ➋ ➂ ❽ ➐ ❼ ❻ ❸ ❷ ➍ ⑦ ⑨ ⑩ ❿ ➓ ➀ ➢ ➭ ➫ ➩ ➨ ➧ ➥➦ ➤ ➠➡ ➔ ❍ ➟ ➞ ➜ ➝ ➔➙ → ↕ ❶ ➅➇ ⑦⑧ P ❭ ❱ P ❪ ❫ ⑥ ❴ ❵ ❬ ❛ P❜ ❝ ❫ ❱ ❫ ❞ ◗ ❩ ❢ ❙ ■ ❏ ❑ ❖ P ◗ ❘ ❖ ❨ ❚ ❯ P ◗ ❘ ❱❲ ❱ ❳ ❡ ➎ ❴ ① ✉ ❥ ❣ ♣ ✈③ ② ❤ ✐ t ✇ ❤ ✇ ✈ ❧ ❫ r s q ❤♣ ♦ ➲ ♠♥ ❦❧ ❥ ♣ ➯ ➩ ⑤ ✐ - 3 ▲◆▼ ❣✦❤ ♦✭④ P ❷✢❸ ❻✢❼ ➄❀➅ ➃❇➆ Q: void DoublyLinkedInsert(pointer P,Q) /* insert node pointed to by P just after node pointed to by Q */ − >  P ev   Q  Pr     − > − > P Next Q Next     ←     − > Q Next P          Q − > Next − > ev P     Pr ❹✦❺ P void DoublyLinkedDelete(pointer P) /* delete node P from its doubly linked list */ − > − > − >  P ev Next   P Next  Pr ←      − > − > − > P Next ev P ev     Pr Pr →↔➣ ➛↔➜ 61

  31. ➳ ➆ ➌ ➍ ➅ ➀ ➎ ➀ ➂ ➅ ➏ ➈ ➑ ➀➒ ➔ → ➆ ➉ ↕ ➁ ➂ ➃ ➂ ➁ ➄ ➀ ➇ ➊➋ ➁ ➂ ➀ ➈ ➉ ➅ ➣↔ ➜ t ➢ ➢ ➨ ➦ ➲ ➵ ➩ ➯ ➩ ➡ ➵ ➥ ➸ ➢ ➺ ➳ ➦ ➣➝ ➧ ➞ ➟ ➤ ➥ ➦ ➡ ➫ ➲ ➧ ➢ ➭ ➥ ➢ ➯ ➢ ➁ ❹ ➢ ❱ ❍ ❥ ▼ ❘ r ◗ ◗ ▼ r ❙ ❘ ② ③ ② ❋ ❳r ❸ q ♦ ♥ ♣ ❝ ❢ ❉ ● ❥ ❱ ❲ ❋ ❨ ➵ ❱ ③❷ ① ① ❼ ❸ ⑦❽ ① ❺ ④ ❸ ⑧ ❶ ⑩ ❾ ① ❷ ⑩ ✇ ❼ ① ❹ ❺ ❶ ❸ ⑦ ❷ ❺ ⑩ ❷ ❸ ① ✇ ❼ ① ❺ ④ ❶ ➫ ➲ ♠ ↔ ➸ ➢ ➾ ❐ ❒❮ ➣ ➪ ➵ ➶ ➹ ➘ ➴ ➣ ➝ ➥ ➡ ➟ ❰ ➢ ➨ ➦ ➲ ➵ ➩ ➧ ➯ Ï Ð ➦ ➺ ➩ ➢ ➞ ➳ ➩ ➻ ➸ ✃ ➯ ➩ ➢ ➭ ➷ ➵ ➯ ➻ ➧ ➢ ➭ ➾ ➥ ➡ ➺ ➤ ➩ ➢ ➸ Ñ Ò ➢ ➥ ➯ ➫ ➩ ➦ ➱ ➨ ➩ ✃ ➳ ➦ ➢ ➢ ➞ ➟ ➳ ➺ ➩ ➢ ➤ ➪ ➥ ➫ ➩ ➦ ➱ ➨ ➣➝ ➣↔ ✃ ➩ ➳ ➢ ➻ ➠ ➲ ➢ ➫➼ → ➨ ➽ ➽ ➢ ➾ ➚ ➩ ➯ ➲ ➝ ↔ ↕ ➙ ➛ ➜ ➣ ➞ ❒❮ ➟ ➤ ➥ ➢ ➯ ➢ ➣ ❐ ➡ ➢ ➵ ➥ ➸ ✃ ➯ ➩ ➭ ➾ ➻ ➷ ➯ ➻ ➧ ➢ ➭ ♥ ● ❘ ✦ ✫✬ ✧ ✦ è ✥ ✪ ✩ ✤ ★ ✧ ✥ ç ✤ ✣ âé ✢ ✜ ✛ ✚ à âê ✙ ✔ ✫ Ý ✘ ✫ ✸ ✥✷ ✶ ✵ ✥ ✴ ✳ Ò ✲ ✥ Ó Û Ô ❮ ✱ ✦ ✰ ✯ ❒ ✮ ✭ Õ Ú ✟ ë ä ó ä ö÷ ò é à ó ô á ü û ê ö ø ö ì ë à â î ä Ü Ý Ü é ✄ ✕ ✝ é ✑✒ ✏ ✎ ✍ ✌ ì í é ✞ ✆ ã â Ý ü ✄ ö ö÷ í ø ä î ☎ ✳ ✫ ✳ ➽ ➾ ❤ ➚ ❉ ❢❣ ➪ ➼ ➶ ➹ ❵ ➾ ❥ ➚ ➘➴ ➘ ➷ ❬ ❩ ➬ ➮ ❘ ❨ ❏ ❍ ▼ ❳ ➻ ➸ ➺ ● ❙ ❖ ❚ ❙ P ❘ ▼ ◆❖ ❘ ❨ ➼ ❳ ◆❖ ▼ ➽ ❏ ■ ❋ ❨ ▲ ❑ ❋ ✹ ❰ ➘ ❍ ● ➽ ❐ ❒ ❮ ❉ ❈ ➽ ❃ ❖ Ï ➽Ð Ñ ❒ ➘ ❒ ✳ ✴ ✻ ✺ ✃ ▼ ◆❖ P ❚ ➱ ➾ ❑ ◗ ❋ P❖ ❘ ❲ ❱ ❯ ● P❋ ❘ ◗ ◆❖ ● ❙ ❖ ❚ ❙ ❘ ◗ â 64 ⑥✠⑦ ④✗⑤ 3 P❧❦ ✈✂✇① s✉t node degree) tree degree) ➃✂➅➆ node height) node depth) tree height) tree depth) ❑✗❏ k ❻⑨⑦ ➋❧➓ ❿❡➀ ❑✂▲ level) ❆☞❇ ➨✗➩ k ❝❡❞ ❄✗❅ ➏✂➐ ➘❡➴ ❛☞❜ ➨✂➮ ➙✗➛ ❑✂▲ ➶☞➹ . ■✠❏ ❁☞❂ ❫☞❴ -35 ✿✗❀ ❭✗❪ ➠✂➡➢ ➠✂➡➢ ⑩✗❶ ➷✂➬ ➷✂➬ ❚✂✐ ✼✾✽ ❊✂❋ ⑥⑨⑧ parent, 63 leaf, non-leaf) ancestor, descendant) 3 Þ✗ß tail, head) u’ Ø✦Ù ✓✗✖ Ö❀× ✓✂✔ – path) (nodes) �✂✁ (edges) child, sibling) – ✡☞☛ – – ù✦ú Þ✗ßà ✞✠✟ ã❇ä -35 �✂✁ Ü✦Ý õðö÷ á❇â ýÿþ å❇æ ïðßñ o o o o o o o

  32. Ó ❶ ⑤ ⑥ ⑦ ⑧ ⑨ ⑦ ❸❹ ❼ t ❽ ❾ ❽ ❼❿ ➀ ❺ ➁ ➂ s ④ ➈ t ♣ ❢ q ❦ ❡ r s ❡ ③ ✉ ✈ ❡✇ ① s ❦ s ② ➄ ❍ ♥ P ❊ ● ❋ ◆ P ❊ ❯ ❊ ▼ ❏ ❊❋ ● ❋ ◆ ◗ ◆ ➌ ❋ ❈ ❉ ❑ ● ■ ❑ ❯ P ❑ ❑ ❏ ❈ ❇ ❑ ❇ ❋ ❑ ■ ❋ ❯ P ♦ ♠ ◗ ■ ❍ ❉ ● ■ ❑ ❲ ❫ ❈ Ô ❖ ❊ P ❑ ▲ ❩ ❇ ❊❋ ● ❉ ❑ ❄ ❍ ❇ ❚ ❊ ◗ ▼ ❍ ❚ ❭ ❈ P ❘ P ❇ P ❑ ❩ ❇ P ❲ ❦ ❢ ❴ ❵ ❛ ❜ ❝ ❞ ❡ ❣ ❩ ❤ ❞ ✐ ❥ ❡ ❢ ❣ ❦❧ ❲ ❑▲ ❩ ❪ ❊ ❩ ❭ ❲ ❲ ❲ ❭ ❩ ❇ P ❊❋ ▲ ❏ ❑ ● ❍ ❉ ❊ ❇ ❘ ❚ ➠ ➤ ➦ ➛ ➥ ➜ ➛ ➜ ➟ ➙ ➤ ➢ ➤ ➟ ➠ ➫ ➝ ➠ ➛ ➞ ➜ → ● ❇ ➊ ❈ ❆ ❲ ➓ ➔ ↕ ➝ ➝ ➟ ➙ ➜ ➠ ➡ ➙ ➝ ➟ ↕ ❉ ➽ ➻ ➩ ➝ ➻ ➼ ➤ ➝ ➝ ➥ ➟ ➦ ➡ ➙ ➢ ➤ ➙ ➥ ➟ ➛ ➥ ➟ ➟ ➠ ➟ ➜ ➠ ➟ ➩ ➭ ➟ ➦ ➠ ➩ → ➣ ➯ ➵➸ ➺ ➛ ❊ ❍ ▼ ❑ ❉ ❑ ● ❍ ❉ ● ■ P ▼ ❑ ▲ ❑ ❏ ❑ ❈ ❑ ▲ ❍ ❊➐ ◆ ➏ ❙ ❑ ▲ ❲ ❼ ➅ ❻ ➎ ➅ P ➁ ➂ ➄ ➈ ❋ ❊ P ❇ ◗ ➌ ❋ ❊ ● ❋ ◆ ➒ ❍ ❉ ❊ ■ ❊ ❋ ❯ P ❇ ■ ❘ ◗ ❊ ❋ ❏ ❇ ❇ ◗ ❘ ▼ ❙ ❑ ❚ ❍ ➉ ❋ ❑ P ❑ ❏ ❑ ❬ ➑ ● ➑ ❉ ❊❋ ❋ ✤ ✮ ✯ ✥ ✩ ✤ ✦ ✯ ✤ ✥ ✩ ✦ ✰ ✱ ✤ ✲ ★ ✳ ✭ ★ ✵ ✚ ✎ ✝✍ ✑✗ ✘ ✙ ✔ ✏ ✑✗ ✪ ✘✛ ✤✥ ✦ ✧ ✢ ★✩ ✧ ✦ ✴ ✴ ☛ ✤ ✤✼ ✩ ✤ ✼ ✮ ✯ ✽ ✳ ✯ ★ ✻ ✷ ✶ ✢ ✷ ✮ ★ ✢ ✴ ✯ ✺ ✶ ✢ ✤ ✷ ✶ ✢ ✷ ✮ ✩ ✥ ✧ ✦ ✪ ★ ✥ ✤✻ P ✧ ✎ ✝✍ ✸ é å ß Ù æ ç è Ù ê ä Ùë ì ç ß ç í î ï Ú ã ç Ü Õ Ö × Ø Ù Ú Û Ø â Ý Þ Ù Ú Û ßà ß á è ö÷ ✌ ✓ ☞ ✟ ✑ ✎ ✒ ✟ ✑ ✑ ✎ ✡ ✌ ✔ ☞ ✟ ✡ ✟ ✕ ✏ ✡ ø ☎ ÷ü ö ù ú ÿ � ✁ þ ✆ ✝✍ ✝ ✡☛ ☞ ✟ ✡ ✌ ✝ ✌ ✻ ✳ ❨ ❱ ❋ ▼ ❘ ◗ ❚ ❑ ❍ ❉ ❚ ❚ ❊ ❉ ❍ ❊ P ❑ ❚ ❱ ❖ ❇ ❚ P ❑▲ ❲ ◗ ❳ ❯ P ❚ ❍ ▼ ❙ ❚ ■ P ❑▲ ◗ P ❘ ❏ ❘ ▼ ❙ ❊ ❑ ▲ ❋ ❚ ❊ ❈ ❲ ❊ P ❋ ❉ P ❑ ❩ ❊ ◗ ❏ ❑ P ❇ ❬ ❇ ❚ ❈ ❋ ● ❪ ❭ P ❇ ❑ ● ❑▲ ❏ ❉ ❊ ❍ ❑ ❉ ❍ ❊❋ ❉ ❀ ❊❋ ✳ ✭ ◗ ✼ ❑ ❙ ◗ ❀ ❁ ▼ ❄ ❅ ❘ ❑ ❩ ▼ ❋ ◗ ❍ ❑▲❚ ❙ ▼ ❩ ❭ ❯ ❑ ❘ ◗ ❊❋ P ❉ P ❑ ❖ ❱ ◆ ❲ ❲ ❲ ■ ❉ ❭ ❋ ❊ 66 ➊❷➋ 3 ➔➧➦ ➙✿➥ ➢✣➤ ➟✹➾ ⑩❷❶ ➙❷➞ ➙❷➛➜ ❍✣➉ ➅➇➆ ➡✿➩ ➣✠↔ NULL ➅➇➆ ➁➃➂➄ ➢✣➤ nill -35 ➢✣➨ ➲✿➳ ❺✣❻ ❺➇➍ 65 ❈✣❉ 3 r 1 , ..., r k ●✿■ 1 ❈✣❉ û✗ô r ❆✠❇ r ù⑨ú <r,r 1 >, <r,r 2 >, …, <r,r k ❂✠❃ ✫✠✬ ❈✣❉ ó✗ôõ 1 , ..., ✞✠✟ r ð✗ñò ✔✄✖ ★✹✸ ●✿■ ✧✿✳ �✄✂ -35 1 ✜✣✢ ✜✿✾ • • ý✠þ

  33. ➚ ❸ ★ ✦ ❁ ❂ ✽ ❇ ✦ ✪ ★ ✽ ✛ ✚ ❃ ⑥ ✩ ✼ ❉ ✢ ✿ ❂ ✛ ✢ ✥ ✦ ✦ ✿ ❉ ❹ ✛ ✚ ✦ ✼ ★ ❃ ✼ ✣ ✥ ✛✜ ✛ ✚ ❂ ✧ ✛ ✜ ✢ ✚ ⑤ ✦ ✿ ❉ ✙ ✤✥ ✦ ✦ ✦ ✦ ✣ ✦ ⑧ ✩ ✣ ❺ ✛ ✧ ❆ ❅ ★ ❁ ❉ ❂ ✢ ✧ ★ ✢ ✩ ✦ ✦ ✿ ✙ ✧ ✼ ✣ ★ ✛ ✜ ✧ ★ ✛ ✢ ✩ ★ ✜ ✩ ⑧ ❈ ✼ ✣ ✤✥ ✜ ❉ ✛ ✜ ✼ ✤ ✥ ✼ ✩ ❂ ✥ ✼ ✩ ✼ ✥ ❈ ❈ ✿ ✪ ⑦ ✛ ❄ ❅ ❃ ❃ ✛ ✛ ✥ ✦ ✾ ✧ ❆ ✦ ➪ ★ ✥ ✦ ✚ ❉ ✧ ✧ ❃ ✰❷ ⑩ ✷ ① ❸ ★ ✛ ✚ ⑥ ❶ ✩ ✼ ❉ ✛ ✢ ✜ ❁ ❂ ✢ ✹ ① ✤ ✦✿ ✼ ✜ ❉ ✥ ✦ ✿ ✢ ✧ ★ ✪ ✷ ⑨ ✸ ✴ ✵ ✹ ✱ ⑩ ✰ ✶ ✼ ✼ ✦ ➫ ➵ ➚ ➦ ➠ ➻ ➸ ➺ ➤ ➻ ➧ ➚ ➹ ➢ ➤ ➥ ➦ ➧ ➨ ➢ ➭ ➥ ➦ ➥ ➢ ➲ ➵ ➠ ➧ ➤ ➤ ➫ ➻ ➧ ➳ ➢ ➤ ➯ ➠ ➾ ➧ ➩ ➠ ➡ ➠ ➩ ➥ ➥ ➯ ➠ ➾ ➧ ➩ ➠ ➡ ➳ ➧ ➢ ➢ ➲ ➵ ➠ ➧ ➤ ➤ ➫ ➻ ➤ ➧ ➫ ➥ ➭ ➤ ➯ ➯ ➦ ➤ ➠ ➡ ➳ ➧ ➥ ➭ ➵ ➚ ➦ ➠ ➻ ➤ ➫ ➻ ➧ ➳ ✣ ➓ ➓ ➔→ ❽ ❿ ❾↔ ➅ ❿ ❿ ➆ ➒ ➆ ➣ → ❽ ❼ ➆ ➛ ➔ ➒ ➁ ❼ ➎ ➄ ✥ ✼ ❻ ❼ ❽❾ ❿ ❾ ➀➁ ❽ ➅ ❽ ➃➆ ➇➈ ➉ ➊➋ ➌➈ ➍ ❼ ❽❾ ❿ ➔ ➛ ➡ ➨ ➝➞ ➟ ➠ ➠ ➢ ➤➥ ➦ ➧ ➢ ➛ ➠ ➦ ➭ ➤ ➯ ➯ ➦ ➤ ➠ ➏ ❽ ❽ ➜ ❾ ❽ ❼ ➅ ➄ ❾ ❽ ➓ ➎ ❽ ➓ ↔ ❽ ❾ ❻ ➣ ❼ ➆ ❿ ❿ ❉ ✿ ⑤ ✤✥ ✜ ❁ ❂ ✢ ✧ ★ ✦ ✦ ✛ ✦ ✚ ✦ ✣ ✦ ❂ ❃ ✛ ✢ ✦ ✛ ✰ ✦ ✢ ✧ ★ ✬ ✭✮ ✯ ✱ ✽ ✴ ✵ ✰ ✶ ✷ ✲ ✱ ✺ ✥ ❄ ✣ ❏ ❍ ■ ❏ ❑ ▲ ▼ ◆ ❖ ❋ P ❑ ▲ ▼ ◗ ❘ ◗ ❙ ● ❊ ❅ ✥ ❃ ❃ ✛ ✧ ❆ ✦✿ ★ ✦ ✪ ✣ ❅ ✜ ✦ ❇ ✾ ❈ ✦❉ ✤✥ ✢ ❯ Ö Ò ❒ ➷ Ó Ô Õ ➷ × Ñ ➷Ø Ù Ô ❒ Ô Ú Û Ü ➬ Ð Ô ➱ ➶ ➹ ➘ ➴ ➷ ➬ ➮ ➴ Ï ✃ ❐ ➷ ➬ ➮ ❒❮ ❒ ❰ Õ Ý ✛✜ ✑ ✁ û ú ✡☛ ☞ ✎ ✟✏ ✘ ý ✓ ✟ ✓ ✒ ✡ ✖ ✙ ✚ � ú Þ êë ß à áâ ã ä é è î ÷ å æ ï ì ô ñ õ ö ❚ ✢ ❱ ❁ ❃ ✼ ✜ ✦ ✛ ✢ ✜ ❂ ✥ ✢ ✧ ★ ✢ ✩ ✦ ✦ ✿ ✧ ✛ ✙ ✼ ✹ ✱ ✱✳ ① ④ ✚ ✦ ✥ ❃ ★ ✼ ▲ ✥ ✛✜ ✛ ✚ ❂ ✧ ❉ ✤✥ ✲ ✼ ✚ ✼ ✩ ✜ ✤ ✤ ❂ ✥ ✩ ✛ ✛ ❄ ❅ ❃ ❃ ✛ ✽ ✼ ⑥ ★ ✦ ✿ ✦ ✚ ✦ ✣ ✦ ✧ ❆ ✦ ★ ✧ ✚ ✩ ✦ ✤✥ ✼ ❈ ✼ ✣ ✵ ✣ ❩ ❨ ❣ ❞ ❴ ✯① ✹ ❳ ❤ ✇ ✮ ❨ ✈ ◗ ❦ ❩ r t✉ ❨ ❞ ❢ ❑ ❬ ♦ ❭ ❑❪ ❨ ♠♥ ❫ ◗ ❞ ❞ ❡ ❢❣ ❤ ❛ ✶ ✐ ❝ ❜ ❤ ✷ ❲ ❑ ❵ ❡ ✰ ✵ ✴ h (complete h>1, 68 ❅✠⑤ ➸✠➺ ➸✠➺ h-2. 3 ➼➶➪ h- s❀r ↕✠➙ ➂✠➃ q❀r ➎➐➏ ➸✠➺ ❅✠⑤ ❅✠⑤ ②✠③ ➂✠➃ ✢❀✿ ♦✔♣ ➲✠➳ binary tree of height h) ➲✠➳ ✭☎✸ h ➩✠➫ ❿❀➣ ❥❧❦ ✢❀✿ ➂✠➃ ✲✝✳ ➼✠➽ ➼✠➽ -35 ✢✔✩ h- h- ➡✔➢ ✢✔✩ ➑✠➃ o o o h- 67 ✦✫✪ perfect binary tree) ✢✔✩ 3 (full binary tree) å✿ó ò❷ñ ð❷ñ ì✿í þ ✝✆ ✵✝✹ ✢✔✩ ✂☎✄ ✭☎✸ ç➧è ✕✗✖ å✠æ ✒✔✓ þ✣ÿ û➇ü ✢❀✿ ✲✝✳ ✌☎✍ -35 ø✿ù ✾✔✥ ✞✠✟ ✻✝✼

  34. ➘ ÏÐ ➬ ➮➱ ➸ ➺ ➹ ❒ ❮ ❰ Ñ ➴ ÐÒ Ó Ô Õ Ö × Ï Ø Ï ➷ ➼ × ➢ ➡ ➢ ➤ ➓ ➥ ➦ ➓➧ ➨ ➙ ➘ ➢ ➩ ➫ ➭ ➤ ➢ ➳ ➺ ➹ Û Ú ➙ å â Ï ÝÞ ã è ä Ï ß é ä Ú Ò Ý Ù å ä Ù â Ý Ù å Ò Ú Ý Ò à á Þ â Ò à × Ï Ù Ò à Ñ à ß Ô ã ÐÒ Ï ➓ ➠ æ ➅ ❼ ❿ ➀ ❹ ❾ ✶ ➂➃ ➄ ➅ ❾ ➁ ⑥ ➁ ➂ ● ➀ ❈ ⑤ ❹ ❻ ⑩ ❼ ④ ♣ r✇ s ❁ ❡ ❦ ❄ ② ③ ▼ ❽ ▼ ▲ ❑ ❄❅ ❹ ⑥ ❺ ❻❼ ⑩ ❾ ➁ ➔ ↔ ➐ ➑ ➒ ➓ ➔ → ➣ ➒ ↕ ➎ ➓ ➔ → ➙➛ ➙ ➜ ➝ ➞ ➟ ➏ ➍ ➂ ❾ ❹ ➆ ❻ ❺ ❹ ➇ ➂ ⑥ ⑩ ⑥ ❏ ⑦ ⑧ ❇ ➂ ❸ ⑥ ➈ ➉ ➴ ê ç ♣ ✝ ô ñ � ✂ ð ñ ú ô ï ù ó ✝ ✕ ï ô ✠ ✚ ✛ ✜ í ó ✦ ✖ ✂ ù ö ✑ ñ ó ù í ô û ñ ò ò í ✝ í ô ö ✑ ì ✢ ✧ ï ❆ ❆ ❇ ❈ ❆ ❄ ❅ ❇ ✹ ❃● ❂ ✽ ❂ ✽ ❈ ❃ ● ❍ ■ ■ ❄❅ ❄ ★ ✳ ✩ ✪ ✫ ✬ ✭ ✫ ✮ ✱ ✜✲ ✴ ❀ ✵ ★ ✶ ✷ ✸ ✹ ✿❀ ❁ ❃ ó î Ý ✝ ïýþ ÿ ð ✂ ✄ ☎ ✄ ✆ í ø ô ð ✞ ú ó ✟ í ð ☛ í í õ ó Þ ã ë î ï ð ñ ò ñ ð ô í ô í î õö ð ÷ ò ñ î ñ ï ì ✄ ☎ ñ ô ö ✑ ✒ ✓ ✝ í ✕ ïý þ ☎ ñ ✝ ð ✂ ù ✄ ð î õ í ☞ ✂ ☞ ô ✝ ✌ ý î ✍ õ ñ � ✞ ✝ í ✍ í í î ✺ ❼➁ ❂ ú þ û ✆ û � ✍ ◆ ÿ ✕ û ✠ � ÿ ✗ ✓ ü ÿ ✞ � ✆ þ ✖ ✕ ú ❇ ✏ ✁ ✄ ❆ ÿ ✆ ü ✓ ✁ � ✔ ✁ ✕ ✓ ú ✍ ❃ ÿ ✕ úû ✁ ✆ ❖ ✯ ✗ ü ✁ ❈ ♠ ❇ ✪✫ ✬ ✭✮ ✮ ✥ ✸ ❏ ✶ ✷ ✸ ✫✹ ✴ ✶ ✭ ✺ ✦ ✓ ✝ ô ✎ ✏ ✁ � ✘ ❇ ❄❅ ✙ ✚ ✛ ✥ ✜ ✢ ✣ ø ● ✤ ÿ ÿ ✔ û ✆ ✍ ✼ Ý ✃ ❈ × Ø Ù ✃ Ú Û ✃Ü Ø Ö Ð Ø Þ ß à Ù Ø ❇ ❆ Ð ❐ á ❰ ➷ ➬ ➮ ➱ ✃ ❐ ❒ ❮ ➱ Ï Õ ✃ ❐ ❒ ÐÑ Ð ❄❅ ❆ Ò Ó Ô ❄❅ ❂ � ý ✹ ✁ ❈ û ✂ � ✆ ✝ ✂ ú ü ✽ ✁ ✠ ✡ ✁ ✄ ☛ ✹ ❅ ÿ ý ✺ ❄ é æ å çè ❀ ë â ã ì ❃ ï ✻ î ñ î ❁ ❅ ò ó ö ÷ ø ✹ ✧ ✽ ✴ ✶ ✹ ❍ ❅ ❈ ✼ ✽ ❄ ❅ ✱ ❄ ❄ ✹ ■ ❊ ▲ ▼ ❉ ❄❅ ❆ ❏ ✭ ✮ ❈ ✺ ✭ ✹ ❂ ✫ ✩ ✯ ✶ ✫ ✿ ✹ ✴ ★ ❀ ✱ ✼ ✽ ❄ ❅ ✴ ✻ ● ✿ ■ ❘ ✴ ✐ ❄ ❤ ❢ ✐ ❥ ❝ ✐ ❦ ❀ ❧ ❢ ♦ ❡ ❦♣ ❃ ♥ q ❦s ❝ q ❧ ❣ ❂ ❙ ❱ ❚ ◗ ❯ ❱❲ ◆ ❳ ❲ ❇ ❬ ◆ ❆ ❝ ❄❅ ❭ ❪ ❫❴ ❙ ❴ ❱❲ ❵ ❛ ❜ ✼ ✼ ✸ ❅ ✬ ✽ ✷ ✾ ✭ ✼ ✽ ❄ ✴ ✴ ✿ ❃ ❆ ✭ ✼ ✭ ✹ ★ ✭ ✾ ✭ ✬ ■ ✴ ■ ✴ ✻ ✱ ❂ ✭ ✴ ✽ ✶ ✫ ✸ ✫ ❃ ✶ ✭ ✹ ❄ ✿ ■ ✴ ✼ ✬ ✴ ✫ ❅ ❃ ❄ ✻ ✾ ✭ ✹ ✽ ✸ ❈ ❉ ✹ ✫ ❀ ❍ 1 , 70 ✼✾✽ ✺✁❂ ✗✙✘ 3 ✼✾✽ ✺✁✻ 2 õ ✁✔ ÷✌ù Û✌ä – 1 ì✌í Ô✌ã ✃❖❐ k ÷ ✁� – k –1) + 1 = 2 k+1. ✯✁✰ ❉◗P õ ✁✔ ➳➶➪ û✌ü Ô✟ß – – ➽➚➾ ❉❋❊ = 2 ø✟ú – ➻❖➼ ÷✌ù æ✌ç Ü☎ÝÞ + 2 2*(2 2*(2 -35 Ù☎Ú ÷ ✏✎ ➵✟➸ ✺✁❂ 2 ✣✏✤✥ 2 ïý ✡✠ ➯❖➲ h+1 –1 h –1 69 ❧✌① 3 ý ✌☞ ✉✌✈ ✱❋❊ â✔ð h ➊➌➋ ✷✌❇ q✟t í❀î ❀✌❁ h ⑨❖⑩ ✂✟✞ ✺✟✿ 2 h+1 –1 = 2 0+1 –1 = 2 –1 = ❨✟❩ ✷✌✾ h = 0 q☎r é✔ê - h. ❞☎❡ ▼❖◆ ❷✒❸ ✰✌✱✒✲ ✂☎✄ ä❧å ✺☎✻ ⑨❖⑩❶ ❧✌♠♥ â✗ã ❏✌❑ ⑦✟⑧ P✌◗ þ✗ÿ � ☛✒✑ 2 h = 2 0 -35 ⑤✌⑥ – ô✔õ 2 h – – ★✟✩ ✝✟✎ ù✠úû ✳✵✴ 2 2

  35. ❘ å Ù Ø Ü Û Ý â ß ã æ é ❮ è í ð ê ó ò õò ö ÷ ú Ö ❰ ✁ ❮ ➬ ➮ ➱ ✃ ➚ ❐ ➴ ➾ ❒ ❰ Õ ➾ Ï Ð ➾Ñ Ò ❮ ➴ ❮ Ó Ô � ù ➴➷ ✡ ✞ ù ✟ ✁ ✄ þ ☞ ✄ ✌ þ þ ✆ ✍ ✂ ✌ ✡ ù ✍ û ú ✆ ☞ ✝ ✂ ✠ ✆ ✝ ù ✞ ✟ þ ú ✆ ✂ ✞ ☛ ù ✟ ú ✡ ù ÿ ☛ ú ✆ ✂ ➴ ➪ ✌ ➟ ➛ ❙ ➒ ➙ ➔➩ ➫ ➒ ➣ ➔ ➙ ↔ ➠ ➡ ➢ ➒ ➔ ➙ ➛ ➭ ➫ ➯ ➢ ➔ ➙ ➒ ➛➜ ➝ ➞ ➒ ➣ ➔ ➟ ➙ ➠ ➔ ➑ ↔➤ ➛ ➒ ↕ ➤ ↔➔ ➢ ➥ ➦ ➦ ➛➲ ➛ ➚ ➺ ➛ ➑ ➡ ➤ ➑ ➠ ➛ ➵ ➸ ➻ ➟ ➼ ➽ ➾ ➚ ➪ ➶ ➽ ➹ ➘ ➾ ➙ ➜ ➜ ➣ ➟ ➙ ➛ ➑ ➡ ➤ ➑ ➠ ➛ ➒ ➔ ➛ ➟ ➙ ➠ ➒ ➔ ➙ ➛ ➛ ➲ ➙ ✂ ✡ ↕ ✂ ö ✍ ✆ ✡ ÷ ü ù ú ✆ � è ✄ ✆ ✡ ✍ ✟ ✡ þ ✞ ù ✟ ✤ ✢ ✡ æ ✂ ✡ þ ☞ ✍ ✔ ✟ ✆ ✜ å è î å ✢ ï ð ✣ ã æ å é ú ✛ ✓ ✝ û ✡ þ ☞ ú û ü ✧ ✂ ✓ ✂ ÷ ✫ ✯ ✰ ✫✱ ✲ ✬ ✰ ✫ ✱ ú ✍ ✌ ù ù ú ÷ ú û ü ✝ þ ☞ ✞ ✟ ✡ ✁ ✄ ù ✂ ✌ ✡ þ ✄ ✝ ✆ ù ✛ þ ✍ ✁ ✒ ✂ ✓ ù ✂ ✡ þ ☞ ✔ þ ✟ ù ÿ ✕ ☛ ú ✆ ✂ ☞ ✓ ✂ ✡ ✝ þ ✎ ✂ û ✝ ✆ ✂ ✞ ✟ ✡ ☞ ✌ ú û ü ý þ ☞ ✆ ✄ ✡ ✟ þ þ ✡ ✟ þ ✏ ✙ ✟ ✄ ✆ ✂ ✡ þ ✞ û ✂ þ ✆ ✓ ✚ ✂ ✆ ✔ ✂ ✆ ✄ ✍ ✂ ù þ ✖ ✄ ✡ ✆ ✂ ✝ þ ✓ ✖ ✖ ✁ ú þ ✓ ✆ ✌ ù ✗ ✕ ✘ ☛ ➙ ➠ ➑ ❶ ❼ ⑨ ⑩ ❶ ❷❸ ❹ ❺ ❻ ⑨ ❸ ➊ ❼ ⑩ ❹ ➍ ❸ ❼ ⑩ ➃ ❶ ➌ ➊ ➉ ❸ ❽ ➅ ❸ ❼ ➎ ➌ ⑨ ❶ ❼ ❷ ➉ ❼ ❻ ➊ ❼ ➃ ❽ ❼ ⑨ ➆ ❽ ➃ ❶ ➆ ❼➄ ❸ ❾ ⑨ ➏ ❶ ❿ ➃ ➌ ⑨ ➊ ❻ ❶ ❿ ↔➔ ❾ ❿ ➀ ➁ ❼ ❸ ❼ ➄ ➅ ➃ ➌ ❸ ➅ ➊ ➃ ❽ ❼ ⑨ ➆ ❽ ➃ ❶ ❻ ➉ ➊ ❶ ❼ ⑨ ⑩ ❶ ❷ ❸ ❹ ❺ ❻ ⑨ ❼➄ ❶ ❼ ♠ ❣ ❳ ❤ ✐ ❳❥ ❦ ❢ ❫ ❢ ❧ ♥ ❡ ❣ ❢ q r st ✉✈ ✇ ♣① ② ♣ ❢ ❳ ⑦ ❪ ❚ ❯ ❱ ❲ ❳ ❨ ❩ ❬ ❲ ❭ ❳ ❫ ❨ ❩ ❫❴ ❫ ❵ ❛ ❜ ❝ ❨ ❞ ③ ✉⑧ ➌ ❼ ➈ ➃ ⑨ ⑩ ❶ ❷❸ ❹ ⑨ ❶ ❸ ❷ ❹ ➉ ❽ ❼ ➊ ❼ ❸ ➋❽ ⑩ ➃ ❶ ➆ ❶➇ ⑨ ❼➂ ⑩ ❶ ❷❸ ❹ ⑨ ❶ ❸ ❼❽ ❾ ➁ ❼ ➃ ❽ ❸ ❼➄ ➅ ➃ ❽ ❼ ⑨ ➆ ❽ ❽ ❼ ❸ ➊ ❷❸ ⑨ ❹ ❿ ➃ ➌ ⑨ ❻ ❹ ❶ ❶ ❾ ❿ ❽ ❾ ❿ ❶ ➃ ➂ ❹ ⑨ ➃ ❶ ❷ ❸ ❸ ➉ ❺ ❽ ❻ ⑨ ❶ ❸ ❼ ❼ ❾ ❼ ❼➄ ❼➄ ➈ ⑨ ❸ ❶ ❿ ➇ ❹ ➆ ➃ ❽ ➅ ❹ ⑨ ❷❸ ❶ ❸ ❼ ➃ ➆ ❸ ➌ ➀ ➆ ➁ ❼ ⑨ ❼ ➄ ➅ ⑨ ❶ ➃ ➃ ❽ ❼ ⑩ ❼ ❹ ❼ ⑩ ➁ ❹ ➏ ❶ ❿ ❸ ⑨ ❹ ➃ ➌ ❼ ⑩ ❼ ➃ ❶ ➀ ❶ ⑨ ❼ ➌ ➇ ➆ ➐ ❿ ❽ ➃ ❹ ❶ ❹ ❶ ➆ ⑨ ➈ ➃ ➅ ❸ ❷❸ ⑨ ❸ ❹ ⑨ ❿ ❾ ➄ ➄ ❾ ➁ ❼ ❸ ➉ ❼ ❼ ❾ ➄ ❹ ➂ ❸ ❷❸ ❼ ❶ ❼➄ ❶ ❽ ➆ ⑩ LC RightChild() 72 Parent()? ý✾þ 3 ✕✦✥ ý✾þ ø⑥ù Parent() RC (Right Child Info à✁á Þ⑥ß LeftChild îìô ✞ ⑥ù ✵✷✶ Ú⑥Û ñ✙ò �☎✄ ç✙è ✬✴✳ ø⑥ù î✏ï ×✁Ø ä✙å êìë (Left Child ý✾þÿ ✆✑✏ û✁ü ç✙è ✬✮✭ -35 ✬✮✭ ø⑥ù ä✙å ★✪✩ nill false 71 v ➍✏❽ v ➛➨➧ 3 ❺✏❻ ❿✁➀ ❺✏➃ v v nill) ④⑥⑤ v RightChild(v): v v ❿✁➀ v v ➥✏➳ true nill nill ❺✏❻ ❺✏❻ ➡✏➢ ➡✏➢ RightSibling(v): →✁➣ ♦✁♣ LeftSibling(v): FirstChild(v): v v LeftChild(v), ➓⑥➔ Children(v): Parent(v): Height(v): v ➑⑥➒ IsLeaf(v): Depth(v): -35 v nill v

  36. ✸ å è î ê î ê ì ò å ö ü ò ç æ å é ï á æ ë ê ò û ✽ ð æ á á î ñ ç ñ ã ì ò ã ð á î ê ç î ö ✺ á ä â ç å æ ï á ä õ á ✾ á ã ï î ê ï á ò ç ñ î ✺ á é á á é ï á æ ë ê ü á ❀ ❁ ❄❅ ❆ ä è è ä î ê ï á ä â ç ñ î ✺ á â ê ✿ û û ç å æ å ö á ✾ í ç äå â ❉ ✡ � ✁ ✂ ✄ ☎ ✆ ✝ ✞ ✟ ☎ ✠ ✆ û ✝ ✞ ☛ ☞ ☛ ✌ ✍ ✎ ✏ ✝ ✑ ☛ ÿ þ ✒ ð ò ú ø ã êò ä ö á é ï í ü ç ý ê ï å æ å ö á ✹ á ï æ ✆ ✓ ê ë ✱ ✷ ★ ✵ ✯ ✰ ★ ✸ ö þ ð ê ✶ æ î ñ ò á ì ò å ö õ ✹ ð ★ ✦ ✔ ✛ ✆ ✕ ✖ ✆ ✗ ✘ ✓ ☛ ✓ ✙ ✚ ✔ ✥ ✓ ✤ ✥ ✦ ✥✪ ✫ ✬ ✭ ✰ ★ ✴ ✵ ❊ ❋ ð ❑ ❊ ❘ ❏ ❡ ❄ ❅ ❉ ◆ ◗ ❖ ❏ ❇ ❄ ❃ ❉ ◗ ❲❢ ❊ ❘ ❏ ❡ ❄ ❢ ✐ ❥ ❅ ❃ ❈ ❤ ❑ ◗ ❄ ❊ ❘ ❏ ❡ ❄❢ ❣ ❋ ❜ ❄ ❇ ❉ ◆ ◗ P ❃ ❑ ◆ ❘ ❆ ❉ ◗ ❨ ❑ ❙ ❞ ➀ ❬③ ✇ ⑨⑩❶ ⑧ ❸ ❺ ⑧ ❶ ❾ ❽ ❿❶ ❺ ① ➂ ⑩ ❽ ⑧ ❻ ⑦➃ ➄ ❿➅➆ ➇ ➈ ➉ ➊ q② ✇ ❃ ❄ ❄ ❨ ❑ ♥ ❇ ❆ ❉ ◗ ❨ ❇ ❃ ❅ ❴ ❲❢ ❊ ❘ ❏ ❡ ❄❢ ♦ s q ❭ ❴ ❵ ❉ ❇ ●❍ ❑ ❃ ❄◗ ❄ ❃ ◆ ❘ ❆ ❄ ❙ P ❃ ❈ ❃ ❉ ❆ ❑ ❃ ❄ ❅ ❏ ❑ ❃ ▼ ❈ ❑ ❍ ❑ ❃ ❅ ■ ❄ ❅ ❏ ❑ ❈ ❉ ❊ ▲ ❈ ❄ ❏ ❏ ❑ ❃ ▼ ❈ ❉ ◆ ❇ ❖ ❏ ▼ P ▼ ❊ ❍ ◆ ◆ ❄ P ❑ P ❃ ❉ ❖ ❑ ◆ ❘ ❯ ❘ ❨ ❈ ❄ ❋ ❩ ❬ ❭❪ ❫ ❴ ❵ ❴❛ ❜ ❝ ◆ ❈ ■ ❑ ❑ P ▼ ❯ ◆ ❱ P ◗ ❏ ❲ ❈ ■ ◗ ❍ ❃ ❄ ❅ ◆ ❳ ◗ ❙ ❈ P ❃ ❄ ❨ ñ ã ë ➈ ❻ ➀ ❷ ➁ ➈➉ ➊ ➋ ➆ ➏ ➊ ➐ ⑩ ➓ ➈➔ ➈ ➓ ↕ ➜➝ ➜ ➠➡ ➢ ➥ ➧ ❶ ⑨ ➭➯ ❽ ♦ ③ ⑤ ⑧ ⑨ ⑩ ⑨❹ ❺ ❻ ⑨ ❼ ⑧ ➃ ❽ ❼❾ ❿ ➀ ❷➁ ➂ ❹ ❼❷ ❻ ➂ ❼ ➨ ➲ ① ✃ ➭ ➚ ➹ ➹ ➮ ➘ ➱ ➶ ❐ ❒❮ ❐ ❰ ➼ Ï ➶ ➹ ➹ ❮ ❰ ➷➬ ❐ ➶ ❰ ➷ ø ➦ ➳ ➭ ➨ ➭ ➥➵ ➧ ➸➺ ➻ ➼ ➽ ➦ ➼ ➵ ➥ ➽ ➾ ➻ ➨ ➭ ➼ ➯ ➥ ➼ ➸➺ ➻ ➼ ④ ♦③ ➹ ● ■ ❏ ✾❑ ▲ ● ❄ ● ▼ ◆ ❖ ❍ P ❍ ❙ ❘ ❱ ❯ ❨ ❩ ❬ ❩ ❨❭ ❪ ❵❴ ✾ ● ❵ ✾ ✺ ✻ ✼ ✽ ✾ ✿ ❀ ❁ ✽ ❂ ❃ ✿ ❋ ❀ ❄❅ ❄ ❆ ❇ ❈ ❉ ✿ ❊ ❄ ✾ ❲ ❴ ❞ t t ❥ ❞ ❦ ❞ ✉ ❣ t ♦ ♣ q ♦ ❧ ♠ ❤ ❢ ❣ ❥ ❤ ❞ ① ② t ♦ ❣ ✐ ❡ ♠ ✐ ❥ ❞ ❥ ❣ ❦ ❤ ❣ ❡ ❣ ❧ ❢ s ♣ q ❤ ✐ ♦ ❣ ❥ ♠ ❞ r q ➬ ➵ ➷ ë Ñ Ð ❐ Ò ➱ ß àáâ á ê é ì ❐ é í î ê ï á å ð ë æ ➘ Ó ñ ➮ Ú ✃ Ï Û ➹ Ó ❐ ➴ ❰ ❮ ❒ Ð ❰ ➶ ❐ ➴ ➘ ➱ ➶ ➘ ✃ ❐ ➴ ç ã Ð æ á ä ç á é ï á è ö å ã í æ á è ï á ä é ê ÷ ú ï é ã ï ð á ó õ ä ö õ á ç ñ í ê ï å ÷ ë ð ñ ò á ê è ð Ú Þ Ý ➷➬ ➘ ❰ Ö ➷ ➮ Ò ➹ ➘ ❰ ➮ ➹ Ò ➬ ➮ ❒ Ù ❐ Ò Ó Ú Ò Ó ➶ Ï ➘ ➶ ➹ ➘ Ñ Ð ❐ Ò ➹ ❮ ➮ ➷ ➱ ➹ Ó ❰ Ó ➹Ô ❒ ➱ ➶ ➹ ➘ ✃ Õ Ó ➹ Ï Ï ❰ ➷ ➶ ❮ Ò Ú ✃ Ð ➹ ❰ ➹ Û ❮ ❰ ➴ Ó Ü ➶ n n 74 LeftChild(i): if (2i+1 < n); return (2i+1) else return nill; RightChild(i): if (2i+2<n) return(2i+2); else return nill; RightSibling(i): if (i != n-1 and i not even) return(i+1); LeftSibling(i): if (i != 0 and i not odd) return (i-1); 3 Parent(i): if (i != 0) return(  (i-1)/2  ); i ❇✼❈❉ ã✮æ .n ❂✳❃ ✱✳✲ i ➁❹➂ IsLeaf(i): return (2i+1>n); ④r⑤ ✮✩✯ ❚✼❯ ❼❹❽ ❦♠❧ t✈✉ i ❺❹❻ i ❇✼❈❉ ✧✩★ 2i+2. ❷❹❸ ü ✼✻ -35 ✜✣✢ ♣rq ⑥✼⑦⑧ v 73 ➃✪➄ ×✦Ø 3 ❛✴❜ ì☎ò Ï✴Ð ❫✮❴ s✴q (1) ❫✮❴ - ➘✴✃ FirstChild(), RightSibling(): ♥✮♦ →✦➣↔ ✈☎✇ ➩➎➫ ❲✦❳ ➘✴✃ ➑➎➒ : ➤➇➥➦ è✴é ø✦ù K th -child(k, v ❚✮❯ á✑û ã✮æç ➌➎➍ ➘✮➷➬ H Parent ➞➎➟ ◗✴❘ ❶❸❷ ã✮äå k -35 ➘✮➴ ❢☎❣❤ ➙✮➛ ⑥☎⑦ ô✪á ➅➇➆ ❝✪❞ ➪☎➶

  37. ➋ ù Ö × Ö ý á Ø Ï â ✃ ü û ö î Ô ú î ÷ ó ïò ð ñ ð ï ëì ê Ò Ò â × Ü è ❐ ✃ ß þ ❐ Ô á × Ï ❰ Ô ß ã ❐ × Ö ý á Ø Ï â ✃ ü Ò Ü Ö Ø à Ô Ñ Ð Ô Ü Û á ß ❐ ✃ Ü â ❮ ❐ß á × ➌ á ❒ Ö × á Ü Õ ß Ü ➯ ã × â ❐ á è ❐ ❰ á à á è ❐ â ❐ ✃ à Ü ❐ × Ï ❐ ç Ð Ñ Ø à ❐ × Ï ❰ ä è ➫ ❐ß è ß ❐ × Ö ý á Ø Ï â ✃ ü Ô Þ è à ä è ❐ ✃ ß þ ❐ Ô á × á ❰ ❰ Ï Ò Ü ã â Ô Ò Ö × Ö ý á Ø â ❐ ✃ ü ❐ß Ô è à ä è ❐ ✃ ß þ Ï Ô Ô á Ô è à ä è ❐ ✃ ß þ ❐ ❰ Þ è ÿ ß ❐ × Ö ý á Ø Ï â ✃ ü ❐ß ❐ß ñ Ü ö ß ❐ × Ö ý á Ø Ï â ✃ ü û ù ð î ú ù ø ö ó ò ï ð ñ ð ï ï ➮ ❰ ➽ ØÛ Ð Õ ❰ Ô Ü ❐ Ò Ö Ï Ü Ô Ú Ö Ò Ù Ø ❰ × Ö Ð Õ ❰ ÏÐ ❐❮❰ Ù ✃ ❐ Ï ã â ❮ ❐ ❒ ❐ ✃ ß à ❐ × ❰ Ô Ü × Ûá Ý Ô Ü Üà ❐ß ✃ Ö Þ Ý ❒ ✃ ❰ ➓ ➟ ➑ ↔ ➞ ➒ ➝ ➜ ➛ ➙ ↔ ↔↕ ➒ ➡ ➑ ➣ → ➐ ➔ ➓ ➒ ➑ ➐ ➏ ➎ ➍ ➠ ➑ ➱ ➸ ➯ ➬➮ ➳ ➚➹ ➶ ➚ ➺ ➾ ➽ ➺ ➳ ➵ ➢ ➫ ➠ ➡ ➩ ➨ ➧ ➠ ↔ ➠ ➦ ➑➥ ➤ á ❮ × ➑➥ ➠ ➡ ➑ ➍ ➌ ➋ ➢ ➤ ➦ ➎ ➠ ↔ ➠ ➧ ➝ ➩ ➡ ➠ ➟ ➏ ➺ ➒ ➛ ➙ ➒ ➞ ↔ ↔ ↔↕ ➓ ➑ ➐ ➣ → ➐ ➔ ➓ ➒ Ö ➑ ➑ ➫ ➨ ➘ à × ➺ Ð Ü ❮ ❐ ❒ Ý Ü × å ➴ Ö Ï Ü Ô ØÛ ä à ❐ á ❰ ❰ ➴ æ ❐ × ➜ à á â Ü -35 3 -35 3 ➭✩➯ ➲✳➳ ➻✣➼ ➪✩➯ ➘➷➴ ➭✩➯ ➲✳➳ Visit(P): ârÔ P â✼é í✼î ôrõö ø✳ù Ñ✼Ò ÓrÔ Procedure Preorder(pointer P): Label(v): Ñ✼Ò /* P is a pointer to the root of a binary tree */ v Visit(P) ApplyOp(op: operation, x,y: numbers): foreach child Q of P, in order, do x <op> y Preorder(Q) op. í✼î ôrõ í✼î function Evaluate(pointer P): integer /* Return value of the expression represented by the tree with root P */ if IsLeaf(P) then return Label(P) else Procedure Postorder(pointer P): x_l = Evaluate(LeftChild(P)) /* P is a pointer to the root of a binary tree */ x_r = Evaluate(RightChild(P)) op = Label(P) foreach child Q of P, in order, do return ApplyOp(op, x_l, x_r) Postorder(Q) Visit(P) 75 76

  38. � ❏ ▲ ◆ ❍ ◆ P ❉ P ❋ ● ❇ ▲ ❈ ● ■ ❍ ● ❵ ▲ ❏ ❄ ❏ ● ❫ ❄ ❏ ❉ ❵ ▲ ● ❉ ● ❇ ❏ ❬ ❫ ● ❄ ❈❉ ❇ ❄ ● ❉ ❪ ❭ ❑ ❍ ❬ ❋ P ❖ ❬ ❩ ❏ ❏ ● P ❏ P ❖ ❬ ❩ ❏ ● ❉ ● ❏ ✁ ◆ ● ❬ ❉ ❬ ❋ ● ❇ ❍ ● ❵ ▲ ◆ ▲ ❋ ❍ ❉ ❇ ❬ ❍ ❴ ❬ ❫ ❈ P ❍ ● ❉ ● ❏ ❑ ❬ ❫ ● ❄ ❉ ❈ ❇ ▲ ❄ ❉ ❪ ❭ ❍ ❈ ❄ ✪ ♦ ♥ ♠ ✐ ❤ ❣ ✫ ✰ ✮ ✭ ✫ ✢ ♠♣ ✩ ✦ ✣ ★ ✦ ✥ ✔ ✕ ✜ ✛ ✚ ♥ q ☞ t ✾ ✲ ✽ ✼ ✸✻ ✹ ✺ ✹ ✸ ✲ ✱ s ✉ r q ♠♣ ♥ ♦ ♥ ♠ ❦ ❥ ♠ ♥ ♦ ✔ ✔ ❫ ❏ ✆ ☎ ✄ ✂ ✁ � ❍ ● P ❞ ▲ ❄ ✞ P ❝ ● ▼ ❙ ❏ ● ❉ ❬ ❍ ❴ ❬ ✝ ✟ ✙ ✞ ✘ ✝ ✗ ✖ ✝ ✕ ✔ ✓ ✝ ☞ ✒ ✑ ✠ ✏ ✎ ✍ ☞ ✌ ☞ ✟ ✞ ✝ ☛ ✡ ✆ ❏ ▲ ❏ ✶ ▲ ❑ ❏ ❍■ ● ❊❋ ❈❉ ✾❂ ❀ ❁ ● ✶ ✳ ✲✾ ✽ ✼ ✸✻ ✹ ✺ ✹ ✸ ❇ ❉ ✱ ■ ◗ ❏ ❘ ❆ ❑ ◗ ❋ ● ❊❋ ❈❉ ❍ ❑ ❑ ◆ P ❖ ◆ ❍ ❄ ■ ▼ ● ✲ ✫ ❊❋ ✝ ✑ ✏ ✎ ✍ ☞ ✌ ☞ ✟ ✞ ☛ ✒ ✡ ✆ ✠ ✟ ✞ ✝ ✆ ☎ ✄ ✂ ✞ ☞ ✰ ✔ ✢ ✩ ✦ ✣ ★ ✔ ✕ ✜ ✛ ✚ ☞ ✝ ✔ ✙ ✘ ✝ ✗ ✖ ✝ ✕ ✔ ✓ ❈❉ ❍ ● ❉ ❪ ❭ ❑ ❍ ❬ ❋ P ❖ ❬ ❏❩ ● ● ❄ ❍■ ▲ ◆ ❍ ❑ ❄ P ❋ ● ❨ ❲ ❉ ▲ ✲ ❈ ❫ ❄ ❏ P ▲ ❑ ❄ ❍ ❴ ❬ ❫ P ❇ ❍ ● ❉ ● ❇ ❏ ❬ ❫ ● ❄ ❉ ❈ ✸ ❏ ❱ ✾ ▲ ■ ✼ ❍ ▼ ■ ❄ ✲ ❉ ❚ ❍ ◆ ❖ P ◆ ❑ ❍ ❙ ✴ ◆ ❯ ✲ ✲ ✸ ❳ ✶ 78 3 11 10 7 9 8 ✳✷✶ ✳✵✴ ✢✤✣ r❧st -35 ❥❧❦ 77 /* P is a pointer to the root of a binary tree */ 3 ■❜❛ ✭✯✮ ◆✬▲ if P = NULL then return Procedure Inorder(pointer P): ■❢❡ ✪✬✫ Inorder(P->RC) Inorder(P->LC) ✿✯❀ ◆✬▲ ✥✧✦ Visit(P) ✢✤✣ ❱❅❲ else -35 ✳✷✶ ✳✵✴ ❆✯❇ ❆✯❇ ❆✯❇ ❃❅❄ ❃❅❄ ❃❅❄

  39. ✈ ø í å ✂ ù è ÷ ð í è ñ ä ✌ ☞ ☛ ä é ü ë í ü å þ ø ó è ç ✏ ú ó è ä ì æ è î ÷ í ó è ó è ÷ è ✍ è é ê å î ó ð ô ä ó ✇ í ê ñ êë õ é è â Þ Û ➦ Õ ➑➎ ➇ ö ÷ ç ð å ÿ ✁ � íÿ õ ð ê ò þ õ ê ý í è ä ü ç û ñ ø ê ä ➌ ✧ ✭ ✬ ✚ ✥ ✫ ✛ ✪ ✩ ★ ✥ ✚ ✥✦ ✜ ✛ ✚ ✤ ✣ ✙ ✢ ✜ ✮ ✯ ✚ ✿❀ ✺ ❆ ✾ ❅ ❄ ✷ ❃ ❂ ❁ ✽✾ ✰ ✭ ✮ ✵ ✴ ✳ ✭ ✥ ✭ ✲ ✚✱ ✛ ✙ å ê ✓ ê äå ù ñ ê è é ø ç î ü ✍ î å ù è êë é è è ï ✘ é ✗ ✖ ✕ ✔ ú õ é ô ä ñ å ð ù è êë ÷ õ ÷ ê å ✓ Ô è ➊ ➻ ➵ ➽➾➚ ❼ ➺ ➲ ➼ ❶ ➺ ⑤ ➹ ➵ ➳ ➲ ❽ ➈ ❾ ➫ ➩ ⑥ ➽➾ ➡ ➹ ➺ ➲ ➼ ➵ ➚ ➽ ➮ ➘ ➽➬ ➚ ➷ ➴ ➼ ➘ ❻ ➺ ➼ ➵ ➝ ➔ ❾ ➇ ➣ ➀ ➁ ⑤➂ ➎ ➑ ➃ ❾ ➌ → ❶ ➊ ➈ ➋ ❾ ➄ ➅ ➆ ➓ ↔ ➨ ➦ ➤ ➔ ➡ ➠ ➟ ➧ ➠ ➛ ➤ ➒ ➥ ➤ ➢ ➔ ➡ ➠ ➞➟ ❿ ⑤ ➽ ➾➚ ➵ ✃ ⑦ ⑧ ④ ⑨ ➲ ➷ ➻ ➼ ❰ ➺ ➚ ➽ ➪ ❐ ➯ ⑤ ⑥ ⑤ ❐ Ó ➋ ➊ ➉ ➈ ➇ Ô ➈ ① Ñ ② ➊ Ò ➈ ③ Ð ④ ➘ ⑩ ➻ ❶ ➾ ➽ ➺ ✃ ⑦ ➱ ❶❷ ❸ ➯ ➾ ➽ ❹ ➺ ❺ ➽➾➚ ➹ ➶ ➻ ➲ ➲ ➯ ➳ ➲ ❐ ➼ ➺ ➪ ➼ ➼ ➽ ➯ ➸ ✃ ➾ ❿ ⑥ 80 /* Initially Q points to the root of the tree to be traversed */ 3 while (P != NULL and Tag(P) = 1) do Procedure LinkInversionTraversal(pointer Q): if (P == NULL) then return while (Q != NULL) do ascend from right ascend from left descend to right descend to left Tag(Q) = 0 Tag(Q) = 1 Visit(Q) Visit(Q) Visit(Q) ✹✻✺✼ ✶✸✷ P = NULL; else while (1) -35 79 æ✯ç 3 ✂☎✄ ù❮ú ➏✯➐ ➏✯➐ ï ✞✎ ➍✬➎ æ✬õ ➍✬➎ n), n ➪✯➶ ì✷ò ➉✧➊ ➇✤➈ ßà❅á æ✯ç ➛✵➜➝ ï✯ð ➘❮❒ Ü✵Ý ↕❅➙ ì✷í Ù✯Ú ➸✯➯➺ -35 ÖØ× æ✯ç ✟✡✠ ➔✯→ Ï✯Ð ì ✞✍ ✆✞✝ ã✷äå å ✒✑ ➭❅➯ ➒✷➓

  40. ❇ ➞ ➄ ➒ ➌ ➋ ➑ ➍ ➟ ➃ ➍ ➒ ➄ ➒ ➆ ➉ ➡➢ ➤ ➥ ➦➧ ➍ ➄ ➩ ➍ ➍ ➈ ➈ ➟ ➍ ➑ ➍ ➉ ➝ ➋ ➄ ➒ ➆ ➉ ➎ ➁ ➆ ➈ ➒ ➨ ➫ ➆ ➁ ➉ ➋ ➑ ➆ ➉ ➇ ➐ ➇ ➐ ➁ ➔ ➄ ➁ ➉ ➏ ➒ ➍ ➄ ➯ ➇ ➇ ➋ ➁ ➐ ➊ ➒ → ➓ ➁ ➄ ➓ ➓ ➊ ➋ ➎ ➙ ➍ ➈ ➔➭ ➌ ➐ ➑ ➌ ➑ ➉ ➣ ➎ ➀ ➈ ➔ ➄ ↕ ➍ ➇ ➎ ➁ ➐ ➊ ➒ ➁ ➓ ➔ ➌ ➄ ➁ ➏ → ➣ ➇ ➉ ↕ ❈ ➔ ➉ ➏ ➁ ➏ ➐ ➄➆ ➉ ➆ ➊ ➌ ➉ ➙ ➣ → ➉ ➇ ➋ ➁ ↕ ➟ ➟ ➍ ➈ ➓ ➊ ➎ ➄ ➙ ➍ ➈ ➓ ➛ ➠ ➆ ➇ ➏ ➈ ➎ ➆ ➍ ➑ ➍ ➉ ➍ ➝ ➄ ➒ ➉ ➍ ➞ ➉ ➆ ➄➆ ➇ ➁ ➐ ➔ ➄ ➁ ➋ ➉ ➒ → ➆ ➄ ↕ ➇ ➁ ➐ ➊ → ➋ ➞ ➉ ➆ ➄➆ ➯ ➁ ➔ ➭ ➎ ➊ ➐ ➆ ➆ ➯ → ➛ ➵ ➌ ➁ ➉ ➸ ➆ ➋ ➎ ➆ ➏ ➒ ➁ ➃ ➄ ➆ ➌ ➸ ↕ ➝ ➊ ➆ ➄ ➍ ➄ ➉ ➊ ➍ ➓ ➈ ➇ ➁ ➐ ➊ ➒ ➁ ➓ ➛ ➓ ➍ ➍ → ➈ ➎ ➁ ➒ ➍ ➈ ➓ ➄ ➎ ➒ ➆ ➒ ➉➊ ➍ ➝ ➓ ➆ ➑ ➋ ➒ ➉ ➎ ➆ ➆ ➞ ➎ → ➇ ➉ ↕ ➒ ➯ ➁ ➁ ➄ ➳ ➍ ➇ ➁ ➐ ➔ ➒ ➒ ➁ ➎ ➁ ➄ ➋ ➒ ➍ ➈ ➊ ➋ ➙ ➆ ➍ ➏ ➒ → ➄ ➁ ➄ ➇ ➍ ➄ ➉ ➇ ➉ ➒ → ➄ ➁ ➄ ➇ ➍ ➇ ➆ ➍ ➒ ➁ ➒ ➆ ➞ ➎ ➃ ➄ → ➏ ➙ ➏ ➍ ➒ ➍ ➄ ➑ ➌ ➍ → ➞ ➝ ➎ ➎ ➁ ➄ ➋ ➒ ➍ ➈ ➊ ➋ ➊ ➏ ➎ ✇ ❶ ⑥ s ⑤ ⑥ t s ⑩ q ⑦⑨ ⑤ ⑧ ❏ ⑦ ⑤⑥ ✇ ④ ⑤ ④ ❑ s ❷ t ● ❘ P ◆ ◆❖ ❏ ■ ❍ ▼ ▲ ❑ ■ ❏ ■ ❍ ● ❋ ❍ ❊ ❉ ❈ ❇ ● ✉ ■ ❞❡ ❱ ◆ ❱ ❭ ❪ ❫ ❲ ❱ ➌➍ ❢❣ ❍❩ ❤ ✐ ❥ ❵ ❦ ❧ ❡ ♠ ❜ ❬ ❨ t P s qr ▲ ▼ ❍ ■ ❏ ◆❖ ◆ ◗ ❳ ❘ ❙ ■ ❚ ◆ ❍ ❯ ❱ ❲ ❍ ❙ ◗ ❚ ➉ ❉ ➍ ➑ ➐ ➌ ➁ ➍ ➏ ➎ ➇ ➉ ➍ ➇ ➋ ➆ ◆ ❋ ❿ ● ➈ ➁ ❢ ➉ ➇ ➆ ➄ ➆ ↔ ❿ ➣ ➎ ➒ ➐ ➄ → ➒ ➄ ➈ ➍ ➁➓ ➒ ➊ ❝ ❊ ❽ ❭ ❫ ❲ ❱ ❜ ❱ ◆ ❱ ❬ ❺ ❝ ❍❩ ❡ ❦ ❧ ❻ ❨ ❧ ❍ ❾ ❥ ❻ ❯ ❪ ❝ ❱ ❦ ❽ ❲ ❍ ❳ 82 ➃➅➄ . ➃➅➔ 3 →➜➛ ➃➅➄ ➇✸➉ ➐✸➑ bit NULL ❥✻❼ ➇✸➌➍ ➃➅➄ NULL. LC ➇✸➉➊ ❸✸❹ ➇✸➈ →➜➲ -35 ➃➅➄➆ RC ➀➂➁ descend to right:      81 RC      ascend from right: RC > − > Q Q P 3 −      Q P P ←      ←      RC      RC > − > P Q Q −      Q P P                LC LC ascend from left: descend to left: ❛✻❜❝ > > − − P P Q ❴✸❵ Q Q P ②☎③           ← ← ✈①✇           LC -35 LC > > > ♥♣♦ − − Q Q P Q P P          

  41. ➺ Ï ❒ ➶ ❮ ➬ ➪ ❰ Ð ✃ ➪ Ñ Ò ➻ Ô Ï ❐ ➱ Ï ➹ ➼ ➽ ➾ ➚ ➪ ➶ ➘ ➬ ➚ ➴ ➷ ➪ ➶ ➹ ➬➮ ➬ Õ ➺ Û Û ÜÝ Þ î à á â Ù Ý Þ î à ã ä ð Ú Ø Ö Ûê × Ð Ï è Ý é ë ñ ê ì í î Ú á é ➻ ➪Ó ç Ò ➬ ➪ ❰ Ï Ð ➪ Ñ ➪Ó ➶ Ô Ï ➬ Ï Õ Ö × ❮ ❒ Ï ➚ ➼ ➽ ➾ ➚ ➪ ➶ ➘ ➴ ❐ ➷ ➪ ➶ ➹ ➬➮ ➬ ➱ ✃ Ð ➹ Þ ÜÝ á â à ß Ý ß Þ Û Ú Ù Ø à Û Û -35 3 -35 3 ï✻ð ãåä â✻æ â✻æ procedure ThreadedInsert(pointer P,Q): /* Make node Q the inorder successor of node P */ P − > RC child Q         ( )         function InorderSuccessor(pointer N): pointer ← Q − > LC thread P         ( ) /* returns the inorder successor of node N, or if       Q − > RC P − > > RC         N has none */ if Q->RC is not a thread then P = N->RC if (P = NULL) then return NULL R = InorderSuccessor(Q); else if (P is not a thread) then R->LC = (thread) Q while (P->LC is not a thread or NULL) do P = P->LC return P 83 84

  42. ò ✻ ❂ ❆ ✾ ✿ ❃ ❁ ❆ ❉ ❀ ❉ ✹ ▼ ✹ ❉ ❍ ❀ ❄ ❉ ✶ ✹ ❍ ❆ ❁ ✶ ✻ ❀ ✶ ✾ ▲ ✾ ❆ ✸ ✻ ❅ ✾ ✿ ❃ ❁ ❂ ❅ ❉ ❍ ❆ ✾ ✹ ❁ ✶ ❇ ✾ ❆ ✼ ó ✾ ✿ ✶ ✻ ❑ ✺ ❉ ✹ ❀ ❑ ❉ ✹✺ ❞ ❱ ❯ ❨ ❩ ❳ ❯ ❬ ◗ ❘ ✵ ❵❛ ❜ ❝ ❝ ❯ ❡ ❢ ❣❤ ✐ ❥❦ ❧ ♠ ♥ ♦ ♣ q r s ❘ P ▼ ❇ ❁ ◆ ✹ ✻ ❆ ✾ ✿ ❍ ❆ ✾ ✹ ✸ ❁ ✾✿ ◗ ❉ ✿ ❆ ✾ ❖ ❊ P ◗❘ ❙ ❚ ❯ ❱ ❲ ❳ ✺ ❑ ❆ ✑ ø ✡ ☛ ✝ þ ✝ ✌ ✍ ✞ ✝ ✎ ✓ ✗ ✟ ✕ ✛ ✗ ✜ ✙ ✗ ✕ ✢✣ ✤ ✥ ✛ ✦ ✠ ø ✔ ù ô õ ö ÷ ø ù ú û ÷ ü ý ø ú ✞ þÿ þ � ✁ ✂ ✄ ù ☎ þ ø ✆ ✝ ✓ ☞ ✓ ❃ ❁ ❂ ✶ ❅ ✿ ❂ ✸ ❁ ✹ ✾ ✶ ❆ ✼ ✶ ❇ ✴ ● ❆ ✼ ❍ ✧ ✸ ✹ ✾✿ ❅ ★ ✤ ✩ ✪ ❁ ✮ ✼ ■ ❉ ❀ ✴ ❁ ✹ ❀ ✸ ❉ ❉ ✹✺ 85 Deque 3 ❈✖❉❋❊ ✻✽❏ ❄✽❅ ❙✷❴ ✘✚✙ ✻✒❀ ❭❫❪ ✔✖✕ ✲✚✳ C, D, E, F H, I, J, K ✻✽✼ <empty> ✯✖✰✱ B, C, D ✏✒✑ D, E, F E, F, G G, H, I -35 I, J, K K, L F, G J, K ✵✷✶ ✫✭✬ L

  43. t ✮ ✮ ✹✾ ✲ ✮ ✲ ✺ ❄ ✲ ✼ ✴ ✻ ✰ ✷ ✸ ✺ ✺ ✮ ✾ ✱✲ ❄ ❁ ❀ ✰ ✮ ✳ ✰ ✲ ✯ ✮ ✬ ✫ ✪ ✾ ✹ ✮ ✹ ✷ ✯ ✺ ✮ ✼ ✮ ✲ ✹ ✷ ✷ ✹✾ ✯ ✹ ✼ ✹ ✸ ❆ ✲ ✰ ✯ ✮ ✷ ✮ ❂ ✮ ✷ ✺ ✰❇ ✷ ✹ ✲ ✱ ✷ ✲ ✿ ✯ ✼ ❁ ✹ ✰ ✷ ❂ ❄ ✼ ✴ ✻ ✷ ✲ ✸ ✺ ✮ ✾ ✹ ✯ ✷ ✰ ✹ ✺ ✬ ✮ ✳ ✱✲ ✰ ✯ ❋ ✫ ✾ ✪ ✾ ✹ ✲ ✰ ✷ ❂ ✸ ✮ ❅ ✯ ✲ ✹ ✯ ✷ ✾ ✹ ✹ ✼ ✼ ✹ ✸ ❆ ✱ ✹ ✲ ✮ ✮ ❂ ✱✲ ✮ ✷ ✺ ✮ ❇ ✰ ✹ ✷ ✺ ✲ ✿ ✯ ✷ ✰ ✷ ✲ ✱ ✲ ❀ ü þ þÿ ú ù ø ý ÷ ✁ û ú ù ø ÷ ô ó � ✂ ñ ✟ þ ✝ ☛ ✡ ø ✠ ø ✄ ✞ ✝ ✆ ø þ ☎ ù ò ð ☞ í ï Ü âæ Ù ß Ûï è ß æ â ß ç â æ ì à è ✉ Ý â ã â í Ø ß ï í à ï Ü æ ì î Û ✝ ✌ ❅ ❀ ✮ ✳ ✹ ✷✵ ✰ ❁ ✾ ✰❂ ✹ ✼ ✿ ✸ ✾ ✱ ✹✾ ✲ ❃ ✹✽ ✵ ❁ ✷ ✼ ❀ ✰ ✮ ✴ ✪ ✳ ✱✲ ✰ ✯ ✮ ✬ ✫ ✲ ✼ ✍ ✏ ✤ ✚ ✓ ✗✘ ✣ ✢ ✛✜ ✛ ✕✚ ✗ ✙ ✗✘ ✕✖ ✝ ✞ ✥ ✩ ✴ ✲ ✻ ✰ ✷ ✸ ✺ ✮ ✹ ✷✸ ✤ ✰ ✮ ✳ ✱✲ ✰ ✯ ✪ ✰ ❁ è ❵ ❚ ❣ ❖ ❯ ❤ ❛ ❴ ❯ ❙ ❖ ❖ ❯ ❭ ❣ ❚ ❭ ◗ ❙ ❥ ❭ ❤ ❦ ❘ ◗ ❳ ❖ ❘ ❘ ❪ ❭ ❯❬ ❘ ◗ ❩ ❝ ❭ ❡ ❘ ❩ ❘ ❪ ❭ ❬ ❯ ◗ ❘ ❩ ❘ ◗ ❯ ❭ ◗ ❝ ◗ ❩ ❖ ❙ ❪ ❲ ❖ ❡ ❯ ❭ ❢ ❘ ❡ ❙ ❭ ▲ P ❘ ◆ ❙ ❢ ❩ ◗ ❘❫ ◗ ❯ ❭ ◆ ▼ ❯ ❣ ❭ ♠ ❳ ❲ ❚ ❖ ✐ ❚ ❪ ❛ ❘ ❫ ❘ ◗ ❘ ❪ ❯❬❭ ◗ ❖ ❩ ❭ ❱ ❢ ❡ ❭ ❩ ❧ ❵ ❙ ❖ ❩ ❯ ❥ ❧ ❴ ❙ ❖ ❚ ◗ ❘ ❥ ❪ ◗ ❯ ❭ ◗ ❝ ❴ ✐ ❢ ❡ ❯ ◗ ❭ ❚ ❘ ❖ ✐ ❯ ▲ ❏ ❙ ❖ ❯ ❭ ❱ ❘ ❄ ✾ ❍ ✪ ✹ ✲ ✱ ✸ ✹ ✯ ❂ ❅ ✸ ✷ ✸ ✮ ❁ ❋ ✰ ✼ ✮ ✹ ✸ ❆ ✱ ✿ ✸ ❂ ✱✲ ✹ ✲ ✰ ✮ ✵ ✴ ✳ ❃ ✼ ✹ ✸ ✱ ● ✲ ✹ ✼ ✹ ❆ ✹ ✱ ✹ ✲ ❅ ❁ ✷ ✼ ✲ ✰ ✷ ❅ ❀ ✹ ✱✲ ✸ ✾ ✹ ❂ ✷✸ ❇ ✾ ✹ ✯ ✸ ● ✺ ✮ ✼ ✪ ❚ ✱✲ ■ ✰ ✮ ✵ ✴ ✳ ✰ P ✯ ❋ ✬ ✫ ✪ ❉ ❃ ▲ ◗ ✲ ❯❬❭ ❜ ❴ ❘❫ ◗ ❨❙ ❪ ❘ ❘❙ ◗ ❩ ❨❙ ◗ ❳ ❯ ❖❚ ✰❂ ✮ ❋ ✰ ✰ ✷ ✸ ✺ ✮ ✷✸ ✮ ✹ ✵ ✴ ✳ ✲ ✱ ✰ ✯ ✲ ❂ ✳ ❀ ✹ ✷✵ ✰ ❁ ✰ ✷ ❂ ✹ ✮ ✼ ✹ ✸ ❆ ✱ ✿ ✸ í Ô ß ➣ ➾ ➜ ➛ ➏ ➚ ↔ ➑ ➛ ➲ ↔ ➣↔ ➝ ➓ ➪ ↔ ↔ ➞ ➣ ➠ ➓ ➢ ➟ ➣ ➑ ➒ ↔ ➩ ➣ ➟ ➝ ➞ ➵ ➣ ➡ ↕ ➝ ➡ ➜ ➝ ➞ ↕ ➞ ➞ ➑ ➜ ➝ ➟ ➦ ➠ ➝ ➟ ➡ ➑ ➩ ➣ ➵ ➢ ↕ ➜ ➞ ➝ ➡ ➟ ➣ ➠ ➟ ➓ ➢ ➝ ➟ ➝ ➟ ➡ ➢ ➞ ➣ ➜ ➞ ➞ ➑ ➔ ➝ ↔ ➲ ➘ ➓ ➝ ➎ ➔ ➒ ➡ ➢ ➡ ➛ ➥ ➞ ➜ ➑ ➔ ➔ ➝ ➞ ↔ ➭ ➙ ➩ ➠ â ➾ ➶ ➝ ➩ ➣ ➢ ➣ ➝ ➡ ➛ ➝ ➢ ➪ ➞ ➒ ➓ ➟ ➟ ➝ ➹ ➣➙ ➏ ➐ ➑ ➟ ➣ ➡ ↕ ➛ ➛ ➢ ➓ ➜ ↕ ➣ ➝ ➞ ➑ ↔ ➦ ➜➻ ➺ ➣ ➝ ➧ ➣ ➸ ➓ ➔ ➡ ➔ ➦ ➝ ↔ ➑ ↔ ➓ ↕ ➞ ➢ ➝ ➑ ➑ ↔ ➣ ➒ ➲ ➠ ➒ ➟ ➨ ➠ ➢ ➝ ➡ ➛ ➝ ↔ ➩ ➑ ➒ ➢ ➣ ➤ ↔ ➣ ➧ ➳ ↔ ➢ ➟ ➑ ➑ ➔ ➝ ➲ ↔ ➛ ➦ ➭ ➠ ➟ ➫ ➑ ➢ ➝ ➝ ↔ ➩ ➣ ➝ ➑➢ ➤ ➔ ➢ ➝ ↔ ➲ ➛ ➑ ↔ ➑ ➓ ➓ ➣ ➡ ➝ ↔ ➣ ➒ ➩ ➤ ↔ ➠ ➢ ➡ ↔ ➑ ➑ ↔ ➢ ➳ ➑ ➓ ➔ ↕ ➤ ➥ ➞ ➢ → ➒ ➢ ➑ ➤ ↕ ➒ ➸ ➵ ➑ ➑➢ ↔ ➩ ➣ ➡ ↔ ➽ ➏ ➤ ➑ ➞ ➭ ➼ ➤ ➝ ➢ ➻ ➡ ❹ ➡ ➝ ↔ ➞ ➩ ↔ ➑ ➤ ➑ ➛ ➜ ➒ Ó ❺ ❷ ➩ ➒ ➧ ➔ ➻ ➑ ➡ ➣ ➜ ❷❸ ➝ ➡ ↔ ➧ ➔ ↕ ➛ ➪ ➢ ➥ ➛ ➡ ➜ ❾ ➜ ➩ ↔ ➑➢ ➡ ➝ ⑥ Ñ Ò ➞ ➟ ➣➢ ➡ ❻ ➔ ❼ ⑦ ➤ ➝ ↕ ➠ ❽ ➡ ➜ ❷ ➑ ➣➢ ➡ ➛ ➝ å â å ë æì æ ß ì æ Û Ù ã Ý ä â ⑥ Û ÛÜ ß ✈ à ß Þ Ý Ú ì ✇ ① Õ ⑤ Ô æ ê ê ➡ Ù ➾ Ô ⑧ Õ ⑦ Ú ⑥ ❶ ÛÜ ÝÞ ß ⑩ â Û ã è ç ç Ø ⑦ é è ß ⑧ Ý á ⑨ â Ûæ ⑤ ä ➜ Ð ❿ ➅ ➱✃ ❷ ➮ ➷➬ ➴ ❿ ➆ ➬ ➫ ➇ ➀ ➑ ❿ ➔ ❿ ❒ ↔ ➣ ⑥➃ ➻ ➒ ➟ ➑ ➹ ➙ ❮ ➓ ➔ ➔ ➝ ❰ ➄ ➝ ➲ ➙ ➢ ➢ ➝ ➡ ➛ ➜ ➝ ➪ ➣ ➩ ➜ ➝ ➡ ➑ ➞ ➠ ➩ ➛ ➑ ➈ ➜ ➔ ➝ ➞ ➢ ➡ ➝ ↔ ➝ ➻ ➓ ➔ ➜ ➓ ➝ ➠ ➂ → ➾ ➡ ➑ ➁ ↔ ➜ ➞ ➪ ➑➢ ↔ ➩ ➣ ➟ ➑ ➑ ➥ ↔ ➧ ➡ ↔ ➧ ➣ ➛ ➾ ➡ ➟ ↔ ➣ ➀ → ⑥ ➛ → ➡ ↔ ➢ ➛ ➤ ➝ ➞ ➲ ➝ ↔ ➒ ➲ ↔ ➝ ➔ ➢ ➑ ➢ ➑ ➲ ➝ → ➡ ➢ ➑ ➞ ➑ ➛ ➢ ➝ 88 S ∅ . S 3 ❱✭❲ ❞✶❙ F ✦★✧ S ∩ T S=T S x x x x ∈ S. S \ T S ❁✑❄ true ✺✶✸ ◆✑❖ false. S ∪ T ❈❊❉ true true ▲✑▼ ✴✶✵ ✴✶✵ ■❑❏ S x ∉ S. ✴✶✵ ❢✶✐ S false S. ✫✭✬ Intersection(S,T ❈❊❉ MakeEmptySet S. Difference(S,T ✫✭✬ IsEmptySet(S ✫✭✬✮ Member(x,S ❵✭❛ S Union(S,T Delete(x,S Iterate(S,F ❵✭❛ Equal(S,T Insert(x,S ✒✔✓ Size(S ✎✑✏ false. ❈❊❉ S õ④ö - , S 87 multi-sets). ➞➯➭ 3 x ↔➯➵ x ∉ S. Sets) à✒å ➓✽↔➧ - ➔✚➝ x ∈ S»; ë✽æ x ∈ S ë✽î ➌✒➍ ↔➯➵ ➉✖➊➋ ↕✽➙ à✽á ➔✚➝ ➔✚→ S ❐✒❒ ➔✚Ï Ø✷Ù Ø✷Ù ②④③ - Ö✷× Ö✷×

  44. ♥ ❄ ✾ ✿ ✻ ✼ ✹ ❀ ✽ ✾ ❂❁ ❃ ✳ ✸ ❅ ✹ ❆ ❄ ❇ ✶ ÿ �❋ ü ã ✻ ✷ ● ✖✭ ✗ ✧ ✜ ✖ ★ ✩ ✪ ✖ ✫ ✬ ✮ ✷ ✩ ✜ ✩ ✯ ✰ ✱ ✪ ✩ ✵ ✶ æ ó ✥ ❏ Ý î á à ä Þ é ✞ à ■ ❑ ë Ý á å á à ï ó ä Þ é é æ ï á Ý è é â ✞ ✟ ß é ✞ æ í ï Ý Þ ✟ ß ✞ æ è ä Ý ❍ á ✦ ✤ à æ ê Ý ê æ î ÷ ß ✟ ß ã è æ è ã æ é ✞ â å Þ ✠ Ý õ ã î � é ì å ÷ ç Ý á þ ñ ÿ ☎ ë ✆ ç â å â î ÷ ß æ ♦ ✞ á ô ✣ ✖ ☛ � ☎ ☞ ✌ ✍ ✎ ✏ ✑ ✕ ✗ à ✘ ✙ ✕ ✚ ✛ ✖ ✗ ✘ ✜✢ ✜ á ✞ ß å æ ✞ á õ ä Ý á ✞ æ è ✡ Ý Þ Ý õ ø ï à á ø ä ÷ ✡ ✞ ▲ ã ❩ Þ é ✞ à ❏ ❨ ❏ ❨ ❏ ❨ ï â à õ ❏ ◆ æ î á é ✞ æ þ ß ÷ ✂ è õ ñ P ❘ ◗ ❉ ❙ ÿ � ❊ ÷ ◗❚ ❯❱ ✂ ❑ è ß ê Ý ê Ý î î ✡ ✞ ❏ é ✞ ô Ý ❍ à å í ó õ ❴❵❛ ß ❜ ❝ ❫ ❣ ❞ ❫ ❛ ❥ ✐ ❦ ✟ ✡ Ý ï ● æ õ ✞ æ è ä Ý ❍ á æ á ë ã å ÷ ã Ý á ß à Ý à æ õ ê Ý ó ê á ã ä ý é è ß ê ê ä Ý î ÷ ß â ▼ ñ ◆ à é ý ã æ ö â ä à ê ó ã ï à ▼ ❏ è à ßà î á ï ó þ Þ é ✞ à ▲ ✞ áí ó ß ✡ î ÷ ßà ï à ✞ ý ✞ æ ✞ ✞ å ô ã æ Ý á é à ✡ ✟✡ à Ý Ý à Þ à ❖ å Þ é ï æ ß ✞ á ✞ é ✞ â ä Þ é ✞ à ê á à á õ Ý ➸ ➧ ➞ ➟ ➪ ➩ ➥ ➯ ➤ ➙ ➳ ➶ ➚ ➥ ➢➤ ➟ ➙ ➜➥ ➦ ➟ ➤ ➟ ➠ ➥ ➟➠ ➠ ➞ ➙ ➢➤ ➞ ➥ ➙ ➤ ➟ ➩ ➙ ➳ ➯ ➤ ➤ ➥ ➞ ➟ ➢ ➤ ➟ ➙ ➜➥ ➦ ➟ ➢ ➨ ➥ × ❮ Ñ Õ ❰ Ö Ñ Ð × ✃ Ø ✃ Ó ÑÙ ❮ ❰ Ó Ð Ñ áâ ã ä â å Ô ø ➯ ➘ ➹ ➟➠ ➥ ➦ ➨ ➛ ➙ ➽ ➨ ➛ ➥ Ï ➪ ➲ ➟ ➡ ➴ ➷ ✃❐ ❒ ❮ Ñ Ò ➼ ➻ ç ➁ ✇ ❻ ❼ ✇❽ ❾ ❹ ⑥ ❹ ❿ ➀ ❺ ❹ ❹ ➆➇ ➈➉ ➊ ➈ ➆➋ ➌➍ ➃ ➎ ➏ ❺ ❸ ➅ ⑤ ♣ q r ✈ ✇ ① ② ③ ✈ ④ ✇ ✇ ① ② ⑥⑦ ⑥ ⑧ ⑨ ⑩ ❶ ① ❷ ⑥ ➈➉ ➋ ➡ ➩ ➤ ➫ ➤ ➛ ➤ ➫ ➡ ➲ ➞ ➛ ➩ ➙ ➙ ➳ ➵ ➡ ➭ ➙ ➸ ➤ ➛ ➺ ➫ ➯ ➭ ➐ ➛ ➑ ➌ ➔ ➐ ➊ ➈ → ➆ ➣ ↔ ➜ ➙ ➟➠➡ ➢➤ ➟ ➙ ➜➥ ➨ ➩ ➥ ➤ ➫ ➨ æ Ð æ à é ì å ÷ ç Ý á î á ã ã ä â å æ ç æ å Þ à á Ý ø è ñ ö Ý ê æ î ÷ ß æ Ý ø ß ï ä Ý á ê á å ë ð Ü ì ï î á é ì æ á í Ý Þ æ Ý í ï ä Ý á ê Þ ð é ì æ Ý å ÷ ð ì æ á à ù é ì Ý ú ø ý Ü û ñ ü ß ê Ý ß è ã ó æ á é Ý á ê Ý áí æ ì æ ï ì á ê æ ë Ý ß ä Þ é ò Þ Ý ä ð ñ ï Ý î è ì á î ó ì æ æ Ý á ì ã Ý æ Þ è ßà æ á è î æ à Þ ßà ê é á áí Ý Ý è ë ß Þ æ æ ê 90 Þ✭ß ❭❬❪❫ 3 Delete: ✿✄❁ Þ✭ßà ❈✺❉❊ ❲✴❳ Insert, ; (n) ✼✺✽ (n) (n) LookUp(): ❤❢✐ ✷✺✹ LookUp, Delete(): Insert(): ÷ ❬✡ ❈✴❙ ❞❢❡ ✒✔✓ - ✲✴✳ ☛✴◗ Þ✭ßà MakeEmptySet, IsEmptySet, 89 S. 3 . - ✁✄✂ ô✭õ ➒★➓ ❰✭ÏÐ LookUp ô✭õ ➦✭➧ o LookUp(K,S): ➽✭➾ o Insert(K,I,S) Þ✭ßà ➬➱➮ o Delete(K,S) ∈ S Ü✶Ý Insert, Delete K,I S Min(S): ➝✑➞ ✃ÛÚ s✉t ➄✔➅ - ✁✄✂ o ➂✑➃ ✝ ✭à ↕✶➙

  45. ❧ ➎ ❺❻ ❿ ➀➁ ➂ ➀ ➃ ➄ ➅ ➈ ➉➊ ➏ ➇ ➏ ➈ ➌ r ↕ ➦ ➧ ➤➥➦ ➛ ➢ ➞ ➜ ➒ ➞ ➛ ➣ → ➔ ➓ ❹ ⑦ ➩ ❢ ❡ ❣ ⑧ ❥ ① ❧ ❣ ❥ ❦ ① ❶ ⑩ ❡ ❣ q ♦ ⑧ ❦ ❢ ❡ ♥ r ❥ t ❣ ❥ ❥ ① ♥ ② ❤ ❦ ➝ ➝ ✈ ➦ ➝ ➛ ➯ ➳ ➼ ➻ ➛ ➹➘ ➭ ➥ ➺ ➝ ➩ ➦ ➚ ➴ ➯ Ï ↕ Ð ↕ Ð ➩ ➤ ❮ ➷ ❒ ➘ ➬ ➘ ➮ ➬ ➛ ↕ ➫ ↕ ➦ ➩ ➳ ➠ ➝ ➛ ➧ ↕ ➲ ➦ ➯ ➦ ➛ ➭ ➯ ➛ ➠ ➵ ➦ ➡ ➨ ➤ ➵ ➞ ➳ ➵ ↕ ➥ ➢ ➵ ➛ ➯ ♥ ⑥ ➨ ❦ ❢❣❤ ✐ ❣ ❥ ❦❧ ♠ ❧ ❣ ❥ ❣♥ ♦ ✐ ♠ ❡ ♥ ❞ ♠ ❣ ♥ ② ① ♥ s ♥ q ✉ t ❣s ❥ r ❥ ❡ ❝ ❤ P ❉ ❙ ❘ ◗ ❉ ❏ ❊ ❯ ❖ ◆ ▼ ▲ ❏ ❏❑ ❚ ❉❱ ❜ ❘ ❛ ❫ ❴❵ ❫ ❪ ♠ ❙ ❲ ❩ ❨ ❳ ❘ ❏ ❘ ❦ ♥ ⑩ ♦ ① ✐ ❞ ❷ ③ ✉ ❥ ⑩ r t ♠ ❢ r❶ ❥ ⑥ ❥ ❥ ❦❧ ❥ ♥ ♦ ✐ ♥ ❸ ❥ r ♦ ♥ ❣ ♣ ❦ ❡ ⑧ ❣ ✈ ⑦ ❢ ♦ ❡ ✐ q ❦❧ ① ② ⑥ ♦ ⑤ ❝ ❥ t ① ♥ ❧ ✐ s ♦ ♣ ① ❤ ⑩ ❦ ❣ ① ⑨ ⑧ ⑥ ⑥ ❣ ➧ ➩ ❊ ✒ ✖ ✏ ✩ ✒ ✌ ✍ ✔ ✣ ✍ ✘ ✭ ✍ ✮ ✩ ☞ ✒ ✓ ✦ ✖ ✱ ✰ ✌ ✎ ✦ ✫ ✘ ✌ ✎ ✕ ✍ ✌ ✌ ✎ ✌ ✪ ✥ ✍ ✣ ✔ ✘ ✎ ✛ ✖ ✖ ✏ ✩ ✏ ✖ ✌ ✌ ✒ ✎ ✖ ✬ ✦ ✦ ✖ ✌ ☞ ✏ ✩ ✖ ✜ ✥ ✫ ✣ ✔ ✮ ✒ ✒ ✖ ✶ ✵ ✚ ✣ ✒ ✛ ✌ ✸ ✎ ✖ ✏ ✩ ✒ ✎ ✷ ✹✺ ✖ ❂❃ ❁ ✻ ❄ ✿ ✺ ✼ ✺ ✻ ❀❁ ✿ ✾ ✽ ✺ ✻ ✏ ✌ ✣ ✫ ✎ ✫ ✜ ✍ ✒☞ ✩ ☞ ✍ ✒ ✩ ✖ ☞ ✖ ✍ ✌ ✥ ✓ ✒ ✒ ✏ ✍ ✔ ✍ ✣ ✛ ✖ ✖ ✌ ✲ ✰ ✫ ✔ ✓ ☞ ➦ á à Ù Ó á × â Õ Ô à Ü Ó ã Ó ä å Þßà ñ çõ ï ç ì ìò î ð æçè ç è ì è ìí ë ÚÜ × é ➺ ➳ Ñ ➭ ➠ ➝ ➢ ➢ ➛ Ï ➝ ➛ ➯ ➳ Ñ ➯ ➝ Þ Õ Ó Ü Ý Ó Ù Ó × ➽ Ö Ø ×Õ Ö Ô ÔÕ ö ÷ ✏ ✒ ✍ ✌ ✏ ✖ ☞ ✓ ☞ ✣ ✒ ✢ ✒ ✜ ✘ ✛ ✔ ✤ ✌ ✌ ✜ ✖ ✌ ✒ ✩ ✖ ✎ ✌ ✓ ✦ ✥ ✘ ✌ ✍ ✚ ✖ ï í ☎ � ÿ þ õ ç è ✝ î ð çõ ï ø í ✆ ✞✟ ☞ ✍✏✑ ✖ ✍✕ ✔ ✍ ✓ ✒ ✌ ✆ ✎ ✍ ✌ ✡ ✁ ✠ ❋ ↕ ❉ ➐ ➙ ➛ ➟ ➝ ➢ ➠ ➔ ➤ ➐ ➠ ➥ ➋ ➼ ➝ ➑ ➝ ➐ ➨ ➛ ➑ ➒ ➨ ➛ ➑ ➝ ➡ ➒ ➞ ➓ ➝➳ ➎ ➓ ➟ ➝➧ ➝ ➡ ➒ ➞ ➓ ➝➳ ➍ ➝➧ ➍ ➤ ➔ ➵ ➸ ➑ ➨ ➞ ➛ ➜ ➎ ➐ ➑ ➐ ➺ ➵ ➝ ➻ ➨ ➛ ➓ ➝ ➛ ➑ ➎ ÓÔ ➶ ■ ➷ ❐ ➘ ➪ ❒ ➷ ❮ ❐ ➱ ❰✃ Ï Ð ÕÖ× ➘ Ø Ò Ù Ü Ò Ý ÓÞ ß Ø à Ø × Û á ➹➮ ➷ ➐ ➣ ➎ ➓ ➝ ↕ ➏ ➵ ➝ ➎ ➞ ➨ ➐ ➓ ➲ ➑ ➜ ➴ ➪ ➬ ➘ ➷ ➴ ➪ ➘ ➹ ➎ ➶ ➛ ➏ ↕ ➡ ➐ ➓ ➐ Ó ➉ ❼ ❽ ❾ ❷ ❶ ➁ ➂ ➃ ➀ ➄ ➂ ➃ ➀➅ ➆ ➅ ③ ➊ ➋ ➎ ➒ ➓ ➔ ➓ ➐ ➔ →➣ ➑ ➐ ↔ ➑ ❶ ❶ ➎➜ ③④ ♥ ♦ ♣ s t ✉ ✈ ✇ s ① ② t ✉ ✈ ③ ❻ ⑩ t❺ ❹ ❸ t ❷ ❶ t ⑤ ③ ⑨ ✉ ⑧ ⑦ ⑥ ➛ ➎ ➑ ➟ ➦ ➙ ➝➧ ➨ ➛ ➞ ➓ ➝ ➎ ➙ ➛ ➟ ➐ ➛ ➑ ➥ ➐ ➎ ➩ ➫ ➭ ➯ ➞ ➓ ➲ ➒ ➓ ➎ ➛ ➟ ➐➑ ➠ ➐ ➎ ➓ ➝ ➞ ➓ ➝ ➎ ➙ ➛ ➟ ➝ ➛ ➟ ➑ ➐ ➔ ➐ ↕ ➤ ➔ ➠ ➢ ➛ ➏ ➡ ➟ ➐ ➎ ➜ ➐ ➎ ➠ Ò ➹✃ Ù ➘ ➘ ➬ ➴ ➷ ➘ ➹ ✃ ✟ ➾ ➬ ➘ ➹ ✃ ➪ ➪ ➴ ☛ ✗ ✏ ✔ ✕ ✖ ✕ ✍ ✠ ✔ ✓ ✏ ✌✎ ✍ ✌ ➷ ➪ ✒ è ë ✄ ù îï ✝ ï ë ê ó æ ç æ ç è è ➘ ó ➪ ✞ è ë ☎ ï ò ç ø ê ù ë ç æ ☛ ✔ ✁✂ ✳ ✪ ✫ ✬ ★✩ ✭✮ ✲ ✳ ✴ ✲✵ ✳ ✲ ✰ ✺✻ ✼ ✰ ✦ ❈ ❍ ❈ ● ❋ ❊ ❉ ❄ ✽ ❃ ❂ ❁ ❀ ✾✿ ✳ ★✩ ✥ ✘ ✓ ☛ ✒ ✙ ☛ ✒ ✍ ☛ ✕ ✔ ✓ ✏ ✗ ✒ ✚ ✕✛ ✧ ✣ ✧ ✥✧ ✦ ✥ ✤ ✌ ✒ ✙ ☛ ✢ ✔ ✜ ✌ ☛ ù ê ê ë ç æ ñ ê ë å ó ù ü è ä ý ê û ò ûþ ë ÿ Ò è è ê ÷ ô ï ì õ ò î ï ö óú ç ò ë ð ï ð ëì ê é æ õ ë û ÿ ï ÷ Ò ë ù õ ù ê û é ê ë � × å ã â ï ë Move-To-Front 92 ✇✸① MoveToFront ➋➍➌➎ ✼✙✽ ♣✸✈ 3 ✖★✯ ➆❾➇ ): ➨✱➩ ó✆ô ✳✂✴ MoveToFront. ♣✸❣ ❼❾❽ ➸✸➺ ➡✸↕ �✂✄ Heuristic “Move-To-Front îÛï �✂✁ ➟✱➠ ➪➍❰ ✗✙✘ ❬✸❭ Heuristic Transpose ( – ➜✱➝ é✆ê ØÛÚÜ ↔➙↕ ✃➍❐ ♠④③ ➪➍➱ ➐❾➑ ûýü ➻➾➽ ✥★✧ ❅❇❆ - ➪✱➶ ÷úù ☛✂☞ Ò✱Ó 91 n). LookUp ✡☞☛ ø❬ù 3 c j LookUp(). ↕❬➙ x 2 / ➄➈➇ í➚î ) i 1 LookUp i ip n n + C opt = ∑ 1 c j = j. n n = ( = ñ✺ò n ➬❢➱ 2 / ) í➚îï i j. p 1 ≥ p 2 ≥ ... ≥ p n p j ∑ 1 n = 1 i ( ❿✺➀ ✴✸✹ n n x j Ú✴Û x j n ✶✸✷ ô✄û ➏✄➐➑ (c j *p j 2 q✔r - ñ ✆☎ 1 ➌✴➍ 1 ✑☞✒ ➽➚➾➪ Ñ❬Ò ✯✱✰

  46. ❅ ✌ ý ✑ ✌ ✍ �✄ ☞ ✒ ✠ ✍ ✞ ø ✆ ø ÿ ✌ � ☞ ✠ ✌ ú ý ø ✆ ✏ þ ø ☛ ✞ ø ✟ ✝ ✠ ☛ ☞ ☞ � ý ☞ ✞ ✂ ü � ý ✌ ★ ✂ ✠ ✌ ü ý ✆ � ✍ ✥ �✄ ý ✛ ✑ ✌ ✍ ✘ ✙ ð ✚ ó ✠ ✜ ✢ ✞ ✠ � � ✁ ✁ ✌ û Ý Ý Û à á â Þ ã ì ä î æ ç ä è Ý Þ í Þ ß ç ä è Ý Û Þ é á ä ç è êë ❆ Ý å Û ß ☞ ✌ ✁ ✌ ✆ ✠ ✌ ù ✍ ☞ ✎ ☞ ✄ ú ✆ ü ✍ ☛ ☛ ï �✁ ð ô õ ö ÷ ø ý ø ✟ ✂ ý ✆ ✝ ✁ ø ✞ ø ✆ ÿ å ✍ ✑ �✄ ÿ ✌ ✂ ✌ ✬ �✄ ✌ ý ✒ ★ ✁ ✫ ✓ ✥ ✞ ü ✄ ÿ ✩ � ý ✠ � ✠ ✥ ✥ ù ✌ ý � ✄ ✂ ✭ ø ✆ ✳ ✣ ✆ ✩ �✄ ✮ ✯ ✴✵✶ ✆ ✷✸ ✹ ✷ ✽ ✸ ✶ ✾ � ý ✦ ✩ ÿ ✥ ú ý ☞ ✠ � ✏ ü þ �✁ ✌ ☞ ★ ✠ ✌ û ý ✩ ✎ ú ★ ú ý ✆ ✏ û ✒ ✠ ý ø ✞ ✝ ✆ � ø ✌ ✞ ✌ � ý ø ✒ � ✂ ý ✪ ✆ ✞ ✣ ✌ ✠ ø ý ✄ ☞ � ✫ ü � ÿ ✌ � ✁ ✓ ✞ ✢ ✞ ✌ ✍ ✞ ø ý ✂ ✥ ✌ ý ✥ ✞ ø ✠ ✥ ù ✌ � ✂ ✂ ✄ ø ✂ ý ✆ û æ æì ä ⑦ ❸ ➄ ① ② ➀ ⑨ ❼ t ➅ ⑦ ✇ ❺ r ⑨ ③ ➂ ➂ ❶ ⑨ ⑦ ⑦ s r ➁ ⑩ ⑨➃ ➂ ⑥ ✇ ✉ ✇ ❺ r ➃ t ⑦ ⑦ r ➇ ✉ ➇ ➈ ❸ ➇ ⑨ ➂ ➉ ➊ ✇ ✉ ❺ ❸ Ýã ⑦ ⑦ ❺ r ⑥ s t ❸ ⑩ ⑦ ⑦ ➂ t ❸ ⑦ ⑨ ❺ ➀ ⑩ ❺ ❲ ❚ ❏ ❯ ❖ ■ ❱ ❳ ❘ ■ ❨ ❩ ■❬ ❭ ❲ ❙ ◗ ❲ ❑ ❇ ❈ ❉ ❍ ■ ❏ ▲ ❖ ❍ ▼ ◆ ■ ❏ ❑ ❖P ❖ ❪ ⑦ ✇ ③④ ⑦ ❶ ⑦ ❹ ✉ ✉ ✉ ❼ ❶ ✇ ✉ ❿ t ❸ ①② ✇ ❫ ❞ ❴ ❳ ❲ ❝ ❞ ❡ ❝❢ r ❣ ❡❦ ❧ ❞ ❡❦ ♣ s r ➆ ➼ ➶ ➾ ➚ ➯ ➪ ➺ ➭ ➹ r ➘ ➭ ➴ ➷ ➳ ➮ ➽ ➺ ➺ ➸ ➯ ➫ ➭ ➫ ➵ ➧ ➦ ➺➻ ➥ ➤ ➢ ➭ ➯ ➲ ➹ ➭➬ ➹ Õ Þ ⑥ â Û ß Þ Û ↔↕ ➙ Ò ➱ Ø ➛➜ ➝ ➲ ➠ ❐ Ö Õ Ï ➘ ✃ Ð ➹ Ð ➡ 94 h- BinarySearchLookUp �✄✧✦ logn. ✣✗✤ 3 ✂ ❻ü ✝✡✠ û❻ÿ ✣✗✤ ø ✔✓ ✂☎✄ Ñ♦× h n = 2 h – ✺✼✻ n Ó✙Ô à❻á þ❜ÿ Ñ❥Ò û✙ü n O(logn ù❜ú ➨●➩ Ü❜Ý ✰✲✱ - ñ✙òó ✕✗✖ ❒❻❮❰ Ù➒ÚÛ 93 function BinarySearchLookUp ( key K, table T[0..n-1]): info /* Return information stored with key K in T, or NULL if K is 3 else if (K < T[middle]->Key) then ❶❻❺ if (K == T[middle]->Key) then ➞➒➟ return T(middle]->Info; ❷✈❸ middle =  (left+right)/2  ; ♠♦♥ LookUp(). ⑧❥⑨⑩ right = middle-1; not in T */ else left = middle+1; ⑤✈⑥ ➔➒→➣ ❤❥✐ if (right < left) then ➐➒➑➓ return NULL LookUp ④➌➋ ➍➏➎ BinarySearch - BinarySearch Key ❵❜❛ repeat forever right = n-1; else left = 0; t✈✉ ❊●❋ - ❽✂❾ q❜r

  47. ✿ ➹ ❒ Ø ➷ Ï Û ❒ ➹ Õ Ö Û ➮ Ô Ð Ï ➹ ❐ ➹ ➹ Õ ➴ ➮ ❮ Ü Ï ➱ á Ø Û ß ❐ ➮ Ô ➴ ➷ ➱ Ý Ô ➹ Ï Ö ➷ ➱ Û ❒ Þ ➴ Ü ➹ Ï ❒ ❐ ➹ ➘ Ò ➹ ➴ ❰ Ï ➷ ➹ã ✃ å ➘ ➱ ➴ ➹ Õ ➱ ➷ ➱ Ò ➱ ❐ Ý ➷ ➮ ❒ ❮ ❐ ➹ ➱ ➘ ❒ Ò ❒❮ Ú ➬ ➷ ➮ Ñ ➘ ➱ ❒ Ò Ù ➮ ➹ Ô ❒ Õ ➴ Û Ú ❒ ❰ Ô ❐ Ï ❒ Ù ❐ ➹ ➘ ➱ ➮ å Ö ➷ ➱ ❰ ➹ ❐ ➹ ✃ ■ ❖ ■ ❈ P ◗ ❘ ❈ ❙ ❚ ❈❯ ❱ ◗ ◗ ◆ ❲ ❳ ❨ ❘ ◗ ❩ ❬ ❫ ❴ ❵ ❞ ❡ ❢ ❉ ▼ ❣ ❃ ➁ ➌ ➁ ➔ ➄ ➂ ➃ ➣ ✿ ❀ ❁ ❂ ❇ ▲ ❈ ❉ ❊ ❋ ❇ ● ❀ ❈ ❉ ❊ ■❏ ■ ❑ ❴ ❤ ➮ ❮× ➮ ➹ Ô ❒ ❮ ➹ Õ ➴ Õ Ö ❰ ❒ Ø Ò ➘ ❒ Ò ❒ Ù Ú Ñ Û ➹ ❰ ➹ Ï ➘ Ó ÐÑ ✐ ➷ ❵ ❥ ➦ ➭ ➳➵ ➦ ➸➺ ➻ ➾ ➚ ➾➪ ➴➷ ➬ ➮ ❮ ➱✃ ❐ ➹ ➮ ❒❮ ✃ Ï ➘ ➮ ➹ ✃ ➮ ❒❮ ➱ ❰ ❽ ✥ ✫ ✧ ✛ ✗ ✌ ✔ ✌ ✚ ✬ ✌ ✗ ✔✦ ✗ ✔ ✛✮ ★ ✬ ✛ ✭ ✏ ✜ ✚ ✭ ✏ ✓ ✗ ✦ ✢✪ ✧ ✌✍ ✗ ÿ ☛ ÿ ☎ ☞ ✂ ü ✌✍ ✓ ✔ ✌ ✕ ✛ ✔✦ ✎ ✓ ✔ ✓ ✜ ✔✣ ✎ ✚ ✗ ✚ ✔ ✢ ✗ ✕ ✎ ✡ ✥ ✗ ✥ ✮ ✥ ✬ ✏ ✣ ✗ ✌ ✭ ✲ ✗ ✮ ✌ ✥ ✬ ✏ ✯ ✚ ✔ ✌ ✰ ✭ ✧ ✮ ✛ ✪ ✯ ✭ ✏ ✓ ✓ ✦ ✗ ✣ ✚ ✌ ✣ ✬ ✏ ✯ ✚ ✎ ✛ ✭ ✭ ✦ ✧ ✓ ✚ ✮ ✯ ✎ ✓ ✔ ✢ ✔ ✢ ★ � ✠ ➹ Ï Ö ❰ ❒ ❮ á Ø ➷ Ö ➷ ➱✃ ❰ ➹ ➘ ➴ ➮ Ñ ✃ ➹ Õ ➴ Õ Ö ❰ ❒ ❮ æ ❒ Õ Õ Ñ ➹ Ï ➘ ➮ ➹ ✃ ➮ ❒ ❮ ➹ Õ Ó ❐ ➷ ➹ ❒ ❮ Ü ➱ ❐ Ñ Ò Ó ➮ ➹ Ô ❒ ❮ ❰ ã ✂ ý î ïðñ ò ó ð ô õ ò ñ ô ñ ú þÿ ✃ � ✁ ✂ ü þÿ ✄☎ ✆ ÿ ✝ ✞ ý ✁ ✝✟ á ➬ ➹ Õ Ï Û ➹ ➘ Ò ➹ ➴ ❰ Ï Þ à ä ➹ Ö Ü Û ➮ Ô ➹ Ð ➹ ➱ ➷ ➹ Õ ➴ ➮ ❮ ❽ ❍ ➥ ❽ ❽ ➄ ❽ ➐ ➔ ➉→ ➑ ➀ ➋ ➆ ➈ ➐ ➋ ➉ ➆ ➂ ➌➍ ❽ ➀ ❻ ↔ ➈ ➁ ❽ ❿ ➌ ➋ ➄ ➃ ↕ ➆ ➂➌➍ ❽ ➀ ➂ ➍ ❿ ❽ ➌ ➊ ➎ ➂ ➉ ➈ ➋ ➌ ➏ ➌ ➈ ➐ ➌ ➈ ➁ ➑ ➒ ❿ ➓ ➇ ➋ ➆ ➈➉ ➄ ↕ ➋ ➊ ➃ ➇ ➁ ➂ ➑ ➑ ➂ ➙ ➇ ❽ ➉ ➀ ➅ ➆ ❽ ➇ ➛ ➜ ↕ ➉ ➂ ➂ ➌ ➐ ➆ ➙ ❽ ❿ ❿ ➈ ➋ ➆ ➂ ➌➍ ❽ ➀ ➈ ➄ ❽ ➉ ➍ ❽ ➂ ➌ ➊ ➎ ❽ ➆ ➈ ➌ ➉ ➈ ➅ ❿ ➈ ➆ ➆ ❽➇ ➋ ❞ ❱ ◗ ■ ◗ ❲ ❳ ❨ ❘ ◗ ❩❬ ❫ ❴❵ ❡ ❚ ❢ ❴ ❣ ❤ ✐ ❵ ❥ ❦ ♣ ♠♦ s t ♠♦ ❈❯ ❙ ♦ ❉ ❁ ❂ ❃ ❇ ❈ ❉ ❊ ❋ ❇ ● ❍ ❈ ❊ ❈ ■❏ ■ ❑ ▲ ▼ ◆ ❉ ❖ ■ ❈ P ◗ ❘ ✉✈ ✇ ➆ ① ② ✉ ① ♣ ♦ ② r ♠ q ✉ q ✉ ❸ ② ♠ ❷ ❹ ❧ ② s ♦ ❺ ❻ ❿ ➂➃ ➄ ❽ ❷ ❷ ① ♣ ✉ ♣ ③ ⑥ ♣ ⑥⑦ ⑧ ♦ ⑤ ⑥ q ⑥ s ③ s ④ ⑨ q r ♦ q ♣ ③ ⑩ ♦ ♠♦ ❶ ⑩ ➌ ➂ ➂ ➉ ➀ ➍ ➍ ➀ ❿ ➈ ➇ ➉ ➂ ➃ ➋ ➌ ➆ ➂ ➉ ➠ ➂➃ ➂ ➋ ➆ ↕ ❽ ➁ ❽ ➄ ➐ ➣ ➇ ❽ ➁ ➌ ➁ ➔ ➄ ➂ ➃ ➟ ➇ ➃ ❻ ➙ ➄ ❽ ➂ ➒ ➆ ➅ ❽➇ ➁ ➂ ↕ ❽ ➆ ➇ ➂ ➌ ➃ ❽ ➉ ➍ ❻ ➢ ➉ ➈ ➅ ➔ ↕ ➐ ➡ ➀ ➂ ❽ ➇ ➄ ➅ ❽➤ ➈ ➄ ➁ ➉ ➆ ➂ ➅ ↕ ➐ ➡ ➂ ➑ ➔ ➏ ➂ ➉ ➈ ➌ ➌ ➄ ❽ ➀ ➅ ➆ ❽➇ ➆ ❽ ➁ ➋ ❽ ➑ ➅ ➀ ➉ ➌ ➈ ➆ ➂ ➉ ➂ ➌ ➐ ➌ ➂ ↕ ➋ ➂ ➀ ➎ ➔ ➉ ➈ ➈ ➅ ↕ ➐ ➔ ➐ ➝ ➅ ↕ ➐ → ➄ ❽ ➅ ➆ → ➉ ➝ ➈ ➇ ➈ ➌ ➈ ➁ ↕ ➈ ➉ ➃ ➆ ➂ ➙ → ➈➉ ➅ ➉ ➌ ➇ ➑ ➀ ➆ ❽ ❿ ➈ ➇ ➂ ➂ → ➐ ➍ ➊ ➎ ➂ ➉ ➆ ➈ ➌ ➄ ❽ ➅ ➋ ➆ ↕ ➂ ➅ ➣ ➇ ❻ ➃ ➂ ➄ ➡ ➐ ❽ ➅ ➆ ❽➇ ➊ 96 Ú☎➹ ✜✘✭ 3 ✎✏✒✑ ✖✩★ àâá ✬✱✰ ✧✘✯ ❛❝❜ Ö✗Ü ✜✘✥ ❰☎➹ ø✼ù ❭✲❪ ö✼÷ ❒✧ä ➼✡➽ ✤✘✚ Skip Lists ✜✘✢ ✜✘✭ ➯✡➲ ë♥ìí ❄❆❅ ➩❝➫ ç✗èéê ➧❝➨ - ✙✘✚ ➶❾➹➘ û✗ü ✖✘✗ n 95 ➀✗➅ 3 i. i – ④✼⑤ ❛❝❜ ➊✡➋ t✼② ❭✲❪ ➀✗➅ ➀✗➅ Skip Lists logn standard ➈✔➣ ➀✗➅ q✼r ➀✡➁ ❄❆❅ ❼❾❽ ➞✗➌ ➞✗➌ - ❧♥♠♦

  48. ✳ s ④ ❥ ❛ ⑤ ❛ ❥ ⑥ ➊ ➔ ✐ ❥ ④ ❻ ❦ ❣ ❣ r ❞ r ❣ ❹ ❦ ❺ ✉ r ✐ ❻ ❛ ❝ ⑤ ❝ ⑤ ♦ ⑨ ❦ ② ♦ ❣ ❝ ② ✇ ❥ ❞ ❦ ❧ ✉ ⑤ r ❦ ❺ ❥ ❢ ❢ s ♦ ❣ ⑨ s ♣ ❥❛ ❝ ❦② ❝ ✐ ❹ ✐ r ❢ s ♣ ✐ ❥ ❥ r ⑤ ✉ ❛ ❛ r ✐ ❻ ❥❛ ⑤ ❝ ✉ ❝ ❥ ❛ ❥ ❣ ❥ ❞ ✉ ✈ ❝ ❥ ❛ ❥ ❣ r ✐ ♠ ❣ ❞ ❝ ❦ ♥ ❣ ❣ r ❞ r ❣ ❹ ❦ ② ❝ ❦② ④ ⑨ ❦ ② ④ ❝ ✐ ❥ ❹ ❣ ❹ ❦ ♦ ❡ ❛ ❥ ✉ ❣ ✐ ♣ ♠ ❥ ♥ ♠ ➏ ♠ ❞ ❛ ❣ ❝ ❥ ✐ ❥ ❧ ④❣ ✈ ♦ s ❳ ❊ ❂ ❊ ❑ ▲ ▼ ❋ ❊ ◆ ❖ ❘ ❙ ❚ ❨ ❥ ❩ ❙ ❬ ✴ ❪ ❚ ❫ ➎ ❣ ❝ ❥ ❛ ❡ ❝ r ❦② ➒ ❥ ♠ ♥♠ ➏ ✈ ♣ ❝ ✉ ❢ ❞ ♣ ❝ ❥ ➐ ✐ ➓ r ❡ ➉ ❣ ❝ ❦ ❣ r ❞ r ❣ ❹ ♣ ❣ ✐ ❝ ❦ ❣ ① ④ ❦ ❢ ✉ ⑥ ➐ ➑ ❧ ❣ ✈ ✉ ✉ ❛ ④ ❥ ❝ s ❢ ❢ ✉ ❢ ✉ ❻ ❡ ♣ r ♠ ✼■ ➯ ➝ ➻ ➯ ➥ ➦ ➼ ➦ ➦ ➻ ➧ ➡ ➭ ➼ ➛ ➟ ➭ ➠ Ô ➶ Õ ➡ ➭ ➯ ➼ ➟ ➳ ➠ ➫➭ ➝ ➡ ➡ ➦ ➝ ➙ ➭ ➡ Ò ↕ ➻ ➭ ➡ ➯ ➦➣ ➣ ➭ ➨ ➯ ➼ ➟ ➳ ➠ ➞ ➭ Ó ➻ ↕ ➢ ➻ ➭ Ö ➭ ➯ ➦ ➦ ➝ ↕ ➠ Ø ➨ ➫ ➝ ➼ ➽ Ù ➼ ➝ ➛ ➧ ➵ ↕ ➝ ↕ ➝ ➡ ➭ ➯ ➼ ➟ ➭ ➠ ➨ ➼ ➯ ➧ ➫ ➥ ➭ ➝ ➞➟ ↕ ➥ ➻ ➨ × ➟ ↕ ➼ ➙ ➦ ➾ ↕ ➟ ↕➢ ➭ ➻ ➝ ➻ ➛ ➡ ↕ ➢ ➣ ➥ ➭ ❣ ↕ ➣ ➙ ➛ ➜➝ ➞➟ ↕ ➭ ➻ ➯ ➻ ➭ ➡ ➼ ➸ ➽ ➩ ➭ ➼ ➧ ➙ ➾ ↕ ➢ ➚ ➽ ➩ ➭ ➺ ➠ ➧ ➨ ❞ ❝ ❦ ❴ ↕ ➙ ➞➟ ↕➠ ➦ ➟ ➧ ➠ ➦ ➝ ➦ ➩ ➙ ➞➟ ↕ ➝ ➫➭ ➥ ➟ ➳ ➠ ➵ ➯ ➞➟ ➼ ➙ ÐÑ ➧ ➦ ➣ ➛ ➥ ➦ ➻ ➯ ➥ ➦ ➼ ➦ ➦ ➻ ➡ ➡ ➭ ➯ ➼ ➟ ➭ ➘ ➮ ➱ ✃ ➱❐ ❒ ❰ Ï ➯ ➭ ➾ ➨ ↕ ➠ ➙ ➝ ➦ ➠ ➵ ➯ ➞➟ ➦ ➠ ➻ ➦ ➽ ➻ ➵ ➭ ➝ ➪ ➳ ➠ ➛ ➫➭ ➝ ➟ ➨ ➯ ➦ ❏ ❭ ❍ ❝ ❥ ❝ ② ✇ ❥ ✐ ⑤ ❝ ✉ ❝ ❥ ♣ ❦ ❦ ♣ ① ♣ ❝ ✉ ♦ ❥⑥ ⑦ ⑧ ❛ ❥ ✈ ④ ✐ r ♦ ✇ ✐ t✉ ✈ r ❦② ❝ ✐ ✈ ③ ❝ ✐ s ❛ s ❦② ♦ ❣ ❣✐ ♣ s ♠ ❦ ❛ ❝ ❥ ✈ ④ ⑤ ⑨ ❧ ♦ ❼ ❺ ❷ ❺ ● ❽ ❽ ❺ ❺ ❦ ❛ ❦ s ❹ ❾ ❣ ❝ ❥ ✐ ④ ⑤ ♦ ⑨ ❦ ✈ ❣ r ❦ ❣ ❦ ✈ ✈ ♦ ❣ ⑩ ❶ ❷ ❸ ❹ ❣ ❞ ④ ❝ ❣ ❺ r ❡ ❛ ❥ ♠ ✐ ❥ ④ s ❻ ❣ ❣ r ❞ s ❥ r ❳ ❏ ❊ ❂ ❊ ❑ ▲ ▼ ❋ ❊ ◆❖ ❘ ❙❚ ❨ ❍ ❩ ❙ ❬ ❭ ❪ ❚ ❫ ❴ ❝ ❞ ❝ ❡ ❥ ✼■ ● ❧ ✽ ✵ ✶ ✷ ✻ ✼ ✽ ✾ ✿ ✻ ❀ ❁ ✼ ✾ ✼ ❂❃ ❂ ❄ ❅ ❆ ❇ ✽ ❈ ❂ ✼ ❉ ❊ ❋ ❦ ♠ r ♠ ♦ ❣ ♦ ✉ ❝ ❡ ❢ ❣ ✐ ❥ ❦ ❧ ❥ ❦ ❛ ♥ ♦ ❡ ❛ ❣ ✈ ❢ ❞ ♣ ❝ ❣ ✈ ① ✐ ❥ r ❛ ♥ ♦ ❡ ❛ ♥ ❛ ❢ ✐ ♣ ❝ q ❛ ❥ ❦ ❧ s ❢ ❣✐ t✉✈ ✇ ❧ ✉ ♣ ✐ ♦ ❦ r ✐ ❽ ❡ ❝ ❥ ❽ ➌ ⑨ ❦ ✐ ♦ ❣ ♠ ❥ ❢ ① ❣ ♣ ❧ ❦② ❣ r ✐ r ❡ ❹ ❦ ② r ❧ ❝ ❞ r ✉ ✐ ❥ ♣ ④ ❦ ❧ r ✐ ♣ ♦ ❡ ❛ ❝ ❢ ② ✇ ❥ ❞ ❥ ♦ ❡ ♣ ❹ ♣ ❝ ✉ ❡ ❣✐ ✐ ❂❃ ✷ ✻ ✼ ✽ ✾ ✿ ✻ ❀ ❁ ✼ ✽ ✾ ❂ ✵ ❄ ❅ ❆ ❇ ✽ ❈ ❂ ✼ ❉ ❊ ❋ ✼ ✶ ✴ ❛ ✐ ❥ ♣ ② ❛ ❥ ❛ ❝ ❦ ① ❛ ❝ ❥ ❢ ✳ ✐ ♠ ⑤ ❝ ❣ ❧ ❦ ♣ ② ✇ ❛ s ❽ ❹ ❥ ❥ ❦ r ✐ r ❛ ❹ ♥ ❛ ❣ ➉ ❥ ④ ❢ ❛ ❦ ② ❻ ❦ ① ❛ ❛ ❥ ② r s ❧ ❣ ♥ ❦② ④ ❦② ❿ ➀ ➁➂ ➃ ➄ ➅ ➆ ➇ ➀ ➈ ✐ ⑤ ❧ ♦ ⑨ ❦ ✐ ❝ ♥ ❛ ❹ ✐ ❥ ③ ⑤ ✇ ❡ ❛ ❥ ❥❛ ❥ ✐ ❹ ❞ ♣ ❝ ♣ ❢ ❦②➋ ❣ r ❞ ❧ ➊ ❣ ❥ ❦♠ r ❞ ♣ ❝ ✉ ❞ ❣ s ❢ ❢ ❢ ❥ ❺ ❥ ❞ 98 NULL. ❢❤❣ 3 ➯❜➲ MaxLevel dummy node ➧❜➠ j+1». ➠➹➶ struct ❯❲❱ Header->Forward[j MaxLevel =  logN  - j. P✱◗ ⑤➍♦ Forward j ❢❤❣ ➡➤➢➥ Forward Skip Lists ÚÜÛ ➴❲❮ , Level ➛❜➜➝ Header Level ❢❤❣ Key Info ✸✺✹ - →↔➣ ➴➷➍➬ j o o o o o 97 j. 3 ❯❲❱ P✱◗ - j j Skip Lists ❢❤❣✐ ❢❤❣ ✸✺✹ - ❵❜❛ ⑤➍♦

  49. Ý ❴ ▲ ❉ ❙ ❚ ❯ ❱ ❉ ■ ❲ ❳ ❨ ❩ ❬ ❵ ❘ ❛ ❜ ❝ ❴ ❞ ❡ ❵ ❛ ❜ ❢❣ ❢ ❤ ✐ ❥ ❍ P ❛ ❇ ✵ ✪ ✺ ✬ ✽ ✱ ✳ ✯ ✪ ✮ ✬ ✸ ✾❄ ❅❆ ❆ ◗ ❅ ❈ ❉ ❊ ■ ❉ ▲ ▼◆ ❉ Þ ❈ ❖ ❈ ❍ ❦ ❧ ✾ ➛ ➀ ➁➂ ➃ ➄ ➅ ➉➊ ➋ ➌➍ ➎ ➓➔→ ➣ ↔ ↕ ➙ ➜➝ ❾❿ ➡ ➢ ➤ ➥➦ ➧➨ ➩ ➫ ➥ ➦ ➭ ➯ ➭➲ ➳ ➻➼➽ ❾ ⑨ ❢ ✈ ❵ ♠ ♥ ♦ ❵ ♣ q ❵r s ♥ ❢ ♥ t ✉ ♦ ③ ♥ ✇① ⑤ ⑥⑦ ⑩ ❶ ❷ ⑥ ❸ ❹ ❺ ⑦ ❽ ❸ ✯ ❍ ✫ ★ å÷ ö ✘ õ ✛✜ å ô ✥ ó ★ ✩ ✜ ø ò å ë ñ æ ð ï î ó ✪ ✫ ✓ ✒ ✭✮ ✞ ô û ú ù ó üý ✴ � ✁✂ ë ✝ ✟ ✏ ✁ ✠ ✡ ☛ ✂ ó ✎ ✠ ÿ ☎ ✏✑ ✬ í ✯ ✻ å ✼❀ ✶ é ä è ç æ å ä á ✭✮ ç ✼ ✹ à ✽ ✫ ✽ ✯ ✾ ✹ ✪ ß æ ê ✺ ✶ ✪ ✬ ✮ ✳ ✮ ✳✴ ✭✮ ✳ ✬✵ ✽❁ ✪ ✬ ✬✿ ✳✴ ✪✸ ë ✪ ✷ ëì ✹ - 3 - 3 â✺ã ❭❫❪ Skip Lists Skip Lists þ✱ÿ ✄✆☎ ☞✍✌ ②④③ ⑧✆⑨ ❻✍❼ procedure SkipListInsert ( key K, info I, pointer ✔✆✕ ✖✆✗ ✙✍✚ ✓✣✢✤ ✦✍✧ L): pointer function SkipListLookUp ( key K, pointer L): info ➆➈➇ ➏➑➐➒ ➞➑➟➠ ➵➑➸➺ L */ /* Info ✰✲✱ ✶✍✭✮ ✬❃❂ P = L->Header; for j from L->Level downto 0 { P = L->Header; while ((P->Forward[j])->Key < K) do for j from L->Level downto 0 do P = P->Forward[j]; while ((P->Forward[j])->Key < K) do Update[j] = P; P = P->Forward[j]; } P = P->Forward[0]; P = P->Forward[0]; if (P->Key == K) return P->Info; if (P->Key == K) P->Info = I; else return ; else { NewLevel = RandomLevel(); if (NewLevel > L->Level) { for j from L->Level+1 to NewLevel do function RandomLevel (): integer Update[j] = L->Header; ❋✲●❍ ❏✲❑ ❖✲P L->Level = NewLevel; MaxLevel */ } P = NewCell(Node); P->Key = K; v = 0; P->Info = I; while (Random() < ½ && v < MaxLevel) do for i from 0 to NewLevel { v = v+1; P->Forward[i] = ((Update[i])->Forward)[i] ((Update[i])->Forward)[i] = P; return v; } } 99 100

  50. ➾ è å æ ä ã ç â áé ã à ê ã ê æ é ä àáâ é ❐ Ð Ö Ô × ✃ ❮ Õ ß Ø Ò ❰ Ï ❿ ➀ ë ❿ Õ æ ç ñ ç â ò ê é ì ãó ì á ô ä å áâ ❺❻ â é ì áî ï é ì æ ë à é î â á ñ é ì ❽ Ô ä ➉ ➷➬ ➪ ➋ ✃❐ ➊ ❰ ✃ ➌ ➮ ❒ ➈ ➆➇ ✃ ❮ ➴ ➶ Õ ➨ ➻ ➼ ➸ ➺ ⑤⑥⑦ ➭ ➽ ➍ ➨ ➾ ➵ ➺ ➥ ➄ ➋ Ô ➅ ❮ ❰ Þ ✃ ➮ ➱ ✃❐ ➂ ➚ Ï ✃ ➮ ❒ ➁ ❸ ✃ Ý ➃ Ð Õ Ö Ô × ✃ ❮ ❐ Ø Ú Ò ❰ Ï ➄ Ù ✃ Õ á ã ➨ ⑥⑧ ③ ❂ ❃❄ ❅ ❆ ④ ● ⑨ ❍ ■❏ ❑ ▲ ▼◆ ❖ ✿ ✬ ◗ ✹ ✷ ✴ ✬ ✸ ★ ✬ ✲ ✩ ✺ ✻ ✸ ✷ ✴ ✹ P ❘❙ ✶ r ❦ ❧♠ ♥ ♦♣ ♦ q ✉ ❤ ✈ ✉✇ ①②③ ④ ② ✈ ✐❥ ❢ ❚ ❭ s ❨ ❳ ❘ ❩ ❬ ❪ ❣ ❘ ❨ ❫ ❴ ❝ ❞ ❡❢ ✵ ✵ ó ✟ ✁ ✆ ✝ ✆ r ✞ ✠ ÿ ✡ � ☛ ✆ ÿ ☞ � ☎ ✍ ❷❸ õ ❹ ö ÷ ø ù ú ✄ ❶ þ ÿ � ✁ ✂ þ ✌ ÿ ✧ ✩✪ ✜ ✗ ✤ ✥ ✦ ✧★ ✭ ⑥⑧ ✮ ✯ ✰ ✬ ⑩ ✴ ✣ ✘ ✎ ✌ ✏ ÿ ✑ ✒ ✌ ✆ ✓ ✚ ✔ ✕ ✍ ✌ ⑥⑧ ✖ s ➎➏ ➊ ➩ ✪ ➡ ✢ ➠ ✧ ★ ✩ ✫ ➨➩➫ ✬✭ ✮ ✯ ✲ ✳ ✴ ➢ ➭ ✶✷ ➯ ☞ ✌ ✆ ➯ ➲ ✟ ✔ ✜ ✕ ✖ ✗ ✘ ✙ ✚✛ ✵ ✸ ✟ ❭ ◗ ❚ ❯❱ ❲ ❙ ➒ ❪ ❑ ❫ ➟ ❵ ❭ ❬ ❴ P ▲❖ ✹✺ ❍ ✻ ✾ ❃ ❄❅❆ ❊❋ ● ● ❏ ■ ❏ ❑ ▲ ● ❑ ➳ ➲➸ ❛ Ö ❐ ➶ ➪ Ó Ô Õ × ÑÒ ØÙÚ Û Ü Ý Þ ß Ñ Ð Þá ➬ ➪ ➶ Ï ➹ ➱ ➷ ➮ Ï ➱ ✃ ❐ ❒ ❮ ❰ à â ☎ û õ ö ÷ ø ù ú ü ó ➻ � ✁ ✂ ➺ ✄ ïô ñ ãä ➚ ã å æ ç è é ➾ ò ➽ ê ë ➼ ïð ñ ❬ ❳ ❬ ➏ ➜ → ➒➓ ➍ ➔ ➣ ➣ ➝ ➍ → ➎ ➑ ↕ →➞ ➔ ➌ ➜ ➎ ➔→➣ ➋➌ ➍ ➐ ❜ ➑ ➒➓ ↔ ➒ ➑ ↕ → ➒ ➙ ➓ ➛ ➛ ➎ ➑ ➒ ➡ ➣ ➒ ➙ ➓ ➛ ➜ ➜ ➝ ➌ ➜ → ➥ ➦ ➐ ↕ ➓ ➟ ➏ ↕➠ ➑ ➣ ➠ ➑ ➣ ➝ ➒ ➠ ➑ ➋➌ ➍ ➐ ➋➌ ➑ ➀ ➋➌ ➉ ♥ ④ ② ⑥ ⑤ ④ ➞ ✇ ➜ ♠ ➣ ♦ r q ♦♣ ♥ ➜ ❬ ❬ ❝ ❜ ❴ ❞ ❪ ❩ ❧♠ ❪ ❵❡ ↕ ❡ ✐❥ ❦ ➝ s ❶ ➅ ② ❽ ❷ ❾ ➒ ➁➂➃ ➄ ➆ ❸ ➆ ❾ ➁➇ ➀ ➈ ➇ ➀ ➈ ⑧ ❼ ➙➛ ⑧ ② ❸ ❹ P): info P): info 102 else return(BinaryTreeLookUp(K, P->RC)); return(BinaryTreeLookUp(K, P->LC)); ❼❲❽❾ function BinaryTreeLookUp ( key K, pointer function BinaryTreeLookUp ( key K, pointer 3 if (K == P->Key) return(P->data); LookUp else if (K == P->Key) return P->data; else if (K < P->Key) P = P->LC; P ❵❲❛❜ ; ✫✢✬ Info if (P == NULL) return else if (K < P->Key) while (P != NULL) { else P = P->RC; ❊✾❋ ❒❰❮ ❯❲❱❳ ❇❉❈ ➳✳➵ ✛✢✜ ➮➦➱✃ ✱✳✲ return ➣✾↔↕ ✗✙✘ ➴✳➷➬ ➓✾➔→ } ûýü ❀✾❁ - ➹✳➘ - ➤➦➥➧ rts ✼✾✽ ey, data, LC, 101 ⑨✞⑩ ❇❉❈ ⑦✞⑧ ➊✥➇ 3 ì✞ð n n ❁✞❂ ❨✥❴ ①③② ýÿþ ✿✞❀ ✉❉✈ ✼✞✽ ❒✞Ó ❒✞Ó ➎✞➏ ✒✞✓ ❨✥❩❬ s✞t ì✞í Ñ✥Ò Ñ✥Ò struct ✠✞✑ ❒✞ÏÐ ✍✏✎ ❒✞ÏÐ ❿✞➀ ❒✞❮ ❘✥❙ ❒✞❮ ✠☛✡ ✰✥✱ ➵✥➸➺ ➮❉➱ ìîí ❢✥❣❤ ✆✞✝ ▼✞◆ ➯✞➲➳ logn. ➒➓➤➢ ❺✥❻❼ n-1. ➫✞➭ ➹❉➘ Û❉Ü ➘❫➴ - - RC. ➧✞➨ ➎✞➏ ✣✥✤✦ ➚③➪

  51. Ð ❛ ❞ ❝ ❜ ❣ ❢ ❛ ❡ ❞ ❝ ❜ ❪ ❤ ❭ ❬ ❩ ❨ ❆ ▼ ❑ ❚ ▼ ◗ ❤✐ ❥ ◆ s ♣ q ① ✇ ✈ ♣ ❤ ♣ ✉ ❜t r ❦ ❜ q ♣ ♦ ❜ ❤ ♥ ❝ ♠ ❧ ❄ ❊ ❀③ ❄ ❄ ❈ ● ❇ ❘ ❄◗ ❑ ■ ● ❖ ❙ P ❊ ❄ ❆ ❅ ❍▲ ❘ ❄◗ ❑ ■ ● ◗ ◆ P ❙ ▲ ❱ P ❯ ▲ ▼ ❊ ❑ ❚ ● Ñ ❊ ◆ ■ ◆ ❍ ▲ ▼ ❄ ▼ ❑ ■ ❑ ② ④⑤ ❅ ➱ ➶ Ï ❰ ➹ ➴ ❐ ➶ ✃ ➶ ➴ ➷ ❒ ➶ ➹ ➾➚ ➲ ➩➺ ➸ ➵ ➲➳ ➯ ➭ Ð ➹ ➩ Ó â ß á Û Þ Ý Û Ü Ø Ô Ò ❰ ❒ ➶ Ñ ➶ Ï Ð ➹ Ï ❐ Ï ➫ ➨ ⑥ ❺ ➉ ➈ ➇ ➅➆ ❼ ❻ ⑦ ❿ ➃ ➂ ➁ ➍ ❸ ❻❼ ⑦ ❽ ❾ ❽ ❻❼ ❶ ⑩ ⑤ ➊➋➌ ➈ ➢ ↔ ➡ ➞➟➠ ➝ ↕➙ ➅ ➆ →➣ ➓ ➍ ➊ ➍ ➉ →➣ ➏ ➋ ➔ ➉ ➐➓ ➏ ➒ ➐➑ ➏ ➎ ❇ ❍ ■ Ùë Ù æ ç è Ù é ê ì å ç ß ✔✕ ✑✒ ✎ ✠ ✝ ß Ú ✠ ✍ ✑ ☛ ✞ ù ✆ ☞✎ ☛ ☞✌ ☎ ☛ ✡ ✆ ☎ ✠ ✞✟ ✝ ✆ ✓ ✑✒ ✠ ô ✄ ✂ ✁ � ÿ ÷ õö ú î ô ï è ç ð ñ ô õö ÿ ☎ ☛ ç ✞ ✏ ✆ ☞✎ ☛ ✍ ❄◗ í ✆ ☞✌ ☛ ✡ ✆ ☎ ✠ ✞✟ ✝ ✒ ✏ ✓ ❁ ❆ ❅ ❄ ❃ Ô ✶ ❂ ✻ ❀ Ó ✶ Ø ✶ ✻✼ Ù ✶ ✝ ✸ ❇ ● Ú ❊ ◆ ❊ P ■ ❆ ❖ ■ ❑ ◆ ❍ ■ ● ❄ ▲ ▼ ❏ ❍▲ Ò ■ ✶✷ ÷ø ✲ á ✙ Ù Ú Û ✰✱ ßà ß â Þ ã ä ✕ ✔ ✑✒ ✎ ✠ ✗✢ ✜ ✥ ✤ ✭✮✯ Û ✦ ✩ ✧ Ý Ø ★ Ü - 3 - 3 Õ×Ö ❫❵❴ ❹✛❺ ò❰ó û✢ü ý✙þ ⑦⑨⑧ ❷✵❸ ❿✿➀ ➁✿➄ function TreeMinimum ( pointer P): info function BinSearchTreeInsert ( key K, info I, /* P pointer R): pointer if (P == NULL) return error; while (P->LC != NULL) P = P->LC; pointer P, Q, Prev = NULL; return(P->data); P = R; while (P != NULL) { if (P->Key == K) { P->data = I; return R; } function TreeMaximum ( pointer P): info /* P Prev = P; if (K < P->Key) P = P->LC; if (P == NULL) return error; else P = P->RC; while (P->RC != NULL) P = P->RC; } ➛✘➜ ➤✘➥ ➦✿➧ ➻✿➼ ➽❋➩➺ return (P->data); Q = NewCell(Node); Q->Key = K; Q->data = I; ✙✛✚ ✣✛✤ ✖✘✗ Q->LC = Q->RC = NULL; if (Prev == NULL) return Q; ✪✬✫ ✳✵✴ ✹✬✺ ✽✿✾ else if (K < Prev->Key) Prev->LC = Q; else Prev->RC = Q; ❈❉❋❊ ❏✘❑ return R; - ❄◗❳❲ ➪✘➶ ➘✘➴ ➬✿➮ ❒✘❮ Ô❋Õ Ù❋ÚÛ ß❋à Ö❳× 103 104

  52. ã á ✫ ✵ ÙÚ ✱ ✫ ✰ ✬ ✯ ✮ ✬✭ ✫ ßà ✣ ✩ ✤ ✣ ✢ ✜ ✛ ✕ ✑ ✟ ✏ ✎ ✡ ✝ ✕ ✱ ✭ ✙ ✫ ❀ ❁ ✼ ❁ Ô Õ Ö ✻ ✼ ✺✻ ✫ ✰ ✱✹ ✱ ×Ø ✩ ✵ ✳ ✶ ✸ ✵ ✩ ✱ ✫ ✵ ✳ ✱ ✷ ✚ ✘ ❃ åæ ô ì í ò ìñ ð ï î ì ë ê ï ç ä ø í ➚ Þ × Ü Ý Ü Ú Ô Û ñ Ø × ÷ ö ✕ ✝✞ ✡ ☛ â ✟✕ ä ã ✎ ✍ ☛✌ ✝ ☛☞ äå ✟ æ ô ç è ✂ �✁ ÿ þ û ü ú ò ë ùú ö ❂ ÒÓ ÒÓ ➜➠➡ ➧ ➞ ➙ ➞ ➙ ➜ ➦ ➥ ➤ ➢ ➔ ➡ ➙ ➟ ➜ ➞ ↕ ➜ ➛ ➙ ↕ ↔ →➣ ➔ ➓ ➒ ➐➑ ➴ ➛ ↕ ➌ ➷ ➚ ✃ ❰ ➴ ➘ ➷ ➹ ➱ ➘ ➮ ❮ ➚ ❒ ❐ ➨ ✃ ➻ ➱ ➘ ➮ ➬ ➴ ➪ ➻ ➵ ➳ ➲ ➩ ➍ ➻ ❇ ❩ ❣❤✐ ❢ ❡ ❞ ❝ ❜ ❵ ❵❛ ❴ ❫ ❪ ❭ ❬ ❨ ❦ ❳ ❲ ❱ ❯ ➻ ◗ P ❖ ◆ ▼ ■ ❍ ● ❥ ❧ ➉ ④⑦ ➅ ➄ ➃ ➀ ❿ ❾ ❺ ❹ ❽ ❻ ❼ ❻ ❹❺ ⑥ ♠ ④⑤ ➼ ① ✇ ✈ ✉ t r rs q ♠♣ ♦ ♥ Ö îð Ò ✌ ✺ ❀ ù ✡ ☛☞ ✺✻ ✹ ✸ ✱ ✧ ✦ ✷ ✳ ✿ ✧✲ ✪ ✒ ✓✔ ✫ ✧✪ ★ ✩ ✧★ ✦ ✥ ✕ ❀ ❁ ✕ ❋ ❉ ✻ ❇ ❍ ❂ ❊ ● ✻ ❇ ❉ ✾ ✽ Ñ ❂ ❉ ❈ ✻ ❇ ❀ ✹ ❅ ✾ ❆ ✠ ✝ ✹ ü ✖ ✗ ❃ í ù ø ë ñ ÷ ì ö õ ô ó ñ ñò ì ë ë ð ï ê î í ì ë ê ç æ å ú û ✔ ✔✕ ✘ ✙ ✣ ✢ ✒ ✜ ✎ ✔✕ ✘ ✖ ✗ ✖ ✢✣ ü ✠ ✞✟ ✆✝ ù ú ✁ � ÿ ù ñ ù þ ëý ■ ❊ ❉ ➒ ➠➡ ➟ ➝➞ ➜ ➛ ➙ ↕ ↔ þÿ ➣ → ➔ ➓➔ ➑ ➥ ➐ ➌➍ ÿ ✺ ➇ ➆ ➅ ➃➄ ➁➂ � ❾ ❼❽ ❻ ❺ ➢➤ ➦ ✂ ÷ ô ❰ ❮ ❐❒ ✃ ➱ ➮ ➴ ➻ ➼ õö ➴ ➸ ø ➧➨ ➪ ➚ ➼➾ ➳ ➼➽ ù ➸ ➳➵ ➲ ➯ ý ➭ ➫ ➩ ✁ þ Ï ❭ q ♣ ♠ ❧ ❥❦ ú ❣ ❡❢ ❞ ✁ ❛ ✂ ✆ ❬ ♥s ❩ ❨ ❳ ❲ ✝ ❘❙ ◗ ◆❖P ✞✟ ❑ ❏ ✿ ✻ ✾ r ♥♦ t ② ① ✇ ù ③ ✈ ✉ ④ ⑤ ⑥⑦ ⑧ ⑨ ⑩ ❶ ❷ ❸❹ ✄☎ pointer z): pointer 106 ➁✒➈ if (z->LC == NULL || z->RC == NULL) y = z; ➆✪➇ 3 úÝûü ➶✆➷ function BinaryTreeDelete ( pointer *R, else if (y == y->p->LC) y->p->LC = x; ➁❆➂ ➶✆➹➘ ✛✏✜ if y has other fields copy them two; òÝó ✘✚✙ ➾❆➚ if (y->LC != NULL) x = y->LC; ➙✒➝ ➼✒➽ ➾❆➚ÑÐ if (y != z) z->Key = y->Key; if (x != NULL) x->p = y->p; if (y->p == NULL) return x; ➼✒➽ else y = TreeSuccessor(z); ➸✪➺ ✍✏✎✑ éÝê else y->p->RC = x; ❷✴❸ ➭❆➯ else x = y->RC; ⑩✔❶ ➩✪➫ ⑧✪⑨ ÛÝÜÞ return y; ❘❚❙ ➎✒➏ R - z ②❆③ ➊✔➋ p 105 - Ï✏Ð ✼☞✽ ❤✶✐ ✜✛✤ ❏▲❑ í✓î 3 ❊✴❋ ❈✴❉ ❜☞❝ è✓é ✶✴✳ ❴✛❵ ❄❆❅ ➷✏➬ ❪☞❫ ➎✏➏ ✙✛✚ â☞ã ✴✶✵ ✖✔✗ ßáà ➘✏➵ ➊✏➋ ✲✴✳ ✓✔✞ ✰☞✱ ✽✴✿❀ ➶✛➹ ➈✛➉ - ✏✒✑ ó✓ý ❚☎❯❱ ✽✴✾ û✓ü ✮☎✯ ✑✓✒ ❄✛❅ ✬✛✭ Ù✏Ú ✍✏✎ ✼☞✾ ★✪✩ ➺✶➻ ❿✶➀ ▲✶▼ ✥✧✦ ✠✆✡ ✡☞☛✌ ❃✶✻ Ô✶Õ õ✓ö è❵é ó✓ô - ✄✆☎ 1) 2) 3) ✂☎✄ ✼☞✾✿

  53. ✤ à ë ä à ï í è ß î ß Ý á í ç ç Ý Þ ì ë ä è åæç ã à ä á ßà ì ä Ð ß ò æ ï à ç ê ä ä ç ë í ë ñ à ç ð ä à ç å î ë ä ë ã ð ß Þ ❮ êó ✥ ➬ ➪ ➬ ➾➚ ➸ ➺ ➴ ➷ ➶ ➾➚➪ ➽ ➸ ➞ ❐ ➥ ➩ ➫ ➡ ➠ ➟ ➲ ➞ ➞ ➥ ➩ ➧ ➨ ✃ ❰ Õ Ó Ñ Ò ❐ Ð Ñ ÏÐ ❰ ❐ ❰ Ï ❮ Ï ❐ ÏÐ Ñ Ï × × Õ Ð Ï ❮ ÓÔ Ò ❐ Ð Ñ ä á ➥➦ ✳ ✾ ✺✽ ✼ ✺✻ ✮ ✯ ✶ ✵ ✴ ✮ ✦ ✮ ✲ ❀ ✛ ✱ ✰ ✛ ✯ ✮ ✭ ✛ ✦ ✬ ✜ ✫ ✿ ❁ ✩ ❇ ■ ◗ P ❖ ◆ ▼ ▲ ❑ ❄ ✼ ❊ ❃ ✷ ✺ ● ❋ ✸ ❊ ✺ ❃❄ ❇ ❅ ❆ ❅ ❃❄ ✸ ❂ ✪ ★ â þ ✟ ✞ ✆✝ û ü û ú ✁ ø � ö öÿ ý ✠ ú û ö ü ÷ öø ö÷ õ ô ó à ä ã ✞ ✆✝ ✦ ✚ ✧ ✦ ✢ ✜ ✛ ✥ ✤ ✚ ✣ ✢ ✜ ✛ ✖ ✌ ✕ ✔ ✓ ✒ ✑ ✝ ✆ ✠ ✡ ✎ ✍ ✄ ✡ ➧ ➵ ➟ ❫ q ❝ ♥ ❝♠ ❧ ❦ ❝ ❣ ❡❢❣ ❝❞ ❪❜ ❪ ❝ ❩ ❨ ◆ ❉ ❙ ▼ ❋ ❄ ❯ ❚ ❅ ❙ ❜ ❞ ◆ ♥ r ❣ ✐ r ✈ ❣ t ♠ ❜ ✉ ♣ ❢ rs s q s ➤ ❣ t ❝ t ❜ q ❜ ♠ ❇ ▼ ✇ ✯ ✭ ✳ ✹ ✮ ✸ ✷ ✶ ✵ ✳ ✴ ✳ ✮ ✻ ✭ ✲ ✱ ✬ ✰ ✯ ✮ ✭ ✬ ★ ✧ ✦ ✺ ✼ ❋ ❇❈ ❖ P ❖ ▼◆ ❅ ▲ ❇ ❑ ❏ ■ ❇❊ ❉ ✻ ✭ ✼ ❃ ❂ ❁ ✻ ✳ ✻ ❀ ✿ ✭ ✾ ✽ ♠ ❢ ➐ ↔ ➙ ➊ → ➓ ➑ ➐ ➔ ➉ ➏ ➌ ➉ ➋ ➐➑ ➊ ➍ ➎ ↔ ➋ ➎ ➏ ➐ ➓ ➈ ➊ ➋ ➈ ➊ ↔ ➉ → → ➉ ➏ ➎ → ➐➑ ➊ ➒ ➉ → ➓ ➒ ➣ ➊ ➈↕ ➎ ➆ ➏ ❶ ➃ ➡ ➠ ⑩ ➀ ➂ ➀➁ ➞ ⑤ ➊ ❻❼❽ ❺ ❶ ➝➟ ⑤❷ ② ⑥ ④ ⑩ ② ➛ ➜ ➞ ➊ ➈ ➋➌ ➝ ➊ ➈➉ ➢ ➇ ① ②③ ➆ ➀ ➄➅ ④ 108 3 ✷❉❈ ❍❏■ ✗✙✘ - ✷✹✸ 107 - Õ❳Ö ✡☞☛ 3 ➲➯➳ ❾❆❿ ÏÛÚ ♦❛♣ ✂☎✄ ❒❳❮ ➮❍➱ ➫➯➭ è❘ã ❸❆❹ ❤❥✐ é❍ê ➒❛➓ ❋❍● ❒❳❮ ❄❘◗ Ø❍Ù ➹❆➘➴ ⑦⑧❛⑨ ❄❆❅ ❴❛❵ ➍❛➎ â❘ã ù❆úû ✩✫✪ - ❬❆❭ - ➺➼➻ ✡☞✏ ❱❳❲ Ü➯Ý

  54. ❘ ❻❽ ➑ ➐ ➎➏ ➍ ➌ ➊➋ ➉ ➆ ➇➈ ➀ ❿ ❾ ❼ ➓ ❻ ❺ ❙ ❸ ❷ ✇ ✈ ③ ④ ⑧ ⑦ ✉ ➒ ➔ r ➤ ➵ ➳ ➲ ➭ ➫ ➲ ➯ ➭ ➩➫ ➨ ➧ ➥➦ ➢ → ➡ ➞ ➠ ➟ ➞ ➝ ➜ ➛ ➙ ↕ ↔ ➣ ⑥ ❹ ❵❛ ❡ ❩❧ ❫ ❴ ❩ ❵ ❭ ❜ ❝ ❞ ❬ ❤ ❢ ❵ ❩ ✈✇ ❬ ❣ ❦ ❥ ❩ ❤ ♠ ❨ ❬ ③ ① ② ① ✈✇ ❚ ❯ ❱ ❨ ❪ ❩ ❤ ✐ ♣ ♦ ♥ ❤ ❵ ❭ ✐ 109 3 ⑩❉❶ ⑥❉⑨ ➅s➆ ➃❉➄ ➁☎➂ ④❉⑤ ❲✙❳ - t☞✉ qsr

  55. ➸ ❥ ②③ ❣ ✉ ① ❤ ❣ t ❧ s r ⑤ ❣q ♣ ♥ ❣ ❧♠ ✐ ❣❤ ❳ ❨❭ ④ ⑥ ❲ ❥ ❼ ❻ ❷ ❶ ⑩ ⑨ ⑧ ❧♠ ❦ ✐ ⑦ ❤ ❣ ♣ ♥ ❤ ❣ t ❜ ④ ② ❈ ❲ ❾ ❊ ❨❭ ❈ ❲ ❲ ❳❩ ❳❨ ❲ ■ ❋ ❉ ❅ ❈ ❈ ❅ ❱ ❯ ▼❚ ❙ ❘ P◗ ❱ ➺ ❳❩ ❈ ❆ ❈ ❲ ■ ❍ ❨ ❳ ❲ ❛ ❬ ❲ ❨ ● ❈ ❉ ❴ ❩ ❵ ❴ ● ❈ ❽ ❿ ❏❑ ❐ Ó Ò × Ô ÒÓ ➸Ñ Ð Ï ❒ ➮✃ ÔÛ ➱ ➮ ➴ ➼ ➶ ➼ ➺ ➚ ➼ Ø Ü ➲ è ò ðñ ï î ìí Ñ Ï ❐ ê Ð Ý â Ï ä ã â ➵ ➸ Ôá à Þß ➸ ➭ ❻ ➇ ➌ ❼ ➋ ➊ ➉ ❼ ➂ ➈ ❽ ➆ ❼➎ ➅ ➄ ➂ ➂➃ ❾ ❽ ❼ ➁ ➀ ➍ ➏ ➯ ➛ ➭ ➧ ➥➦ ➤ ➡➢ ➠ ➞ ➟ ➞ ➔ ➊ ➙ ↕ → ➊ ➋ ➒ ➑ ➐ ➊ ➂ ❏❖ ❈ ☎ ✠ ☛ è ✒✓ ✑ ☛ ✡ ✟ ✓ ✏ ã ✟✠ ✞ þ é ü ✔ ➼ ú ✚ áå ✟✛ ✖ Þ ☛ ✒✓ ✙ ➻ ✗✘ ✖ ✕ ✒ ✑ æ ç ✁ ã ð ➶ ➽ ú ù ß ö ➪ î ÿ � Û ï Ú ➹ Û î ß Þ þ õ ✄ ✝ õ ✆ ì ë Û ➪ ö ✄ ✂ æ ✁ ä ü õ à ✟ ä ✲ ❃ ❀ ❂ ✼ ✳ ❮ ➹ ❒ ❰ ➮ ✽ ➶ Ï Ð Ñ ✽ ❐ ✖ ❋ ➷ ■ ➬ ➶ ➹ ❍ ❈● ❊ ✼ ❉ ❈ ➘ ➮➱ ➮ ✃ ❁ ✶ ➶ Ò ✛ Ø ✤✥✦ ✢✣ Ñ Ð ➴ ✍ ✣ ➘ ✙ ✜ ✘ ✒✓ Þ ß ✢ ✧ Ó ★ ➶Ô Õ ✳ Ð ✴ ✳ Þ ✮ ➮ Ð Ö ✭ × ✪ ✩ 112 v ❮✿❰ 3 ç☞èé ➧✺➫ å✺æ ➨✷➩ ➷✎➬ óõô ➧✺➫ ➨✷➩ – AVL ➺✎➹➘ ➜✷➝ ➪☞➶ balance Ù◆Ú ➺✎➽➾ ➣☞↔ ➺✎➻ ➓✺➔ Õ✿Ö ❸❺❹ - ❮✿ë ➳✿➵ O(logn). LeftHeight(r) 111 ❅❇❆ ✡☞☛ balance(v) = RightHeight(v) – LeftHeight(v). . ❥✿❦ (RightHeight(T)) 3 ❀✺❄ ❳❫❪ ✌✎✍ r êâë ✫☞✬ v v->LC = NULL, ▲◆▼ LeftHeight(v) = 1 + Height(v->LC), ❡✺❢ ❊✺❬ ❀✷✰ ❜❞❝ ❡✺♦ Delete ❀✷❁ ✸✿✾ ❊✺❬ LeftHeight(T) v RightHeight(v) Ù❉ã ✉✇✈ LeftHeight(v) = 0, LookUp(), Insert (RightHeight(r) ú☞ýþ ✻☞✼ àâá Ù❉í ✸✺✹ balance ( û✹ü Ü❉Ý ÷☞ø ✵✷✶ Ù❉ÚÛ ô☎õ ➾✙➚ - ❡✺♦ ñâòó ✯✱✰✲ ✌✎✍ ?

  56. ö ✤ ✛ ✤ ✘ ✚ ✜ ✩ ✛ ✣ ✣ ✪ ✔ ✧ ✮ ñ ð ✍ ó ù ü ✁ ✡ ☎ ★ ✒ ø ✖ ù ✁ ✂ ü ÷ ú û ú ✔ ✕ ✗✘ ✔ ✙ ✚ ✘ ✛ ✙ ✜ ✢ ✜ ✔ ✣ ✔✤ ð ö õ ÷ ✂ ü ✁ ö ÷ õ ÿ ÷ ú ✂ ö ✄ ð ò ø ï ✄ þ ✟ ð ò ø ö ú ñ ÷ þ ✡ ☞ ü û ÷ ò ö ✄ ö ð ð ð õ ð ✡ ☛ ö ✂ ï ÷ ✂ ò ✡ ✁✂ ☞ ò � ✁✂ ü ÷ ú û ú ý ø ü ✁ ò � ÷ � ✂ ï ✄ � ÿ õ ð ÷ ð ð ÷ ñ à â ã á ã Ô æ í î ïð ð ó ò ÷ ô õ ö ÷ø ö ùú õû üö ÿ õ ✝ ú ✍ ï ÷ ö ☛ ñ ñ ù ñ ð ✌ ñ üö ñ ð üö ð ð ✡ ð ☞ ð ó ÷ ð ÷ ý þ ö ù ü ù ø ✞ ✁ ø ö ð ð ✝ õ ð ✡ ✂ ÷ ✡ ✄ ö ù ☎ ☞ ð þ Ü ❳ ❋ ▲ ❍ ❋ ●■ ❬ ❋ ❖ ❙ ❋ ❯ ❍ ❍ ❙ ❍ ❭ ◆ ▼ ❪ ❏ ❪❫ ❲ ● ❙ ▼ ❨ ❍ ▼ ❯ ❋ ❖ ❙ ❯ ❙ ❍ ❘ ❍ ❑ ■ ❯ ❱ ❯ ▲ ❩ ❘ ❑ ❙ ❍ ❘ ❘ ❯ ❘ ❲ ❱ ❯ ▲ ❫ ▼ ❯ ▲ ❫ ■ ❬ ❴ ❘ ❯ ▲ ❫ ❙ ❍ ❖ P ❯ ▲ ❙ ◆ ❬ ❲ ▲ ▲ ❫ ❑ ❯ ▲ ❫ ■ ❬ ❲ ❴ ❯ ❫ ❵ ■ ❍ ● ■ ❱ ❘ ● ❵ ❍ ❱ ❱ ❏❲ ❱ ð ✁✂ ✄ þ ✟ ð ò ÷ ñ ñ ☛ ✝ ü ò ✁ ✰ ✱✲ ✳ ✴ ✷✸ ✹ ✺ ✹ ✻ ï ð ✴ ö õ ð ✡ ÷ ö ✂ ð ✍ ÷ ò ð ✂ ✍ ✎ ✮ ✯ õ ö ð ï ☛ ✡ ð ✳ ✷ ❖ ❍ ✹ ✴ ✻ ❂ ✹ ❅❉ ❊ ❍ ❏ ❑ ❍ ✽ ▲ ▼ ◆ ● ❏ ❍ ❙ ❍ ❚ ❯ ▼ ✺ ✺ ✸ ❃ ✷ ✳ ❀ ✷ ❁ ❂ ✹ ✳ ✷ ❂ ✹❄ ✺ ✲ ❅ ❆ ✹❇ ✳ ❆ ✹ ✸ ✷ ❆ ❈ ì ó Û ❲ ▲▼ ❴ ❫ ▲ ❴ ❯ ❙ ❵❛ ❯ ◆ P ▲▼ ❖ ▼ ❨ ▼ ❳❨ ❯ ▼ ❭ ❫ ❚ ❲ ❱ ❱ ❲ ▲▼ ❚ ❯ ❙ ❳❨ ❱ ❩ ❲ ▼ ❬ ▲▼ ▼ ❭ ▼❪ ❯ ❱ ❩ ❱ ❲ ❱ ❜ ❙ ⑤ Ü ① ❦ ③ ④⑤ ⑥ ⑤ ⑦ ⑩ ⑨ ✉ ❶ ❷ r ❸ ✈ q ❢ ♦ ❹ ❦t ❥ ❦t ❪ ❤ ❴ ❪ P ❨ ❜ ❨ ▼ ❡ ❞ ❣ r ✐ ❞ ❥ ❦ ❞ ❥ ♦ ❦ ✐ ❣ ❯ ❚ ❺ ☞ ☛ ☞ ✌ þ ✍ ✎ þ ✏ ✑ ☎ ☎ ☞ ✒ ✓ ✔ ✌ ☞ ✘ ✖ ✙ ✚ þ ✡ ✘ ✂ ø ù ú ý þ ÿ � ✁ ý ✄ ÿ þ ÿ � ☎ ✆ ☎ ✝ ✞ ✟ ✠ ✛ ✜✢ ◗ ❋❍ ❇ ✿ ❃ ❊ ❋ ● ❆ ❊ ● ❈ ✽ ❇ ✿ ❇ ❁ ❇ ✿■ ❏ ❑ ▲▼◆ ❖ ❁ ❉ ✚ ✬ ✜ ✙ ✣ ✤ ★ ✘✩ ✦ ✪ ✫ ✯ ❈ ✵ ✮✷ ✺✻ ✾ ✿ ❀ ❁ ✽ ❁❆ ❇ ✈ ✈✇ ② ➮ ➥ ➲ ➼ ➽ ➾ ➳ ➲ ➹➘ ➴➘ ❒ ➻ ➴❮ ➶ ❰ Ï ➷ ❮ Ñ Ò ÓÔ ÕÖ ➲ ➞➺ × ➨ ➝ ➢ ➤ ➞ ➟ ➠ ➥➦ ➥ ➧ ➩ ✐ ➫ ➟ ➭ ➥ ➞ ➯ ➲ ➳ ➞ ➵ ×Ø Ù ➠ ç Ö ê â ã Û ä å á Ú Ý ß Ü ç ßë ê ç é Û ç Ø × Ü Ú Õ â ÚÛ ÜÝ Þ × Ö ×ß × Ý Õ Ú ã Ø Û äå æ Ò × Ý ç ß Üè é ➡ ➸ ➆ ➌ ❡ ➅ ➊ ➎ ➋➌ ➈ ➇ ✐ ➏ ✈ ➁ ➐ ➆➑ ➊ ❺ ❽ ❿ ➇ ➂ ➄➊ ➋➌ ➍ ➉ ➈ ❾ ➁ ➇ ➌ ➆ ❿➅ ❿ ➁ ➂ ➒ ➌ ❼ ➝ ➔ ❞ ➣ ➁ ↔ ➅ ↕ ✐ → ➊ ✈ ➟ ➁➓ ♦ ➞ ➇ balance 0, 114 ☛ ✴ö ✥✦✜ ❋✠● ❋✠●■ 3 Insert ❋✆❳ Ï✴Ð ❋✠●❍ AVL ✟✠☎ ý✴þ balances à➀á Insert ✾✠✿ ❋✠● ✾✠✿ ✑✓✒ ➱❐✃ ý ✏✎ ✼✠✽ ✫✭✬ ➷✧➬ balance �✆☎ ❖◗P❘ -1. balance ✱✶✵ ➚✴➪➶ – ➙➜➛ - ✯ ✲ïð ❛✶❜ Key, Info, 113 ❢✴❣ O(logn). 3 ❝❄❞ O(logn). ✥✧✦ ❢✲② AVL AVL struct balance? AVL ✸✲✹ ♣✲s ✰✲✶ ❻✲② ✯✴✳ LC, RC, balance. ✰✲✱ ❘❄❙◆ ♣✲q LookUp()? AVL P✴◗ ❂❄❃❅ ➃❄➄ ⑧✴⑨ ✕✗✖ ✭✧✮ ❧✧♠♥ û❺ü - ✼✲✽ ❾➀❿

  57. ❝ ➨ ➢ ➤ ➠ ➝ ➠ ➞ ➝ ➥ ➥ ➦ ➢ ➫ ➞ ➩ ➞ ➭ ➨ ➯ ➲ ➠ ➨ ➵ ➻ ➼ ➽ ➾ ➚ ➽ ➪ ➝ ➙ ➴ ➊ ❼❻ ❿ ➀ ➁ ➂ ❼➃ ❹ ❞ ➅ ❽ ➃ ➅ ➆ ➇ ➍ ↕ ➎ ➑ ➒➓ ➔ ➏ ➎ → ➔ ➏ ➑ ➒➓ ➏ ➣↔ ➎ ➽ ➱✃ ❹ ß Ô ➮ Ü Ý ❒ ❐ Ô Û ❒ Ó Þ Ï × Ù ❐ à áâã äå æ ç â ê è â å í ì î ï Ó Ò ➱❐ Ò ❒ ➮ ❮ ❰ Ï ÐÑ Ò ❐ Ó Ð ❒ Ô Õ Ö ➱ Ò Ø Ù Ú Ð ➮ ➱ ❐ Û ✃ ➱ ➮ × Ï Ñ ❺❻ ➄ ➆ ❺ ➇ ♠ ➈ ❺ ✇ ➉ ➅ ➄ ❸ ❻ ➁ ✈ r ✉ t r rs ❝ ❞ ♥ ❡ ❢ ❣ r ① ❧ ❧ ❺❻ ⑨ ⑩ ❶ ④ ③ ③ ⑧ ❧⑦ ⑥ ⑤ ❹ ❼ ➃ ❻ ❧ ❿ ④ ➂ ❼➃ ❹ ➄ ③ ➅ ❽ ② ❦ ♠ ♠ ⑥ ♥ r ❧ ❧ ❦ ❣ ② ③ ④ ❧ ⑤ ❧⑦ ♠ ⑧ ③ r ③ ⑨ ⑩ ❢ ❶ ④ ③ ❡ ① ③ ✇ ♥ ❦ ❧ ♣ q q ♣ ❦ ♦ ✈ ♠ ❧ rs ♥ ✉ t ♠ ♦ r Single Right 116 ➸✓➺ 3 ❮✦× ➯✦➳ AVL ➋➐➏ ➧✓➨➩ ➋◗➌ ë✦ì Single Left Rotation, RR Rotation, LL)? ❽✠❾ ➷✠➬➮ è✦é ➪✦➘ ➟➡➠ ❷✆❸ ➶➡➹ ❤❥✐ - ➛◗➜➝ 115 3 ➅✆➆ AVL ➀✓➁ ❽✠❾ ❷✆❸ ❤❥✐ -

  58. ð ❂ ● ❋ ❉ ❉❊ ❅ ❄ ❃ ❈ ❇ ❂ ❆ ❅ ❄ ❃ ✾ ■ ✽ ✼ ✻ ✺ ✹ ✷✸ ✴ ✶ ✵ ✱ ñ ✳ ✲ ✬ ❍ ❄ ✧ ❚ ❡ ❛ ❬ ❝ ❜ ❲ ❩❛ ❵ ❪ ❩❨ ❳❨ ❲ ▲ ▼ ❙ ❏ ❘ ▲ ❉ ▲ ◗ ❃P ❖ ◆ ❃ ▼ ▲ ❑ ❃ ❉ ✫ ✮✴ ø ✝ � ✁ ✂ ✝ ✞ ✍ ✌ ☞ ✝ þ ☛ þÿ ✡ ø ✄ ù ☎ ✠ ✟ ✣ ✞ ✝ ✆ þ ú ø ÷ ò ó ô ÷ ✖ ✥ ø ù ú û ✤ ✒ ✒ ✕✣ ✢ ü ✙ ý ✔ ø ù ✕ ✓✔ þ 118 3 ❝✑❞ Double Rotation, RL (Right-Left): Case A AVL ❫✜❴ ❬✘❭ ❯✑❱ ✿❁❀ - 117 ) 3 ✥✑✦ AVL ✰✑✱ ✭✯✮ ★✪✩ Single Right Rotation, LL ( ✚✜✛ ✖✘✗ ✎✑✏ õ❥ö -

  59. ❢ â Û ❒ Ü Ý ❒Þ ß Ú Ò Ú à á Û Ù Ú å æç èç ë î èï å ð ñ é Ú ❒ ó Ð ➘ ➴ ➷ ➬ ➮ ❐ ❣ ❮ ❰ Ï ❐ Ñ Ò ❒ ❮ ❰ ÒÓ Ò Ô Õ Ö × ❮ Ø ï ÷ ➾ ✚ ✠ ✌ ÿ ✑ ✝ ✞ ✒✓ ✔ ✕ ✖✙ ✗ ✗ ✍ ✖✛ ✗ ✜ ✗ ✖✢ ✗ ✜ ✛ ✣✤ ✥ ✦ ✑ ✏ ø ✡ ù ú û ü ý õ ☎ ÿ ✆ ✠ ÿ � ÿ ☛ ☞ ☛ ✝ ✞ ✌ ☞✍ ✆ ✎ ✠ ÿ ✁ ➸ ❒ ➼ ❸ ♥ ⑦ ⑧ ♥⑨ ⑩ ⑤ t ⑤ ❶ ❷ ⑥ ⑤ ⑤ ❻ ❼❽ ❾❽ ➁ ➄ ➪ ❻ ➆ ➇ ❿ ⑥ ④ ➉ s ❤ ✐ ❥ ♠ ♥ ♦ ♣ q ♠ r ♥ ♥ ♦ ♣ t✉ t ✈ ✇ ① ② ♦ ③ t ➅ ❾➅ ↔ ➋ ➙➛➜ ➞ ➢ ➤ ➛ ➣ ➟ ↕ ➥ ➣➦ ➟ ➥ ➔ ➡ ➝ ➩ ➺➻ ➺ ➽➾ ➼ ➚ ➌ ➍ ➸ 120 3 ñ✑ò AVL ✧✩★ ì✜í ôöõ é✘ê ✝✟✞ ã✑ä ➱❁✃ - ✁✄✂ ✖✘✗ þ✪ÿ � Left-Right: 119 3 ➇✑➈ ➾➹➶ AVL ➑✘➒➓ ➧✑➨ ➎➐➏ ➊✪➋ Double Rotation, RL (Right- ➲➵➳ ➂✜➃ ➫➯➭ Case A & Case B ➞➠➟➡ ❿✘➀ ❹✑❺ ❦❁❧ - ↔✯↕ →✯➣

  60. ✪ ✫ ✬ ✭ ✱✲ ✱ ✳ ✴ ✵ ✶ ✬ ✷ ✱ ✸ ✰ ✹ ✺ ✫ ✻ ✼ ✫✽ ✾ ✹ ✱ ✹ ✿ ❀ ✫ ✯ ✺ ✎✛ ✌ ✔ ✗ ✎✘ ✎ ☞ ✙ ✘ ✔ ✚ ✔ ☞ ✪ ✙ ✜ ✢ ✣ ✤ ✥ ✦ ✪ ✫ ✬ ✭ ✮ ❁ ✹ ✖ ❘ ▼ ❱ ▼ ❲ ❩ ▼ ❛ ❴ ◆ ◆ ■ ❯ ▲ ❲ ❨ ❩ ■ ❚ ■ ❙ ❚ ◗ ❩ ▼ ❱ ❱ ▼ ❉ ❑ ❊ ❈ ❂ ❋ ❉ ● ❊ ❈ ❍ ■ ❏ ▲▼ ❘ ■ ❙ ▼ ▲❯ ■ ❱ ❲ ❳ ■ ❲ ❙ ❬ ☛ ✕ ❩ ☎ ô ô å û ý ö î ï ê ï ê ✆ å ✝ ✞ ð ù ä ë î ï ä ê ò ✟ å ✫ ÷ ê ì è ç í ê ø ä ù ë ñ ú é û ï å ë é å ù ô ù û ø ü ê û é ✔ ç ë ç ù ï ê é ì ä ÷ î ë ê ï ì è ç í ê ÿ ✡ ☛ ☞ ✌ ✍ ✎✏ å ù ù î ï å ë ç è éê î ù ý é å ✠ ú ç ù ÷ ê ð ê þ î ø ù ï ÷ ▼ ❱ ç ➇➈ ⑦ ♦ ♣ ❦ ⑦ ⑥❾ ❼ ➂ ➃ ➄ ➅➆ ➇ ④ ➉ ➅ ➊ ➋ ➂ ➌ ➈ ➇ ➉ ➅ ➊ ➍ ❥ ③ ➁ ⑦ ❦ q ❻ ❼ ❶ ⑥ ③ ✉ q ♥ ❺ ❥ ♦ ♦ ❧ ❽ ❥ ❦ q ♦ ① ❧ ♥ ❥ ❦ ➂ ➆ ♣ ➏ ➇ ➁ ➂ ➊ ➄ ➋ ➂ ➁ ➙ ➆ ➇ ➐ ➂ ➆ ➏ ➇ ➍ ➑ ➃ ➙ ➃ ➂ ➃ ➙ → ➓ ➃ ➎ ➆ ➏ ➁ ➂➐ ➃ ➑ ➃ ➇ ➅ ➇ ➋ ➇ ➁ ➊ ➒ ➓ ➔→ ➣ ➁ ➇➈ ↔ ↕ ➋ ➇ ➐ ➅ ❥ s ❲ ♥ ♦ ♥ ② ♥ ❦ q③ q ④ ⑤ ❦ ⑥ ① ✇ ⑦ ⑧ ♣ ♦ ⑨ ⑥ ① ⑤ ⑨ ⑩ ⑥ q ❥① r ⑥ ▼ ◆ ❲ ❞ ❱ ■ ❨ ▼ ❯ ❘ ❙ ❚ ❯ s ◗ ❚ ❳ ✐ ❥ ❦ ❧ ❦ ♠♥ ❥♦ ♣ qr ① ❶ ❸ ❥ ❦ ⑥ ♥ ❥ ♦ ♣ ❥ ⑦ ❺ ⑤ ⑨ ♥ ⑥ ⑥ ① ⑤ ⑨ ⑩ ⑥ ① q♦ ❹ ♦ ⑤ ⑦ ③ ⑦ ⑥ ❦ ❷ ❸ ♠ ♣ ❥ ① q ♦ ④ ❧ ❶ ♦ ⑥ ④ ⑧ ❹ ♦ ⑥ ⑨ ⑥ ♥ ⑥ ④ ❧ ❦ ♦ ê ë ë ❣ ❬ ❱ ❲ ❞ ◗ ❨ ❚ ❱ ❡ ❢ ❣ ❱ ❭ ❩ ❘ ❝ ❚ ❱ ❛ ❳ ❘ ❤ ❵ ❝ ❳ ✐ ❱ ❤ ❤ ♦ ❣ ♥ ❥ ❭ ❚ ❞ ❝ ❳ ❘ ❵ ❜ ❝ ❩ ❱ ❲ ❳ ❨ ❩ ❬ ❱ ❲ r ❩ ❳ ❦ ④ ❵ ❞ ❥ ❧ ❥ î ✉ ①② ⑦ ✇ ⑩ ②❶ ❭ ① ④ ✇ ④❹ ⑩ ❺ ⑩ ④ ✇ ④ ❹ ❝ ❛ ❨ ❚ ❩ ❬ ❱ ❭ ❤ s ❭ t ❱ ◗ ❨ ❭ ❦ ✐ ❞ ❭ ❝ ❪ ❫ ❴ ❩ ❭ ❚ ❜ ❚ ❘ ❚ ❹ ❁ ❃ ❄ ✳❅ ❆ ❁ ✹ ❁ ❇ ❈ ❉ ❂ ▲ ❂ ▼ ❑ ❊ ◆ ▲ ❖ ▼ ❑ P ◗ ❘ ✳ ❁ ❳ ✳ ✬ ✭ ✮ ✲ ✳ ✴ ✵ ✶ ✲ ✷ ✸ ✴ ❀ ✵ ✹ ✺ ✹ ✻ ✼ ✽ ✾ ✴ ✿ ✹ ✳ ❱❲ ❬ ❵ ❥ ❱ ✐ ❞❥ ❤ ❛ ❭ ❝❦ ❭ ❝ ❵ ❞ ❧ ❱ ❥ ❴ ❚ ❱♠ ❛ ♥ ❱♠ ❳ ❬ ❱ ♠ ❤ ◗ ❱❲ ❚ ❩ ❭ ❪ ❫❴ ❚ ❱ ❩ ❱❛ ❱ ◗ ❜ ❝ ❱ ❞ ◗ ❨ ❚ ❱❡ ❢ ❣ ❣ ❱ ❵ ❚ ❱ ① ❴ ❒ ➬ Ñ ï Ò ➬❒ î ë ❮ Ó Ò Ô ê ❰ í ç Ø Ù Ù Ú Û Ö è ➬ ê ❮ Ó ➹ å ➬ ë ï ñ ➾ ð ➚➪ ➚ ➶ ➪ ç ➘ ➴ ➘ ê é ➷ ➬ ê ➬✃ ❐ ➬❒ Ý ❐ ➺ ß ➬✃ ❮ Ó Ò Ô ➬ ëì ê ✃ ç à ➬ Ö Ñ æ ❐ ➬ å Ï Õ Ñ ➱ Ñà ❐ Ö Ñ Ò Ö ❐ Ú ❐ Ñ ✃ Ý Ö ❰ Ð Ð ❐ ❰ ➬Þ ß ✃ Ü ➬ Ø × ❐ Ï ð ➽ ➳➵➸ ➎ ➌ ➋ ➍ ➃ é ➅ ê ➅ ï ➄ õö ➑ ➊ ➐ ➈ ➆ ➃ ➐ ➃ ⑨ ✇ ❿ ➀ ➁ ➂ ➄ ➋ ➃➅ ➆ ➇ ÷ ä ì ➊ ➄ ➮ ➒➓ ➙ ➙ ➠ ó ➡ ò ➟ ➔ ↕ ➙ ↔ ï ➨ è ➭ ➯ ➲ ➟ ê ➙ → ô ô ➛ ➜ ➝ ➓ ➞ ê 122 balances. ❴❵❚ 3 ❭❫❪ Insert()? ③❤❿ balances? ❨✂❩ ❡❵❱ ❏❫❞ t✈✉ ❅❇❆❈ ❂❄❃ ❏P❚ ◗❄❘ ▼❤❣ ◆P❖ ➎✓➃ ✧✩★ - ▼❝❜ ❨❵❢ ➀✂➁ –1 121 ❺❏❾ ➩⑤➫ ③⑤② 3 ÑqÜ balance �✂✁✄ ➥➧➦ ❽⑤④ þöâÿ ❻⑤❼ ❫❯❵ ➈✟➏➐ – ➛➤➢ balance ♥✟s Õ✄Ö× balance ♥✟s balance ❞q♣ ❷⑤❸ ❍❏■❑ ➈❯➉ è✟éê balance –1? ❊●❋ Ï❯Ð (critical node). –2? ❨❯❩ ã●ä balance ⑧❏⑨ ➣✟↔↕ ✑✓✒ ❨❯❩ –2. ó✟é ➬✃âá ❨❯❩ balance ③⑤④⑥ ❙❯❚ ➻⑤➼ ✯✱✰ - ➮✟➱ ✈✟✇

  61. ➛ ✘ ✒ ✤ ✑ ✛ ✢ ❤ ✓ ✗ ✒ ❤ ✙ ♣ ✒ ✒ ✙ ✙ ❦ ✒ ✙ ✚ ✥ ✧ ✑ ✥ ❤ ✙✚ ✖ ✢ ✓ ❤ ❧ ✕ ✘ ✛ ✢ ❤ ✓ ✗ ✗ ✜ ✒ ✢ ✢ q ❦ ✛ ✢ ❤ ✓ ✗ ✒ ❤ ✙♣ ✧ ❦ ✫ ✢ ✣ ✓ ❷ ✳ ✢ r ✈ ✇ ①② ③ ④ ⑥⑦ ⑧ ⑨ ⑩❶ ❸❹ ✲ ❺ ❷❽ ➁ ❿ ➂ ➃ ➄ ➅ ➆ ➇ ➄ ➈➉ ✖ ✲ ✢ ✦ ✓♦ ❥ ✢ ❧ ✔ ✕ ✓ ✒ ✙ ✛ ❤ ✙ ✜ ✕ ✑ ♦ ✑ ✒ ✙ ✚ ✔ ✜ ✣ ✱ ✙ ✚ ✣ ✕ ➍ ✒ ✧ ❤ ✕ ✒ ✙ ✚ ✣ ✢ ✓ ♥ ✑ ✢ ✕ ✕ ✓ ✔ ✕ ✓ ✖ ❤ ✕ ✔ ✕ ✓ ✒ ✙ ✒ ✛ ✲ ✛ ✫ ❡ ✒ ✙ ✕ ✲ ✲ ✖ ✳ ✢ ✓ ✕ ✜ ✙✚ ❣ ✭ ✘ ✭ ✗ ✢ ❢ ✒ ✙ ➜ ❦ ✒ ✖ ✑ ✒ ✭ ✲ ✲ ✖ ✳ ✢ ✓ ✕ ✛ ✜ ❣ ✭ ✘ ✗ ✙ ✢ ❣ ✪ ✘ ✒ ✜ ✒ ✢ ✦ ❧ ✑ ✢ ✕ ✒ ✛ ✥ ❤ ✙ ✦ ✜ ✑ ♦ ✑ ✒ ✙ ✚ ✢ ✑ ✢✩ ❡ ✙ ✣ ✧ ✑ ♦ ❦ ✕ ✲ ✲ ✖ ✳ ✢ ➌ ➎ ✢ ◆ ❄ ✿ ✾ ❅ ❆ ❇ ❊ ❋ ●❍ ■ ❏ ❑ ❖ ✼ ❖ P ◗ ❘ ❯❱ ❲ ❯❳ ❬ ❭ ❪ ❪ ❫ ✿ ❃ ❩❵ ✲ ★ ✩ ✭ ✮ ✯ ✲ ✮ ✲✳ ✯ ✴ ✩ ✵ ✳ ❀ ✩ ✶ ✧ ✦ ✵ ✭ ✶ ✷ ✶ ✾ ✼ ✿ ❴ ❛ ✤ ❵ ❬ ❭ ❥ ❞ ❬ ❯ ❳ ❬ ❜ ❩ ✐ ❭ ❲ ❭ ❦ ✐ ❧ ❜ ❩ ✐ ❵ ❡ ❲ ✐ ❯♠ ❦ ❵ ❲ ❭ ❲ ❵ ❭ ❜ ❝ ❩ ❛ ❩ ❞ ❛ ❭ ❵ ❡ ❯ ❩ ❩ ❢ ❣ ❱ ❨ ❩ ❬ ❤ ✐ ❩❵ ❫ ❴ ✧ ✛ ➒ ➱ ➸➫ ➺ ➻ ➼ ➹➘ ➴➷ ➬ ➮ ➶ ➴ ➬ ➹ ➴ ➵ ➬ ✃ ➱ ➶ ❐ ❒ ❮ Ñ ÒÓÔ Õ Ù Ú ➦ ➥ Ü ➡ ➓ ➐➑ ➒ ➔ ➛ ➜ ↕➙ ➔ ↕➠ ➡ ➢➤ ➟ ↕ ➲ ➥ ➦➧ ➦ ➨ ➭ ➯ ➨ ➲ ➫ ➦ ➧ ➳ Û Û ✙ ☛ ý � ✁ �✂ ✄ ☎ ✆ ✄ ✝✞ ✟✠ ✠ ✡ ✎✏ ø ✑ ✒ ✏ ✓ ✔ ✕✖ ✗ ✘ ✖ ✖ ✕✙ ✚ ü ûü Ý ëì Üà á â ã â ä å è ç é ã ê í ö ëì î ï ð ô õ ö õ ÷ õø õ ù ú ✓ ✙ ✳ û ✄☎ þ ✆ ✝ û ✄ ✆ þ ü ✞ ✟ ✄ ✂ ☎ þ ú ☞ ü ✌ ✝ ú ✄ ✠ ú þ û � þ Ý å Þà Û õ Þ à Ý ç ä ô ç Ú þ ì â ß ß Þ ö ÷ ø û üýþ û û ✆ ü Þ ✕✦ ✑ ✢ ✑ ✥ ✙ ✥ ✓ ✕ ✒ ✢ ✒ ✣ ❧ ✧ ✑ ✤ ✥ ✓ ✖ ✒ ✕ ★ ✤ ✕✑ ✢ ✤ ✗ ✞ ✒ ✟ ö ✍ ✎ ✒ ✓✔ ✕ ✒ ✒ ✕ ✗ ✘ ✙✚ ✙✚ ✕ ✛ ✜ ✒ ✙ ✑ ✢ ✛ ✢ ✑ ✜ ✒ à Ú ✪ ➬➮ ➾ ➤➚ ➪ ➻ ➫ ➻ ➶ ➹ ➘ ➼ ➻ ➱ ➤ ➬✃ ❮ Ò Ó Ô Õ ❐ Ò Õ ➱Ö × Ø ➽ ➼ Û ➤ ➝ ➞ ➟ ➢ ➤ ➥ ➦ ➧ ➢ ➨ ➩ ➥ ➻ ➦ ➫➭ ➫ ➯ ➲ ➳ ➵ ➥ ➸ ➫ ➤ ➺ ÙÚ Ü ë ñ Ý ë Ú Û ê ë ï ç ë Ø Þð ò ë ó Ú Ý æ Ø Ý ì í å Þ à ô à ä Ý ß ÛÞ á Þ ä å Þ æç è é å ê Ý æ é ê Ø Ú á ã Þ ë Ú Ý æ Ø Ýì ✓ ✒ ✛ ❋ ❄❅ ❄ ❆ ❇ ❈ ❉ ✿ ❊ ❄ ✾ ● ✿ ❍ ✾ ■ ❏ ✾❑ ▲ ● ❄ ● ▼ ◆ ❀ ✾ ❍ ✳ ✥ ✓ ✩ ✓ ✗ ✤ ✕ ✲ ✲ ✖ ✢ ❃ ✵ ✶ ✷ ✸ ✹ ✽ ✾ ✿ ❀ ❁ ✽ ❖ ● ✑ ✕ ✘ ✗ ✢ ✭ ✮ ✒ ✜ ✒ ✢ ✙ ✲ ❣ ✦ ✓ ❥ ✣ ✙❦ ✒ ✢ ❤ ✣ ✕ ✒ ✭ ✢ ❙❚ ❡ ❯ ❙❱ ❩ ❵ ❛ ❜ ❝ ❲ ❵ ❝ ❯❞ ✒ ✗ ✙ ✕ ✲ ✲ ✖ ✳ ✢ ✓ ✕ ✛ ✜❢ ✤ ❂ ✧ ✖ ✔ ✛ ✰ ✙ ✣ ✙ ✑ ✣ ✙✚ ✒ ✙✚ ✓✔ ✑ ✗ ✒ ✜ ✱ ✑ ✍ ✣ ✳ ✲ ✢ ✓ ✫ ✕ ✒ ✙✚ ✙ ✚ ✙ ✒ ✒ ✙✚ ✢ ✤ ✧ ✙ ✥ ✥ ✎ ✢ ✓ ✭ ✓ ✕ ✙ ✒ ✢ ✓ ✕ ✒ ✕ ✦ ✘ ✲ ✪ ✙ ✦ ✙✚ ✒ ✒ ✑ ✥ –2 124 ✻✽✼ ✧☛✩ ✪✬✫ ✰☛♠ ❾☛❿➀ ✸✺✹ ❏▼▲ – 3 – ↔❘↕➙ ✥✍✦ ❻☛❼ →✁➣ – AVL v æßç – ✜✍✣ balance 0 ❰ÐÏ ✜✍✢ þ✁ÿ ❈✬❉ ✰☛♠ ➩✁➫ balance ⑤☛⑥ ❫✁❴ ➏❘➐➑ v ❬❪❭ ÝßÞ ❨✍❩ Balance ❁✱❂ ❲❨❳ balance balance ✲❘♦ s☛t✉ ➊☛➋ ❱♦♥ ✓✴✫ ✓✴✫ Delete()? balance balance Ö❘×Ø ð❘ó ✜✐❤ ➚❪➪➶ P❘◗ ☞✍✌ ❙✬❚ ✺✼✻ ➞✁➟ ñ☛ò ➽❨➾ - ★✱✰ ✏☛✑ ✏☛✑ ✏☛✑ ↔❨➝ v 123 - ✠☛✡ 3 ✖✬✫ v v AVL ✜✁✣ ê❵ã ✓✴✫ ✧☛✩ Ð❵Ñ balance ❰❇Ï â❵ã balance ✖✁✗ ❐❫❒ ßPÞà ÿ ✁� ✖✁✗ ✎✯✮ í✂î Balance ➴P➷ ✎✯✮ ➠✩➡ - ù✂ú ✏☛✑ ✏☛✑ ✏☛✑

  62. ♣ îï ê ë ç ì í è é ê î è ð ñ ò ó é ô î è é ç ö Ó ➘ ➾ ➪ Ï Õ Ò Ô Ö q ä Ò× Ô ÚÛÜ Ý à á â ã õ ÷ ➮ ✕ ✞ ✂ � ✡ ✍ ✎✏ ✑✒ ✓✔ ✙ ✝ ✜ ✤ ✜ ★ ✩✪ ✫ ✬✭ ✮ � ✠ è ý ø ù èú û ö î ö ü þ � ÷ ö ✂ ✄ ☎ ✆ ✝ ✞ ✟ ✃❐ Ô ➚ ➒➓ ➍ ➎ ➏ ➬ ⑩ ➐ ➑ ➔→ ➋ ➣ ⑨ ➙ ⑧ ➞ ❷ ➡ ➌ ➉ ⑦ ❶ ❷ ❹ ❺ ❻ ❼ ❽ ❾❿➀ ➁ ➉➊ ➂ ➃ ➄ ➅ ➆ ➄➇ ➈ ➢ ➟➠ ➤ s ➵ ⑥ ➺ ➻ ➼ t r ② ➹ ➾➘ ➴ ➪ ➾➚ ➷ ① ➸ ③ ➯ ➤ ➧ ➨ ⑤ ④ ❸ ➳ ➲ 126 ✖✘✗ 3 ✯✌✰ ☛✌☞ ✚✁✧ ✥✌✦ ✜✣✢ ✚✁✛ å✇æ ÿ ✁� - 125 3 ➥✍➦ AVL ÒÓ▼Ô Ð✬Ñ ❒❰❮ ➛➝➜ ➩➭➫ ➴✱➱ ↔✍↕ Þ✺ß ✉✇✈ - Ø▼Ù ➪✍➶ 19, ➽✬➾➚

  63. ✱ á è Û ê å ç ê Ú è ã çè ñ î Ù ò × ò Þ ç Ö ó Û Ö Û Ù Ú ã è ä è ê á Û ð á ê å ç ê Ú ï ô è ☞ ✁ û ✟ ✠ ✡ û ☛ û ü ✌ ✍ ✠ ✁ ✠ ✎ ✏ ✑ ✞ ✝ õ ÿ ö ÷ ú û ü ý þ ú � ✆ û ü ý ✁ ✂ ✁ ✄ ☎ Û Ü ✠ Ø ❐ ➘ Ò Ó Ô Ö × Ù ➱ Ö Ú ÛÜ Ü ✲ Ö ã Ù ➳ Ï ä ➵ ➾ ➶ ➾ ➴ ➷ ➬ ➾ ➮➱ ➘ ❮❰ ✃ ❐ ❒ ➘ ➼ ➽ ➾ ➚➪ Ö Ö Ô Û î ä Ö Ú Û å æ Ö ê ê ë á Ü é Ö Û ê Û Û Û Ú ã Û Ö Û Ü é Ö Û ê Ü è è Û Ö Û ê ë á ä ✡ ✒ ➚➪ s s t q ✉ ✈ ✇ ① ♠ ♠ ③ ④ r ⑤ t ④ ⑥ ♠ qr ❲ qr ❴ ❬ ❩ ❴ ❨ ❨ ❜ ❫ ❝ ❨ ❝ ❵ ❴ ❨ ❝ ❞ ❡ ❪ ❨ ③ ⑦ ❩ ➙ ➎ ➣ ➎ ➋ ↔ ➍ ➎ →↕ ➒ ➒ ➙ ➛ ➒ ➝ ↕ ➟ ➢➤ ➥ → ➔ s⑧⑨ ❼ ⑩ ❶ ❷ ❸❹ ❼ ❿ ❾➀ ❾ ➃ ➓ ➄ ➅ ➆➇ ➈➉ ➊ ➋ ➏ ➐ ❨ ❜ ✓ ❑▲▼ ❁ ❂ ✾ ❅ ❆ ❇ ❈❉❊ ◆❖P ❄ ❙ ❚❯ ❱ ❲ ❩ ❨ ❬ ❩ ❀ ❀ ❨ ✮ ✚ ✣ ✤ ✥ ✦✧ ✬ ✩✫ ✭ ✯ ✿ ✳ ✴✵ ✶ ✺✻ ✼ ✽ ✾✿ ✾❀ ❴ ❨ ❝ ❦ ❫ ✐ ❵ ❴ ❬ ❢ ❨ ❥ ❞ ❣ ❡ ❪ ❞ ❢ ❨ ❵ ❛ ❨ ❬❤ ❨ ❜ ❞ ❫ ❝ ❴ ❨ ❵ ❴ ❨ ❝ ❡ ❴ ❪ ❨ ❝ ❲ ❩ ❨ ❬ ❩ ➸ Þâ ⑥ ✇⑩ ➀ ❻ ✉ ⑨ ✇ ➵ r ❿ ⑩ ❼ ❽ q ✺ ② ❹ ① ④ ⑩ ⑧ r ❷ ✈ ⑥ ✇ r ✈ ⑩ ❼ ❽ ❾ ② ❊ ⑤ ③ ⑤ ⑧ ➣ ❆ ➣➅ ➆ ➍ ➇ → ➌ ➊ ✻ ➄ ➍ ➌ ➙ ❅ ➓➝ ➅ ➑ ✈ ➇ ① ❉ ❈ ➃ ➄➅ ➆ ➈ ➊ ➉ ❇ ✺ ❀ ➄➎ ➏ ➐ ❋ ⑩ ➞ ❣ ❴ ❢ ❡ ❳ ❩ ❡ ❳ ❳ ❤ ❪ ❜ ❜ ❪ ✐ ❬ ❩ ❝ ❘ ❭ ❬ ❚ ❪ ❫ ❙ ❪ ❵ ❵ ❩ ❝ ❞ ❵ ❡ ❭ ❞ ▲▼ ⑧ t ✈ ⑦ ❍ t ⑧ ⑨ ✇⑩ ① ❶ ❷ ❸ ❸ ✈ ❹ ✺● ⑥ ✈ ❪ ■ ❥ ❈ ❉ ❑ ❏ ♣ q ⑤ r ❈ ❀ ✈ ❈ ✇① ② ↕ ➇ ↕ ➍ ➝ ↕ ➇ ➄ ➄ ❀❁ ➝ ✼ ➌ ➏ ➎ ➓ ➊ ➇ ➒ ➆ ➞ ➥ ➣ ✺ ➄ ↕ ➝ ↕ ➡ ↕ ➝ ❂ ➞ ↕ ➍ ↔ ➍ ↕ ➝ ↕ ➍ ❀ ✻ ➍ ➇ ✴ ➌ ➐ ➣ ➌ ➞ ↕ ➭ ➯ ➩ ➲ ✳ ➸ ➵ ➺ ➻ ➳ ➳ ➫ ➆ ✿ ➉ ✾ ✹ ✽ ➒ ➟ ↔ ➉ → ➣ ✼ ✻ ✺ ➨ ✹ ✵ ➄ ➐ ↕ ➄ ❬ ➉ ↕ ↕ ➝ ↕ ➡ ↕ ➒ ➆ ➑ ➍ ➇ ➆ ➟ ↔ ➉ ➉ ➇ ↕ ❃ ➠ ➓ ➄ ➣ ➆➟ ➇ ➍ ➈ ↕ ➞ ➎ ➓ ➄ ➌ ➐ ➣ ➌ ➆ ❄ ➇ ↕ ➣ ↕ ➎ ➡ ➌ ➍ ➔ ➈ ➏ ↕ ➤ ➅ ➆ ↕ -3 128 ❁✲❂❃ h? ➌✲➍➎ 3 h LookUp n ❞✪❢ ❞✪❢ ✷✲✸✹ ➠✪➡ ✗✙✘ ◗✪❘ - ➞✪➟ ✔✖✕ ✇✙② ✰✲✱ 2- - ➛✪➜ ■✪❏ - ➁✪➂ ❋✪●❍ ❵✲❛ ❵✲❛ ❽✪❾ ★✪✩✫ logn). ❭✲❪❫ ❭✲❪❫ ❭✲❪❫ ❺✪❻ ø✸ù ♦✢♣ - ➐✪➑➒ ✛✢✜ ➌✲➍ ❳✢❨ ❳✢❨ ❳✢❨ ❧♥♠ 127 ➣➧➦ ↔❛↕ Ôíì ÔÑï 3 r✌⑧ ➏➋➓ ③✣④ - ➔➋→ å❛æ å❛æ ➒✣➓ ③✣④ P✣◗ ➹✌➘ ◆✌❖ à➋á Ý✣ß s❛t 2- Ý❛Þ ❫✣❜ å❛çè ➼❛➽➾ r✣✉ ❿➂➁ ➌✌➍ ➐❛➢ ⑨✣❻ ❴❛❵ s❛t ➉➋➊ å❛æ ➷ÑÐ ➹✌➘ ❴❛❵ ➉➋➊ - - - s❛❺ Õ✌Ö ✶✸✷ ➣➜➛ ➳➋➵ o o - ♠❛♥♦ ➐❛➢ ❯✁❱ ❲❨❳❩ ❦❨❧

  64. ➦ ❉ ◆ ● ▼ ❵ ❆ ❇ ❩ ❛ ❇ ◆❜ ❄ ❄ ❆ ● ❇ ❍ ❃ P ❆ ❂ ❇ ❋● ❄ ❏ ❆ ❜ ◆ ◆ ❇ ❋ ❆ ❭ ❍ ■ ❏ ❑ ❉ ❍ ❇ ● ❉ ◆ ❯ ❫ ■ ❄ ❇ ❨ ❏ ❉ ❊ ❲ ❉ ❆ ❉ ❲ ❄ ◆ ❏ ❆ ● ❉ ❀ ▼ ❘ ❇ ● ◆ ❄ ❉ ❩ ❞ ❡ ❢❤ ❣ ✐ ❥ ❣ ❣ ● ❂ ❁ ◆ ❏ ❆ ❋● ❉ ❑ ❱ ❏ ◆ ❭ ❫ ❇ ❀ ❀ ◆ ❅❝ ❄ ▼ ❋ ❉ ❁ ❏ ❆ ● ❉ ● ❂ ❧ ❇ ❂ ■ ❖ ❂ ◆ ❇ ❘ ❇ ❃ ❍ ◆ ❲ ❇ ❀ ◆ ❍ ❯ ❆ ❇ ❘ ▼ ❳ ❩ ✿ ◆ ◆ ❘ ❏ ❂ ◆ ❇ ❘ ▼ ❆ ❀ ■ ❊ ❍ ■ ❑ ● ❉ ❁ ◆❀ ● ❇ ❍ ➧ ❨ ▼ ❋● ◆ ❀ ❆ ❲ ▼ ❴ ❩ ❉ ❂ ◆ ● ❲ ❄ ◆ ❊ ❆ ❀ ❏ ◆ ❇ ❖ ❍ ❑ ❏ ❆ ● ◆ ● ❄ ◆ ❱ ❀ ❍ ❁ ❂ ❃ ❄ ❅❆❇ ❂ ◆ ● ❄ ❇ ❍ ❑ ❑ ❉ ❊ ❂ ❱ ❫ ◆ ❇ ❀ ❆ ❋● ❉ ❦ ♠ ● ➅ t ♣ ♥ ❻ ❼ ❽ ❾ ❿ ➃ ➄ ➆ ❦ ➇ ➃ ➈ ➉ ➄ ➅ ➆ ➊➋ ➊ ➌ ➍ ❹ ❣ ➏ ❧ ✐ ♥ ❷ ❤ ⑤ r ❦ ⑥ ❤ ❣ ❢ ① ❦ ① ⑥ ❧ ① ❦ ❡ ✉ ♣ ❡ ✇ ① ➎ ➅ ❣ ➷ ➫➭➯ ➵ ➸ ➺ ➻ ➼ ➽ ➾ ➳ ➹ ➘➴ ➴ ➦➧ ➱✃ ❐ ➪ ❰ ➪ ➱ ÏÐ ➹ ➮ ➱ ❰ ➧ ➥ ➐ ➊ ➊ ➄ ➑ ➒ ➓ ➄ ➔ → ➄➣ ↔ ➒ ➒ ➤ ↕ ➙ ➛ ➓ ➒ ➞ ➟➠ ➡ ➠ ➢ ➞ ❢ ❧ ♥ ❥ ① ④ ❢ ❣ ❧ ① ❤ q ♥ ❢ ❡ ✐ ③ ♥ ❤ t ♦ ① ❢ ① ❤ ⑤ r ✐ ✇ ✈ ① ❧ ♦ ❣ ♣ ❣ q ❣ r s t ♣ ♥ ❣ ♥ ❢ r ✇ ❢ ♣ ❣ s ✇ ① ② r ❧ r ♥ ❤ ✐ ❧ ⑦ ✇ ❥ ⑧ ❣ ⑨ ① ♦ ♥ ❣ r ❢ ⑦ ✇ ❥ ⑧ ❣ ✈ ❶ ✇ ❥ ⑧ ♥ ❦ ① ⑦ ♣ ① ♦ t ❧ ❢ ♦ ✇ ① ⑥ ❧ ✇ ① ❥ ⑧ ❣ ⑨ ♠ ♣ ❣ ⑩ ❧ t ✐ ❉ ◆ ▲ ✼ ✩ ✷ ✸ ✷ ✩ ✪✹ ✺ ✪✻ ✪ ✴ ✯ ✩ ✯ ✪ ✽ ✼ ✪ ✹ ✴ ✫ ✪ ✴ ✳ ✪✹ ✕ ✓ ✔ ✕✖ ✗ ✎ ✘ ✘ ✕✖ ✒ ✔ ✕ ✰ ✙ ✕ ✚ ✕ ✛✜ ✢ ✦✧★ ✩ ✪ ✫ ✬ ✾ ✑ ❆❇ ■ ❏ ❑ ❉ ❯ ◆ ❘ ❱ ❲ ❅ ❏ ❖ ■ ❀ ❉ ❲ ❀ ◆ ❍ ❯ ❆❇ ❘ ❍ ❇ ✵ ❄ ✾ ✷ ✫ ✭ ✲ ❁ ❂ ❅❆❇ ❅ ❈ ❉❊ ◆ ❋● ❉ ❀ ❍ ❑ ❉ ▼ ❆ ▼ ❀ ✒ ✍ ❳ ❰ ➮ ➵ ➱ ✃ ➵❐ ❒ ➬ ➾ ➬ ❮ Ï ➷ ➮ ➬ ÓÔ Õ Ô Ö Ò × Ø ÙÚ ➬ ➵ á ➽ ➨ ➩ ➫ ➳ ➵ ➸ ➺ ➻ ➳ ➼ ➵ ➾ ➸ ➺ ➾➚ ➾ ➪ ➶ ➹ ➘ ➸ ➴ à è ☞✌ ñ î é ò ïù ú ó ó ï ñ í è ÷ ý ü ï ð ó í î þ ✂ ✟ í ê éê õö ë ï íð ñ í ò ë ó íô é ò ô è ô ÷ ø æ è ô ê è ô ▼ ▲ P ◆ ❏ ◆ ❆ ❁ ❉ ● ● ❉ ❄ ❀ ❲ ❂ ● ◆ ❂ ❄ ❈ ● ❉ ◆❀ ❲ ◆ ❖ ❏ ❑ ❉ ❋ ❪ ❆ ❘ ❪ ❉ ❉ ❄ ❍ ❍ ❑ ❉ ❁ ❇ ❊ ❅ P ❍ ❂ ■ ❲ ❘ ❯ ▼ ❍ ❍ ◆ ❇ ❂ ❄ ❇ ◆ ❋ ❆ ❉ ▼ ● ❉ ❏ ❆ ❆ ❆ ❀ ● ❉ ❋● ❇ ❲ ❀ ◆ ❀ ❏ ❆ ▼ ❯ ❉ ● ❄ ▼ ● ❘ ❉ ❇ ❍ ❍ ❏ ❋ ❆ ● ❆ ❄ ❏ ● ❁ ❇ ❋ ❍ ▼ ❙ ❨ ❘ ❆ ❀ ❉ ❁ ❂ ❃ ❆ ❇ ❅ ❉ ❄ ■ ❋ ❨ ❆ ❋ ❆ ● ❍ ◆ ❯ ❆ ❇ ❘ ❆ ❍ ▼ ❆ ❍ ❄ ❂ ■ ❈ ❍ ■ ❏ ❑ ❉❊ ❳ P ❅ ❄ ❭ ❇ ❉ ❋ ❆ ● ❑ ❉ ❀ ❍ ❁ ❫ ❆❇ ▲ ❅ ❈ ❄ ◆ ❉ ❊ ◆ ❀ ◆ ❉ ❱ ❍ ❇ ❑ ❉❊ ❆ ▼ ❀ ◆ ❇ ◆ 130 3 ➱❬Ô ➲✆➳ ❰☛Ó Ñ✥Ò ➨❚➩ - ❒✮❮ ➬☛➮ ➀➂➁ ➜☛➝ - ➚➶➪ 129 - - ❊◗P ❘❚❙ ■☛❏ s✮① ■☛❏ ✉✥✈ 3 - ✣✥✤ ■✥❊ - s✮① ■☛❂ extra ÿ ✁� Ý✙Þß Û✖Ü ■☛❏ ❉❊◗P ■☛❏ - ❢❬❺ ■☛❏ û✢ü - ❆❬❩ ❃☛● ✵✆✶ Ð✙ÑÒ ❃✆❄ ì✖íî ✠☛✡ ✱✁✲ ❡❚❢❣ ⑤☛❧ ➭➲➯ ✝✆✞ - ✎✆✏ ✬✮✭✯ ✿✥❀ ✿✥❀ ✿✥❀ ëäæ âäã ❘❚❙ ②✥❸ ✄✆☎ 1. 2. 3. 4. 5. åçæ

  65. Õ ↔ → ➠ ➣ ↕ ➑ ➞ ➝ ➡ ➎ →➣ ➔ ➓ ➒ ➑ ➎➏➐ ➣ ➣ ➙➛ ➛ ➙ ➑➏ ➐ ➣ ➎ ➥ ➙ ➙ ➣ ➤ ➝ ➑ ↔ ➢ ➏ ➜ ➣ ➏ ➀ ➃➈ ➂ ➅ ➄ ❽ ❽➄ ➃ ➊ ❿ ❾ ❻ ❹❺ ❸ ♥ ❬ ➉ ➈ ↕ →➣ ➐ ↔ →➣ ➑ ➐ ➎ ➎ ➔ ➌ ➓ ➒ ➑ ➎➏➐ ➍ ❿ ❾ ➦ ➒ ❭ ➻ ➘ ➪ ➚ ➶ ➺ ➪➹ ➺ ➸ ➚➪➶ ➻ ➺ ➙ ➣ ↔ ➨ ➶ ➺ ➙ ➪ ➼ ➻ ➺ ➬ ➷ ➻❐ ➺ ➺ ➻ ➪ ➮ ➹ ➻ ➺ ➸ ➬ ➦ ➩ ➙ ➙ ➯ ➤ ➐ ➡ ➭ ➛ ➥ ➑ ➍ ➫ ➟ ➫ ➡➩ ➣ ➛ ↔ ➣ ➑ ➩ ➏ ➎ ➣ ↔ ➳ ➐ ➝ ➠ ➎ ➝ ➙ ➥ ➛ ➙ ➦ ➝ ❴ ❪ ➹ t ❛ ❜ ⑦ ⑥ ❛ ❞ ❞ ❢ ❛ ❭ ⑥ ❣ ❧ ✉ t ❣ ❡ ❞ ❛ ❞ t ❢ ❬ ❝ ❢ ❣ ❜ ♥ ❣ ❧ ❞ ❧ ❢ ❞ ♣ ❛ ❡ ✐ ❞ ❛ ❞ ❡ ❧ ❩ ❬ ♣ Ö ❢ ❣ ❧ ♠ ❭ ❛ ❞ ❣ ✐ ❧ ❬❞ ✐ ❬ ❜ ❫ ❩ ⑤ ♥ ❝ ❡ t ❛④ ❤✐ t ✈ ❛ ✇ ❢ ❬ ❭ ⑧ ❣ ❧ ✉ t ❩ ⑤ ♥ ❬ ❛ ✇ ❡ ✐ ❣ ❛ ❝ ❝ r ❞ ❛ ❣ ❦ ♠ ❧ ❢ ❛ ❞ ❞ ♠ t ❬❞ ❝ ⑨ ❣ ✉ ❥ ❧ ❡ ❤ ❧ ❛ ❣ ① ♥ ❬ ❣ ❢ ❬ ❥ ❣ ❛ ❝ ❧ ❞ ❡ ❦ ✐ ❤ ❛ ❩ r ❬ ❝ ❡ ❢ ❣ ❜ ❧ ❬ ❝ ✈ ❥ ❭ ❣ ❝ Ï ➼ ❞ ➬ ❒ ➼ Þ ➻❐ ➺ ➶ Ö ➺ Ð ✃ Ñ Ú ❐ ➻ ➮ ➸ ➻ Ø ❐ ➪➶ ➚ ➻ ➺ ➬ ➷ ➻ ➹ ➹ å ➹ ➬ ➹ ❐ ä Ð ➶ ➻ ➬ ã â ❮ Ð Ø ➶ Ô ➺ ➚ ➻ ➹ Ò ➷ ➪ ➹ ➸ Ñ ➬ ➼ Ô ➚ ➻ ➺ ➪➶ ➚ ❒ ➹ ➻ Ñ ➺ Þ ➪ ➺ ➪ ➮ ➺ ➚ ➬ ➺ ç ➼ ➬ ➷ ❐ ➻ ➱ ë æ ➼ ➽ ➼ ➪ ➺ Õ èéê ì ➸ ï é ï õ ð ò ô ë òó éí í ñ ð ï é î é ➺ ➶ Ô ❒ ➹ ➪ ➹ ➱ ➬ ➸ ➼ ➱ ❐ ➻ ➺ Ð Ø ➶ ➬ ➹ ➻ ➘ Ð ➚➪ ➶ ➺ ➹ ➪ Þ ➻ ➻ ➚ ➮ ➻ ➽ ➻ Ó ➷ Ò ➱ ➷ ➚ ➪ ➼ ➻❐ ➺ Ð Ø ➶ ➮ ➸ ➻ Ô Ô Õ ➺ ➪ ➚ ❒ ➬ ➹ Ù ➪ ➹ ❐ ➻ ❒ ➼ ➪ ❮ ➮ ➼ ❐ ➻ ➺ Ð Ø ➪➶ ➪ ➱ Ò ➸ ➚ ➶ ✃ ➬ ➺ Ñ ➬ Ð ➪➶ Ð ➬ ❒ ➼ ➶ ➪ ➻ ➚ ➶ Õ ➪ ➚ ➬ ➷ ❐ ➻ Ö Ô Ò Ô ➪ ➽ ➶ ➻ Ó ➷ ➮ ➱ Õ à ➱ ➶ ➬ Ú ➱ ➼ ß ➪ ❮ ➶ ➬ Ö Ð ➺ ➪ ➹ ➹ ➬✃ Ó Ô Ô ➪ ➽ ❒ ➼ ➻ ➷ ➪ Ò ➚ Ò ➚ ➶ ➬ Ø ➷ ➺ ➹ ➻ ➪ ➹ ➱ ➘ ➪ ➚ ➺ ➪ Ð Ø ➶ ➬ Ô ➚ ➪ ➽ ➹ ➻Þ ➶ ➷ ➘ ➶ ✃ Ò ➪ ➻ ➪ ➺ ➚ ➽ Ü ➪➶ ➚ Û Ð t ➹ ❛ Ø ✏ ✑ ✒ ✓ ✔ ý öù ø ö÷ ë ì Õ Ö × ó ✎ á ã ãä ß Þ Ý â Ü Ù à ß Þ Ý Ü ò ☞ ✌ ë ✍ ✔ ✏ ✗ ✏ ✗ ✙ ✒ ✭ ✛ ✧ ✕ ✝ ✌ ✟ � ✙ ✟ þ ✛ ✔ ✢ ✌ ✡ ✬ ✒ ✕ ✙ ÿ � ✎ ✛ ✕ ✝ ñ å ✎ ë � ✁ ✂ � ø ☎ ð ✯ Ýï î ✵ í ✸✹ Ý ì ✽ ③ ❄ ❂ ❃ ã ❈ ❇ ✓ ❀ ✾ ❅ Ý ✾❁ ❀ ✿ ê ÿ þ æ ë ð Ýï î í Ý ì ê ã Ý ã é Þ è ç ë ë ý ô û ú öù ø ö÷ õ ❭ ñ ë ã ë ì ó ò ✏ ✕ ✿ ✟ ✗ ✒ ✛ ✙ ✧ ✙ ✎ ✌ ✍ ✪ ✟ ✕✝ ✕ ✢ ✏ ✌ ✛ ✗ ✌ ✝ ✕ ✫ ✙ ✝ ✡ ✠ ✌ ✕ ✑ ✔ ✢ ✢ ✒ ✔ ✗ ✎ ✔ ✙ ✏ ✔ ✏ ✌✙ ☞ ✌✙ ✎ ✑✒ ✘ ✗ ✕✖ ✌ ✛ ✏ ✝ ✦ ✕ ✗ ☞ ☞ ✣✤ ✢ ✠ ✌ ✝ ✜ ✔ ✕ ✝ ✏ ✑ ✠ ✟ ✫ ✔ ø ☎ ✎ ✌ ☞ ✌ ✬ ✔ ✛ ✝ ✕ ✔ ✛ ✕ ✙ ✝ ✧ ✙ ✛ ✒ ✗ ✕ ✜ ✍ ✛ ✔ ✢ ✌ ☞ ✜ ✘ ✧ ✌ ✎ ☞ ✕ ✝ ✌ ✙ ✌ ✘ ✍ ✗ ✙ ✛ ✣ ✜ ✥ ✒ ✙ ✙ ✛ ✛ ✒ ✗ ✌ ✬ ✧ ✔ ✗ ✟ ✟ ✔ ✏ ✗ ✏ ❋ ❆ ✔ Þ ▲ ❊ ✻ Ü à ❨ ❩ ❬ ❭ ß ❴ ❬❵ ❛ ❜ ❬❞ ✿ ❣ ❣ ❧ ♠ ❭ ❬ ❝ ❛ ❬ ❝ ❛❤✐ ❬ ❡ ❬ ❜ ❑ ❊ Ý ❋ ✿ ❊ ✾ ✽ ❊ ▲ ❊ â ❃ ❊ ❉ ❈ ✾ ✽ Ý ❑ ✾ ● ✽ ❊ ◗ ❄ ❀ ❅ á ▲ ❃ ❄ ❈ ✾ ✻ ❊ ❢ ♣ ❏ ❬ ✐ ❫ ❛ ❝ ❡ ✇ Ù ❧ ❣ ♥ ① ❣ ❛ ❦ ✐ ❝ ❬ ❣ ❧ ❞ ❧ ❢ ❩ × ❧ Ø ✈ ❥ ❭ ❣ ❝ ❧ ❵ ❞ ♠ ❛ ❞ ❡ ❧ s ❬r ❬ ❣ ❭ ❛ q ❧ ❩ ❬ ❧ t ❬ ❛ Ü ❣ ✐ ❪ ❤ ♠ ❬ ❞ ❢ ❣ ❧ ♠ ❭ ❛ ❊ ❂ ❚ æ ❊ ✽ ✾ ❊ ✿ ❊ ✻ ✾ ✽ ✾❙ ❈ ç ❃ ✾ ❑ ◆ ❏ å ✽ ❚ ✻ ❊ ✽ ❂ ✾ ã ❂ ❃ ❄ ✽ ✽ ❊ ❉ ❀ ❃ ❊ ✽ ❄ ❃ ❏ ❊ ❈ ▼ ✾ ❆ ◆ ❉ é ❖ ❂ ❃ ● ❅ ❀ ❄ ◗ ● ✽ Þ è ❑ ❍ ❃ ✽ ✻ ✽ ✾ ❄ ❏ ❊ ❈ ❑ ▲ ▼ ✾ ãä ❆ ◆ ß ❉ ❉ Þ ❊ ✽ ❏ ● ✽ ❄ ❈ ❃ ✻ ❖ ❄ ❊ ✽ ❑ ✿ ✾ ✽ ❃ ❚ ❯ ✿ ❏ ❀ ❃ ❃ ❍ ❏ ● ❃ ❂ ❚ ✻ ❍ ● ▲ ❅ ❄ ❊ P. 132 t■✉ S ❡♦✈ ❑✲▲ ❡♦✈ ❪✲❫ ➉✲➋ 3 ❥■❦❧ ❑✲▲ ➱■✃ - ❒❰❮ ö❰÷ ➧✲➨ ➼➾➽ t■✉ ❢✲❣ ●■❍ t■✉ ➎P➲ ➼➾➽ ➆✲➇ ✾P❘ Ô✲Ý ➴✲➷ Ô✲× ➱■✃ - ➱■Ú ❉✲❊ ❢✲❣ ❪✲❝ ✺✼✻ P Ô✲á ➼➾➽ ➀■➁➂ S ✾P❘ - ❪✲❫ ❂✲❃❄ S ❂✲❃ ❡■② ❼✼❽ ➼➾➽ ❒❰❮ P s✞❫ ✶☛✷ ✾P❖ ❡♦♥ ❂✲❃ ➙➛P➟ 3- ⑩■❶❷ ➵■➸ Ú➂Û ✳✲✴ - ❉✲❊ ✺✼✻ ❱✲❲❳ b. a. c. ✰✲✱ 1. 2. 131 ✑✚✌ 3 ✑✚✌ ✁✄✂ ✕✮★ ✘✩★ - úüû ✑✚✌ ô✮õ ✠✞✥ Ú➂Û - ✠☛✡ ✧✄✛ ✆✞✝

  66. ø ❚ ❱ P ❱ ❡ ✐ ❛ ❯ ❘ ❡ ❛ ❭ ❜ ❨ ❪ ❴ ❘ ❝ ❭ ❱ ❡ ❲ ♦ ❯ P ❚ s ❱ ❫ ♦ ❭ ❞ ❝ ❫ ♦ ✐ ❵ ❙ ❭ ❞ ❥ t ✉ ❭ ❙ ❱ ❱ ❘ ❘ ❛ ❤ P ❲ ❘ ❳ ❨ ❤ P ❙ ❚ ❲ ♠ P ❭ ❪ P ❘ ❯ ❪ ❞ ❲ ❩ ❱ ❵ ❡ ❱ ❛ ❭ ❙ ❚ ❲ ❢ ✈ ♦ ❳ s s ❱ ♥ ❱ ❘ ❨ ❯ ❙ ❞ ❯ ❨ ❘ ❯ ❩ ❯ ❱ ❳ ❦ r ❣ ❤ ❫ ❲ ❘ ❛ ❯ ❝ ❲ ❚ ❞ ❙ ❡ ❱ ❲ ❣ ❯ ❪ ❩ ❯ ❱ ❲ ù ❛ ❱ ❨ ❢ ♠ ❲ ❘ ❛ ❱ ❫ ❙ ❚ ❲ ❚ ❢ ❯ ❯ ❫ ❡ ❨ ❬ ❡ ❭ ❯ P ❘ ❙ P ❘ P ❱ ❡ ❱ ❯ ❭ ❪ ❛ ❱ ❬ ❤P ❞ ❙ ❘ ❯ ❘ ❙ ❲ ❭ P ❚ ❙ ❩ ❚ ❙ ❙❯ P ❱ ❫ ❲ ❘ ❱ ❫ ❩ ❱ ❪ ❞ ❭ ❝ ❤ ❛ ❭ ❨ ❯ ❙ ❚ ❙ ❘ ❚ ❲ ❢ ❱ ❞ ② ❙ ❘ ❯ ❯ P P❘ ❘ ❪ P❘ ❯ ❲ ❘ ❘ ❵ ❬ P ❙ ❪ ❚ ❙ ❙❯ P❱ ① ❡ ❱ ❫ ❭ ❬ ❲ ❫ ❙❯ ❱ ❫ ❭ ❪ P❘ ❯ ❙ ❚ ❙ P❱ ❪ ❙ ❘ ❯ ✈ ❳ s s ❱ ❦ ❡ ❩ ❘ ❬ ❳ ❨ ❱ ❡ ❘ ❯ ❩ ❪ ❵ P❘ ❯ ❲ ❱ P ❘ ❩ ❯ ❙ ❚ ❡ ❘ ❭ ❱ ❞ ❱ ❢ ❱ ❞ ❬ ❣ ❭ ❯ ❩ ❳ ❡ ❚ ❘ ❯ ❩ ❯ ✐ ❡ ❱ ❫ ❭ ❲ ❙ P P❱ ❦ t ❖ P❘ ❞ ❙ ❚ ❙ ❙❯ ❞ ❞ ❭ ❝ ❤ ❛ ❭ ❨ ❯ ❙ ❚ ❪ ❘ ❡ ❭ ❚ ❲ ❢ ❱ ❞ ❬ ❣ ❭ ❯ ❪ ❞ ❛ ❭ ✇ ❡ ❘ ❯ ❩ ❯ ✐ ❙ ❚ ❯ ❘ ❲ ❘ ❘ ❦ t ❖ P❘ ❞ ❲ ❳ ❙ ❨ ❱ ❞ ❭ ❝ ❤ ❛ ❭ ❨ ❯ P❘ ❭ ❭ ✻✼ ✎ ✲ ✶ ✴ ✹ ✸ ✺ ❁ ✗ ✾ ❂ ❄ ❆ ❇❈ ❏ ❙ ❚ ✏ ✖ ❙❯ ✏ ☞ ✂ ✌ ✞ ✁ ✍ ✎ ✁ ✕ ✑ ✒ ✁ ✓ ✔ ✎ ✞ ✎ ❙ P❱ ✡ ❜ ❬ P ❱ ❘ P ❘ ❴ ❝ ❚ ❜❞ ❝ ❛ ❱ ❱ ❡ ❱ ❪ ❩ ❙ ❲ ❭ ❘ ❱ ❩ ❬ P ❩ ❨ ❱ ❪ ❯ P ❘ ❯ ❳ P ❘ ❘ ❩ ☛ ✠ ❱ ✑ ✌ ✞ ✁ ✍ ✎ ✏ ✁ ✒ ☞ ✁ ✓ ✔ ✎ ✞ ✎ ✕ ✖ ✂ ☛ ✏ � ú û ü � ✁ ✂ ✄ ☎ ✆ ✡ ✝ ✁ ✂ ✄ ✞ ✟ ✞ ✠ ✗ ✎ ✞ ✄ ù ú û ü � ✁ ✂ ☎ ✛ � ✆ ✝ ✁ ✂ ✄ ✞ ✟ ø ✱ ✛✜ ✩ ✢ ✛✣ ✤ ✥ ✦ ✧ ★ ✩ ✢✭ ✜ ✮ ✯ ✛ ✢ ✰ ✩ ✦ ✙ ❱ ❬ ❯ ❛ ❧ ❱ ❣ ❨ ♠ ❲ ❘ ❜❞ ❦ ❲ ❚ ❙ ❙ ❨ ❪ ❴ ❢ P❱ ❞ ❲ ❬ P ❱ q ❛ ❤ P❥ ❘ ❯ ❫ ❱ ❱ ❙ ❚ ❙ ❙ ❘ ❭ ❘ ❘ ❵ ❲ ❙ ❘ ❭ ♦ ✐ ❙ ❲ ❱ ♥ ❛ ❱ ❣ ❨ ❘ ❞ ❭ ❞ P ❪ P ❘ ❯ ❡ ❚ ✐ P ❛ ❭ ❛ ❘ ❳ ❲ ♣ ❘ ❲ ✐ ❲ ❘ ❨ ❱ ❳ P P ❘ ❣ ❯ ❡ ❨ ❘ ❙ ❛ ❜ ❨ ❪ ❱ ❲ ❤ ❘ ❚ ❲ ❢ ❱ ❯ ❱ ❙ ❯ ❞ ❱ ❯ ❯ ❘ ❙ ❲ ❬ ❩ ❴ ❱ P ❩ ❭ ❙ ❞ ❚ ❛ P ❤P ❭ ❛ ❯ ❳ ❘ ❱ ❣ ❨ ❙ ❙ ❚ ❲ ❘ ❘ ❘ ❯ ❡ ❘ ❡ ❚ ❬ P ❫ ❱ ❢ ♠ 134 - (Red-Black Trees) ❩✬❫ 3 ❳❅❨ - ✿✵❀ ♠◗P ❳❅❨ ✽✬✾ - ▼◆❍■ - ❳❅❨ ❑❅▲ ✷✬✸ ❳❅❨ ❳❅❨ ✳✵✴ ●❅❍■ - ❩✵❨ ❵✵❛ ❉❋❊ - - - o o o ýÿþ - ❩✵❨ ❳❅❨ ❃❅❄ ❖◗P❘ 133 3 ✪✬✫ - ýÿþ ✘✚✙ -

  67. ③ ➁ ⑦ ⑥ ❹ ❸ ④ ↔ ➙ ❸ ⑧ ⑦ ①⑦ ↔ ⑧ ❻ ⑥ ③ ➁ ⑤ ❻ ➛ ⑩ ↕ ➜ ③ ❽ ⑤ ❸➝ ❿ ✱ ✴ ➑ ✯ ✰ ✽ ➣ ⑦ ❾ ➂ ↔ ⑥ ④ ❹ ❻ ⑦ ① ⑦ ⑥ ⑧ ⑥ ❹ ⑦ ↕ ❹ ⑨ ✴ ➺ ➳➚ ➺➪ ➶ ➚ ➹ ➶ ➸ ➘ ➳➵ ➪ ➼ ➯ ➭ ➪ ➼➮ ➳ ➼ ➭ ➮ ➱ ➸ ➶ ➚ ➻ ➳ ➽ ➂ ✫ ➂ ⑦ ❸ ↕ ①③ ⑥ ⑧ ③ ⑦ ✼ ✵ ✲ ➼ ✳ q ➦ ➩ ➫ ➭ ➯ ➸ ➺ ➻ ➳ ➸ ✬ → ➪ ✵ ➈ ✯ ✵ ✯ ✵➉ ✶ ✯ ✵➌ ✳ ✱ ✫ ➌ t ➉ ➌ ✳ ✱ ✯ ✵ ➍ ✯ ✰ ✫ ✰ ✉✈ ✉ ✷ ❾❼ ❻ ❸ ⑩ ⑥ ④ ⑦ ④ ⑤ ⑧ ❽ ⑦ ❸ ➇ ⑥ ⑦ ❼ ④ ⑦ ➁ ⑦ ➂ ⑨➃ s ➆ t ➎ ✱ ➔ ✯ ➒ ✭ ✬ ➒ ➓ ➋ ➌ ✲ ✯ ✯✹ ✲ ✰ ➒ ✫ ✱ ✲ ✳ ➒ ✲ ✯ ✵ ✭ ✯ ✰ ➌ ✰ ✯ ➉ ✰ ✱ ✲ ✯ ✯✹ ✲ ✯ ✰ ➉ ➊ ➉ ✵ ✯ ✱ ✯ ✰ ➋ ✫ ✯ ✰ ✴ ✬ ✱ ✴ ➑ ✃ ➸ ❹ ➸ ✃ ➻ ➺ ➭ ➮ ➹ ➳ ➽ ➚ ➹ ➶ ➘ ➺ ➳➵ ➩ Ú ÛÜ Ý Þ ß à á à Þ ❐ × â Õ ➭ ➯ ➲ ➼ ➻ ➳ ➪ ❮ ➺ ➭ ➮ Ò Ö ➺ ➮ ➻ ➳➵ Ñ ❒ Ò ➮ ❐ ➻ ➳ ➪ à ß ➸ ö å ô ÷ öø ö û ü ý þÿ � ñ õ ß ô ÷ õ ö ✁ ✂ ü ý ü ✄ � ã â ã ç ä Üå Ûç ä è ÛÜ Ý Ü é ê Þ å à ç á í Û àî ß ã ä à ï Ü ä ➺ ➳ ➼ ➮ ➪ ➹ ➶ ➸ ➘ ➳ ❐ ➺ ➳ ❮ ➳ ➳ ➻ ❰ Ï ❮ ➳ ➻ ➺Ð ➽ Ñ Ñ ➳ ➷ ➳ ➶ ✃ ➺ ➽ ➭ ✃ ➪ ➹ ➶ ➸ ➘ ✃ ➪ ❐ ➹ ❮ ❒ ➱ ➺ ➸ ➳ ➪ ➳ ❮ ❒ ➻ ➮ ➼ Ò ➭ ➭ ❒ ➪ ➵ ➸ ➺ ➺ ❮ ➼Ô ✃ Ô ➮ ➹ ➶ ➮ ➻ ➮ ➻ ➳ ➵ ❮ ➳ ❰ Õ ➪ ❰ ➹ ➺ ➯ ➼ ➚ ➪ ➼ ❐ ➵ ➸ ❮ ➺ ➭ ➮ Ñ ➸ ❰ ➘ ➳ ➵ ➸ ➺ ➻ ➳ ➪ ➩ Ó ❮ ➳ ❺ ❻ ❸ ➹ ➞ ➠ ➪ ➜ ➞ ➭ ➸ ➜ ➞ ➶ ➵ ➻ ➸ ➜ ➵ ➳ ➯ ➘ ➴ ➮ ➬ ➱ ✃ ➧ ➠ ❒ ➥ ➻ ➠ ➸ ➜ ➞ ➧ ➯ ➜ ➞ ➵ ➭ ➦ ➚ ➥ ➦ ➧ ➨ ➝ ➯ ➞ ➺ ➯ ➝ ➸ ❐ Ð ➡ é â ä à ç Ú â à è à â í ä î ï ð ñ òó ô õ ò øù ÷ ÷ âå à ➬ × Ñ ❰ ➮ ➬ Ò ➬ Ó Ô Õ Ö Ó Ó Û Ø Ù ➬ ÔÚÛ Ü Û Ý Þ á âã á ➜ ➯ ÷ ➆ ➇ ➈ ❷➉ ➊ ➅ ❽ ➅ ➋ ➌ ➍ ➅ ➆ ➎ ➐ ➑ ➒➓ ➔ ➓ ➓ ↔ ↕ ➒ ➓ ❷ ➅ ➞ ❷ ⑤ ⑥ ⑦ ❶ ❷ ❸ ❹ ❺ ❶ ❻ ❼ ❸ ➄ ❹ ❽❾ ❽ ❿ ➀ ➁ ➂ ❸ ➃ ❽ ❷ ➝ ➥ ➼ ➠ ➞ ➭ ➧ ➠ ➥ ➦ ➥ ➥ ➭ ➜ ➫ ➜ ➞ ➥ ➧ ➸ ➫ ➻ ➲ ➜ ➠ ➜ ➩ ➝ ➯ ➜ ➵ ➠ ➜ ➥ ➦ ➥ ➥➭ ➜ ➯➲ ➧ ➞ ➯➲ ➸ ➸ ➜ ➵ ➳ ➯➲ ➺ ➯ ➲ ➻ ➲ ➜ ➵ ú æ û ❉ ❇ ❈ ❉ ❊ ❆ ❋ ● ❇ ❈ ❍■ ÷ ❍ ❏ ❑ ▲ ▼ ❈ ◆ ❍ ❇ ❆ ❁ P ✯ ✦ ★ ✩ ✪ ✯✰ ✱ ✲ ✯ ✳ ✶ ❀ ✷ ✸ ✼ ✵ ✫ ✲ ✳ ✽ ✾ ✿ ❖ ◗ ✤ ①⑦ ❣ ❢ s t ✉✈ ④ ⑤ ④ ④⑥ ⑧ ♠ ③ ⑦ ❶ ❷ ① ❶ ⑩ ⑦ ⑧ ❸ ❦ ❢ ❇ ❳ ❘ ❙ ❇❚ ❯ P ❍ P ❱ ❲ ◗ ❩ P ❨ ❜ ❞ ❡ ❢❣ ❢ ❣ ❢ ❧ ✧ ❂ ✣ ✁ ☞ ☎ ✆ ✟ ✍ ✁ ✠ ☎ ü ✁ ý ☎ ✆ ✍ ✡ ✁ ü ✁✡ ✍ ✌ ☎ ✎ ☎ ÿ � ✁ ✂ ✄ ✁☎ ✆ ✝ ✞ ✟ ✁ ÿ ✆ ✠ � ✁✡ ☛ ü ✟ � ✝ ☞ ✑ ✢ ✛ ✖ ÿ ✕ ✗ ✔ ✓ ✜ ÿ ✁ ✝ ✎ ✘ ✕✖ ☎ ✎ n log(n+1). 136 ✩➐➏ ➴➬➷ v, bh(v ? 3 - bh(v) - ë②ì ⑥➟➞ ⑨✐⑩ ❥❭❦ ⑨✐⑩ ❨✐❤ ⑨②➀ - v. - LookUp æ♣ç ➽➥➾ ❷②❿ ✩➐➑ v ❝❭❞ - ❒➥➪ ✇②①③ àÙð ù❵ú ➽➥➳ ⑨②❼ ❫❵❴❛ ➊✏➋ ➧②➨ v ➚ÙØ ❩❭❬❪ õ❭ö ➲✐➳➵ ❃❅❄ - q❵r ➢➥➤ ✩➐➏ 1. 2. ♥♣♦ ➠②➡ ñ➥òó ➄②➅ 135 ✃✚Ó ➡❅➳ ➯➾➽ 3 - ê◗ëì ➙✵➣ ✯✹✻✺ ✃✚Ó ➦◗➫ →✬➣ ✴✏✵ üþý - - ➨✚➩ ➦✵➧ ✑✏✒ ❮Ï❰ ✬✮✭ ✝✏✎ ➏✬➐ ✪✏✫ ✘✚✙ ➡◗➢➤ ôÏö÷ ß✚à ➟✬➠ ⑧⑩⑨ - ✥✏✦ ➷◗➬ ➛◗➜

  68. ☎ ü ò ó ì ë ÷ø ù ø ú ö û þ ë õ ÿ � ✁ ÷ � ✆ ✝ ÿ � ✠ ✎ ñ ã ✗✘ è Ü á â Ý Þ ß ãä ã å æ ç Þ ë é ã Ý ê ë ì Ý í î Ýï ð ✒✓ ✛ ß ❇ ✾ ✿ ❃ ❉ ❇ ❇ ❊ ❇ ❍ ■ ❏❑▲ ✼ ❉ ❏ ▼ ▲ ❉ ❑ ◆ ❉ ❋ ❍ ❊ ❀❁❂ ✿ ✜ ✪ ✢ ✣ ✤ ✥ ✦ ✧ ★ ✩ ✩ ✫ ✩ ✩ ✻ ✩ ★ ✯ ✮ ✵✽ ✷ ✾ ✻ ✿ ✼ ❀ à Þ ❑ ✃ ➭ ➚ ➼ ➸ ➫ ➲ ➦ ➪ ➯ ➢ ❐❒❮ ✆ ❰ Ï ➠ ➦ ➦ ➽ ➮ ➳ ➻ ➯ ➥ ➳ ➯ ➳ ➯ ➥ ➹ ➚ ➳ ➼ ➫ ➷ ➥ ➷ ➥ ➸ ➢ ➫ ➷ ➹ ➮ ➳ ➻ ➯ ➢ ➩ ➫ ➭ ➩ ➷ ➾ Ý ➽ ➢ Ó ➥ Ô ➼ ➢➡ ➢ ➦ ➢ ➯ ➲ ➡ ➴ ➳ ➻ ➯ ➥ ➱ Õ Ö × Ø Ù Ü ➸ ➷ ➸ ➯ ➢ ➷ Ñ ➡ Ò ➢ ➩ Ð ➡ ➩ ➫ ➲ ➢ Ñ ➡ Ó ➥ ➩ ➢ ➨ ➷ ➳ ➻ ➼ ◗ ❉ ➾ ◆ ❑ ◆ ❏ ❘ ❊ ▲ ❉ ❍ ❑ ① ② ❘ ❘ ③ ▲ ◆ ❉ ❋ ❍ ③ ▲ ④ ② ❉ ❏ ⑤ ❴ ❛ ❪ ❲ ❱ ❤ ❜ ❪ ♦ ♣ ❜ ❭ ❯ ▲ t ❵ ✉ ❯ ❱ ❲ ❤ q ✈ ❑✇ ▼ ◆ ③ t ❊⑩ ⑨ ❊ ▲ ❘ ❉ ❑ ⑥ ❘ ❊ ❘ ❶ ■ ■ ▲ ❶ ❍ ❊⑩ ❙ ❷ ② ❘ ❏ ❙ ❍ ❇ ▲ ⑦ ⑥ ❏ ④ ⑦ ① ❏ ◆ ❊ ▲ ❊ ❇ ❘ ❏ ❑ ❙ ✈ ❑ ❉ ⑧ ⑧ ❉ ✇ ■ ● ❳ ❲ ❘ ❴ ❫ ❴ ❩ ❳ ❬ ❫❞ ❬ ❡ ❪ ❯ ❫ ❜ ❪ ❯ ❲ ❩ ❢ ❝ ❝ ❩ ❢ ❝ ❩ ❱ ❝ ❳ ❱ ▼ ❙ ❚ ❯ ❱ ❲ ❱ ❳ ❬❭ ❪ ❯ ❫❴ ❱ ❩ ❯ ❳ ❵ ❩ ❱ ❛ ❜ ❫ ❲ ❭ ❯ ❤ ❴ ❩ ❲ t ❛ ❳ ❤ ❜ ❪ ❬ ❡ ❢ ❬ ❬ ❜ ❤ ❫ ❪ ❨ ❴ ❫ ❪ ❯ ❯ ❩ ❲ ❵ ❲ ❜ ❛ ❱ ❳ ❬✐ ❝ ❥ ❲❦ ❳ ❤ ❧ ♦ ❩ ♥ ❦ ❜ ❱ ❲ ❣ ❨ ❴ ❣ ❳ ❲ s ❲ ➬ ➲ ➻ r ✇ ♠ ⑧ ② ❡ ❤ ♠ ✇ ⑨ ③ ♦ ✇ ⑩ ♦ ② ❤ ❡ ❣ ♦ ❣ ② ✇ ❶ t ⑦ ❤ r ❤✐❥ ✐ ❦❧ ♠ ❥ ♥ ❣ ♦ ❤ ♣ ❡ ❧ ✇ ♣ ✇ ❤ t ❤ ① ③ ❣ ③ ❝ ⑥ ❷ ④ ❝ ✇ ❡ t ✇ ④ ➥ ❞ ❻ ♦ ② ⑧ ① q r ⑦ ♦ ❤ t ✇ ❡ ❡ ♣ ❡ ❧ ② r ❧ ❝ ❤ ⑥ ✇ t s ✇ ❸ ⑩ ❹ ❹ ✇ ❺ ✈ r ❤ r ② ③ ✈ ❶ ① ♦ ② ❤ ❡ ❣ ❛❜ ② ★ ✌ ✢ ✣ ✌ ✤ ✥ ✛ ✓ ✛ ✦ ✧ ✜ ✛ ✛ ✬ ✭✮ ✯✮ ✰ ✬ ✱ ✲ ✴ ✪ ✶✷ ✜ ✚ ✭ ✌ ✝ ✞ ✟ ☞ ✌ ✍ ✎ ✏ ☞ ✑ ✒ ✍ ✌ ✎ ✓ ✔ ✓ ✕ ✖ ✗ ✘ ✍ ✙ ✓ ✸ ✷ ❫ ❊ ❖ ❆ ◆ ▼ ❈ P ◗ ● ❘ ▼ ❑ ❂ ❈ ◆ ❯ ❊ ● ❈ ❉▼■ ❋ ❂ ● P ❳ ◆ ❂ ✻ ❋● ✼ ✶ ✽ ✺ ✷ ✾ ✿❀ ❄ ❉ ❊ ❋ ❍ ❊ ❂■ ✿ ■ ❏ ❍ ❋ ❄ ❑ ❄ ❄● ❆❈ ③ ❝ ♦ ➛ ➊ ➋➏ ➐ ➔ → ➣ ↔ ↔ ➙ ➒ ➃➋ ➞ →➝ ➙ ➟ ➢ ➩ ➩ ➭ ➯ ➁➌ ➊ ➳ q ⑧ ♣ ❡ ✇ ✈ ⑨ ❿ t ✇ ⑦ ➁ ♦ ④ ❝ ➀ ➁ ➄ ➅➆➇ ➈ ➉ ♥ ➢ ➭ ❡ ➲ ➶ ➹ ➢ ➥ ➡ ➘ ➩ ➫ ➯ ➳ ➻ ➻ ➴ ➾ ➥ ➨ ➥ ➼ ➨ ➷ ➸ ➳ ➺ ➥ ➢➡ ➤ ➥ ➡ ➻ ➼ ➽ ➸ ➾ ➨ ➨ ➢ ➦ ➧ ➨ ➩ ➩ ➪ ➤ ➪ ➯ r ➅ ② q ♦ ② ❺ ❡ ⑩ ❤ ♦ t ❻ ❤ r ✇ ❹ ✇ ⑨ ✈ ✇ t ❼ ⑩ t ④ ♦ t q t ❤ ✇ ❡ ⑩ ✇ ❣ ⑦ ① ❺ ✈ ✇ ❸ ❤ ❡ ♣ ⑦ q ✈ ❥ ✇ ⑦ ❶ ❡ ⑦ ❤ ② ❣ ❤ ② 138 black-weight ❣☎❤ ✞✟✄ 3 ✺☎✻✼ ❣☎❤ ✶✸✷✹ ✂☎✄ ♣rq ✴✟✵ ●P❖ ❣☎❤ ✲✖✳ - ✫✱✰ AVL ý✵þ ✙✖✚ ✭☞✮ ❀❅❄ ✔✖✕ ✪✟✬ ❥✸❭ ❋✸● ✪☞✫ ✏✟✑ ❨✟❩ ♠✖♥ ✌☞✍ Ú☛Û ô✫õö - ✡☞☛ ❆❈❇ red. ?). 137 ♦❾❽ ♥⑤④ ▲❇▼ - 3 - ❦✫② ➫✉➭ ➥⑤➱ ➦❇➧➨ ✹✵✺ ♣✫✈ ➤✵➥ s✉t - red ➵❬➸ ➍❃➎ ❱❃❲ ✳✵✴ ♣✵q ➜❬➝ ➲❇➳ Ð✫➯ ↕✫➛ ➫✫➯ ➭❾➚ ➲❇➳ ➫✫➯ ↕❬➙ ➫✉➭ ❅❇❆❈ black ➫✉➭ ➦❇➧➨ ➦❇➧➨ ➤✵➥ ❴✫❵ ➑➓➒ ➤✵➥ ❁❃❂ ❙❚❃❯ ➂❇➃ ✠☛✡ ✩✫✪ ❭❬❪ - a. ➤✵➥ ➠✉➡ ➠✉➡ ❨❬❩ ❞❢❡

  69. ❸ ➁ ➊ ➋ ➌ ➂ ➍ ➇ ➎ ➇ ➏ ➐ ➁ ➑ ➒ ➁➓ ➔ ➉ ➇➈ ➇ ➁ ➜ ❸ ❹ ❺ ❻ ❼ ➀ ➂ ➃ ➃ ➄ ➀ ➅ ➆ ➁ ➂ ➏ ➏ ➝ ➜ ➜ ➦ ➫ ➭ ➥ ➦➲ ➳ ➩ ➥➦ ➵ ➯ ➠ ➙ ➝ ➸ ➜ ➧ ➙ → ➜➝ ➣ ↔ ➐ ➏ ↕ ➙ ➛ ➞ ➤ ➝ ➟ ➛ ➠ ➡ ➢ ➤ ➸ ❹ ➙ ➏ ➏ ➐ ➁ ➑ ➒ ➁➓ ➔ ➏ ➠ → ➁ ➣ ↔ ➐ ➏ ➜➝ ➞➝ ➟ ➛ ➠ ➎ ➇ ➤ ➅ ❺ ❻ ❼ ➀ ➁ ➂ ➃ ➄ ➀ ➆ ➍ ➁ ➂ ➃ ➇➈ ➇ ➉ ➊ ➋ ➌ ➂ ➡ ➇ ➙ ➭ ➯ ➥➦ ➩ ➜ ➳ ➦➲ ➩ ➯ ➥ ➵ ➜ ➫ ➧ ➦ 140 ➯✟➩ 3 ➨☎➩ - ❽❿❾ - 139 3 ➨☎➩ - ➢☎➤ ❽❿❾ ↕✟➙➛ -

  70. ➺ ✼ ❅ ❋ ❅ ❄ ✿ ❆ ✿ ❇ ✿ ❁ ❇ ❂ ❃ ✿ ❇● ✾ ❉ ❃ ✿ ❇ ❅ ❆ ❂ ✿ ✎ ✗ ✑ ✜ ✦ ✎ ✘ ✭ ✧ ✕ ✮ ❆ ✺ ✻ ✼ ✽ ✼❁ ❂ ❃ ❄ ✿ ❅ ❍ ✼ ✛ ▼ ❂ ❇ ✼ P ✻ ✼ ❁ ❃ ■ ❇ ◆ ❅ ❂ ❆ ✽ ❃ ❂ ❅ ✿ ❄ ❚ ❖ ✻ ✻ ✼ ❇ ❅ ❋ ■ ✿ ✼ ✿ ✼ ✼ ✻ ■ ❅ ✼ ✿ ❑ ❉ ❇▲ ❃ ✿ ❇ ❅ ❋ ✘ ✑ ✾ ✁ ✛ ✱ ✫ ✩ ✙ ✎ ✮ ✶ ó û ✚ ✄ ✳ ✕ ✙ ✎ ✬ ✑ ✜ ✩ ✏ ✑ ✌ ✎ ✦ ☞ ✑ ✜ ✮ ✳ ✕ ✦ ✛ ✦ ✌ ✗ ➻ ✜ ☞ ✗ ✕ ✥ ✚ ✭ ✑ ✪ ✑ ✒ ✌✦ ✘ ✧ ✕ ✎ ✙ ✜ ✕ ✜ ✚ ✭ ✌ ✕ ✙ ✗ ✙ ✎ ✬ ✑ ✜ ✜ ✣ ✙ ✸ ✜ ✓ ✏ ✘ ✜ ✥ ☞ ✩ ✙ ✌ ✜ ✥ ✑ ✗ ✘ ✔ ✷ ✘ ✜ ✥ ✫ ✭ ✪ ✌ ✤ ❉ ✻ ✧ ❃ ❉ ❂ ❇ ✿ ■ ❳ ✼ ❁ ❂ ❘ ✿ ❭ ✿ ❆ ▼ ✻ ✼ ❉ ❃ ■ ❇ ❃ ❅ ✻ ❖ ✼ ▼ ❂ ❃ ✿ ✼ ▼ ✼ ❂ ❳ ❇ ✼ ❁ ❂ ❃ ❍ ❃ ❂ ❇ ✼ ❬ ❂ ✼ ❁ ❴❡ ✼ ● ✾ ❱ ❘ ❪ ❫❴❵ ❛ ❵ ❢ ✿ ❫ ❢ ❣ ❫❥ ❡ ❧ ❵ ❧ ❡ ♠ ❉ ❍ ❉ ✿ ❃ ■ ▼ ❇ ❲ ❉ ❃ ❅ ❇ ❍ ❃ ❅ ■ ▼ ❇ ▲ ✼ ❁ ▼ ✻ ❃ ❉ ✿ ✼ ❃ ▼ ✼ ✻ ❉ ✼ ❱ ✿ ❇ ✻ ❖ ❯ ❈ ❍ ✿ ✼ ❋ ✾ ▼ ❋ ❉ ✼ ❂ ❉ ❇ ❅ ✼❁ ❅ ▼ ❇ ❂ ❃ ✿ ❇● ✾ ❉ ❂ ❂ ❃ ❃ ✻ ✼ ❂ ✾ ▼ ❯❱ ✿ ✼ ❉ ❃ ✻ ❬ ❇ ✼ ❈ ❂ ❃ ✾ ▼ ❇ ▼ ❄ ❅ ❬ ❬ ❅ ▼ ✼ ❬ ❄ ❂ ❩ ✼ ▼ ❉ ■ ❃ ✿ ❃ ▼ ❇ ● ✽ ❖ ❍ ✼❁ ❂ ❍ ❲ ✿ ❆ ❉ ■ ❳ ❇ ❏ ❨ ❂ ✾ ✕ ✕ ☞ Ò ❰ ➘ Ï ➱ ➹ Ð Ñ ➹ ❒ Ó Ô ➹Õ Ö Ñ ➱ Ñ × ❮ ❐ Ù ➴ ➼ ➽ ➘ ➾ ❰ ➶ ➹ ➘ ➷ ❮ ➶ ➬ ➮ ➹ ➘ ➴ ➱✃ ➱ Ø Ò ➺ ✏ ì ë å æ ➱ ☛ ➱✃ ✑ Ý ✎ ✒ ✓ ✏ é ✘ ✎ ➴ æ ð Ñ Ý ❒ ✜ ê Û ÝÞ é Ý ✠ à ✡ ❐ ã ä ã ä ä Û å æ ➻ Ï ✙ Ø Ü à ßÞ ÝÞ Ñ Ò Ù × â Ñ ➱ Ñ Ö ➹Õ Ô ñ Ó á ä õö Þ î Ý é ï ì á Û ì ð Û Ý å ë ê æ Ý ç åæ ô ÷ ➱ ✆ ó ù ☎ ü ✆ õ ☎ ü ✝ ö ✞ û ù û ✁ û ù ✟ ➹ ✁ ✄ ➹ Ñ ôû ü Ò û ü õ ý ÷ ö ✁✂ ü õ ÿ Ð ñ � ü ó ✌ æí ✗ ✌ ✌ ✏ ✔ ✕ ✜ ✕ ✘ ✎ ✦ ✦ ✎ ✘ ✭ ➷ ✫ ➴ ✫ ✩ ✎ ✌ ✗ ✬ ✚ ✙ ✎ ✘ ✌ ✤ ✙ ✜ ✌ ✪ ✤ ✪ ✬ ✜ ✥ ✕ ✓ ✛ ✫ ✣ ✪ ✑ ✜ ✥ ➽ ✲ ✎ ☞ ✌✎ ✏ ✎ ✣ ➼ ✒ ✗ ✔ ✎ ✚ ✛ ✘ ✚ ✜ ➘ ✗ ✗ ✕ ➹ ✧ ✑ ✏ ✗ ✌ ➶ ✱ ✰ ✤ ✘ ✢ ➾ ✜ ✦ ✛ ✙ ✑ ✢ ✣ ✕ ☞ ✧ ✕ ☞ ✑ ✜ ✎ ✎ ➹ ★ ✓ ✘ ✩ ✗ ✩ ✣ ✛ ✎ ✦ ✘ ✎ ✑ ✛ ✙ ✜ ✌ ✜ ✑ ➘ ✛ ✚ ✜ ✛ ✚ ✢✣ ✤ ✗ ➮ ✤ ✗ ✎ ✜ ✌ ✌ ✎ ✘ ✌ ✎ ✪ ✏ ☞ ✧ ✫ ✗ ✌✎ ✕ ✌ ✗ ✌ ✘ ✤ ➬ ➶ ✘ ✩ ✩ ✤ ✥ ✕ ✘ ✛ v 142 ✬✖✹ ✧✖✪ ? RB-Delete()? ■✖▼ ì✟é 3 ◗❙❘ ✣✯✮ è☎é extra ■✖▼ - ❈❊❉ ❁✵❏ ☞✍✌✎ ❈❊❉ ❦✐❧ ✾✖❲ ❣✐❤ ✔✖✕✗ ✣✵✴ ❩✍❉ ✣✵✴ ❈❊❉ y ❜❞❝ y ➚❿➪ ✾❀✿ - ☞✍✌✦ y ☞✍✌✎ ☞✍✥ 141 RB-Insert()? 3 è☎é ýúþ - black height ã☎ä øúù ➚❿➪ Ú✟ÛÜ - ò✖ó

  71. ♥ ➯ ❮ ➫ ➠ ↔ ➫ ➭ ❰ ➸ Ï ➨Ð ➭ ➯ ➚ ➼ ↕ ➵ ➸ ➬ ➛ ↕ ➫ ➠ ↔ ➯ ❒ ➛ ➜ ➛ ➾ ➯ ➭ ➘ ➨ ➺ ➠ ✃ ↔ ➛➜ ➛ ➵ ➯ ➲ ➙ ➨ ➨ ➻ ➼ ➠ ↔ ➛ ➩ ➚ ➜ ➫ ➺ ➛ ➙ ➪ ➨ ➜ ➜ ➯ ➘ ➺ ➯ ➻ ➪ ➨ ➙ ↔ ➨ ➸ ➠ ➛ ➨ ➵ ➜ ➺ ➭ ➵ ➯ ➨ ➺ ➛➜ ↔ ➛ ➠ ➩ ➵ ➨ ♦ ➵ ➯ ➺ ➨ ➪ ➨ ✃ ➯ ➭ ➩ ❐ ➛ ➞ ✃ ➹ ➜ ➼ ➘ ➫ ➭ ➯ ➼ ➯ ➜ ❐ ➠ ➨ ➜ ➵ ➯ ➪ ➚ ↔ ➫ ➛ ➭ ➵ ➸ ↕ ➼ ➠ ➹ ➜ ➯ ➵ ➛➜ ➫ ➜ ➘ ➬ ➺ ➼ ↔ ➛ ↔ ➛ ↔ ➶ ➙ ➯ ➻ ➷ ✃ ➺ ➯ ➪ ➨ ➱ ➜ ↔ ➼ ➸ ➵ ➘ Ñ ➵ ➯ ➻ ➼ ➯ ➛ ➭ ➯ ➚ ↔➙ ➛➜ ↔ ➙ ➜ ➫ ✃ ➛ ❐ ➶ ➺ ✃ ➷ ➷ ➛ ➺ ➯ ➫ ➠ ↔ ➯ ↔ ↔ ➛ ➙ ↔ ➛ ➫ ➜ ➨ ➻ ➯ ✃ ➪ ➚ ↔ ➯ ➻ ↔ ➺ ➜ ➜ ➯ ➚ ➺ ➨ ➭ ➜ ➙ ➯ ➳ ➷ ➙ ➸ ➼ ➽ ➯ ➻ Ð ➨ Ò ➪ ➷ ➺ ➛ ➾ ➚ ➪ ➛ ➨ Ñ ➙ ➛➜ ➸ ➹ ➜ ➷ ➳ ➬ ➨ ↔ ➭ ➜ ➠ ➹ ➯ ➻ ➛ ➵ ➵ ➪ ➴ ➳ ➙ ➪ ➨ ➜ Ó ➭ ➜ ➨ ➷ Ð ➺ ➨ ➜ ➯ ➙ ➯ ➩ ➙ ➸ ➼ ➽ ➛ ➩ ↔ ⑤ ✇ ⑥ ❷ ① ❶ ⑩ ⑨ ⑧ ⑥ ⑥⑦ ② ① ✇ ④ ❹ ✈ ③ ② ① ✇ ✈ r q ♣ ♦ ♥ ➄ ➎ ➅ ❸ ❺ ➔ ➐ ➠ ➛➜ ↔ ➩ ➨ ➡ ➛ ↔ ➠ ➛➜ ↔ ➩ ➄ ➆ ➌ ✇ ➑ ➂ ❹ ❺ ➁ ➀ ❿ ❹ ⑥ ❹ ❾ ✇❽ ❼ ❻ ➃ ➆ ➛ ⑥ ❻ ✇ ❺ ❹ ❸ ✇ ⑥ ❷ ① ❶ ⑩ ⑨ ⑧ ⑥⑦ ✇❽ ② ① ✇ ⑤ ④ ✈ ③ ② ① ✇ ✈ r q ♣ ❼ ❾ ➄ ➋ ➓ ➍➒ ➆ ➐ ➌ ➑ ➐ ➍ ➆ ➏ ➄ ➎ ➌➍ ➃ ➋ ❹ ➊ ➉ ➄ ➈ ➄➇ ➆ ➄➅ ❹ ❺ ➁ ➀ ❿ ❹ ⑥ ↔ ➣ ➫ ➵ ➪ ➪ ➚ ➾ ➛ ➺ ➛ ➪ ➸ ➛ ➺ ➨ ➲ ➭ ➜ ➺ ➞ ➵ ➛ ➜ ➫ ➜ ➘ ➵ ➵ ➯ ➻ ➨ ↔ ➴ ➨ ➛ ➯ ➨ ➲ ➨ ➭ ➯ ➵ ➼ ✃ ➷ ➨ ➭ ➩ ➛ ➱ ➮ ➯ ➘ ➻ ➻ ➫ ➠ ↔ ➯ ➻ ➯ ↔ ➽ ➺ ➯ ➩ ➫ ➜ ➯ ➜ ➸ ➶ ➛➳ ➨ ➜ ➛ ↔ ➩ ➯ ➠ ↔ ➺ ➺ ➨ ➛ ➻ ➯ ➵ ➩ ➯ ➯ ➻ ➩ ➵ ➯ ↔ ➙ ➯ ➪ ➽ ➠ ➪ ➯ ➚ ➯ ➨➾ ➞ ➙ ➻ ➳ ➛ ➳ ➻ ➲ a ≤ c ≤ b, -3 144 Ò❞Ô ➠✖➹ . 3 ➫❀➭ b ➫❀➭ ➱✍➜ ➡➦➥➧ ➢❊➤ (a,b)- ➸❀➼ c (a,b)- ➝➟➞ ➸❀➺ b ≥ 2a – ➠✖➹ - a = 2, b = 3: ➳✯➬ ➸❀➷ ↕✖➙ a ≥ s✉t - ➫❀➭ ➫❀➭ 143 3 - ➐❀→ ➊✍➋ s✉t ➂❞➃ -

  72. Õ ✘✮ ★ ✳ ✪ ✯ ✢ ✛ ✧ ✬ ✴ ✗ ✘ ✤ ✪ ✛ ✭ ✛ ✤ ✘ ★ ✦ ✘ ✤ ✙ ✣ ✘✙ ✧ ✧ ✥ ✩ ✘✙ ✤ ✵ ✣ ✫ ✬ ✤ ✣ ✤ ✘ ✱ ✓ ✥ ✘ ✤ ✜✢✣ ✛ ✚ ✘✙ �✒ ✧ ✠ ✏ ✁ ✑ ✏ ✎ ✍ ✦ ✧ ✫ ✫ ✩ ✣ ✗ ✢ ✛ ✩ ✬ ✩ ✘ ✘ ✪ ✘ ✩ ✘ ✘ ✤ ★ ✦ ✚ ✂ ✣ ✮ ✪ ✛ ✯ ✘ ✬ ✗ ✗ ✷ ✬ ✗ ✘ ✤ ★ ✛ ✚ ✤ ★ ✤ ✚ ✢✣ ✚ ✛ ✚ ✪ ✻ ✣ ✤ ✤ ✙ ✣ ✘ ✧ ✧ ✦ ✥ ✘ ✘✙ ✬ ✢ ✣ ✧ ✧ ✦ ✥ ✘ ✬ ✗ ✗ ✩ ✬ ✛ ✚ ✦ ✘ ✜ ✫ ✘✙ ✘ ✘✙ ✭ ✰ ★ ✘ ✫ ✩ ✢ ✣ ✮ ✘ ✩ ✤ ✢ ✛ ✕ ✬ ✢ ✫ ✛ � ☎ ✣ ➵ ➻➹ ➶ ➵ ➪ ➚ ➸ ➯➾ ➽ ➲ ➼ ➯ ➲ ➵ ➺➻ ➲ ➯➸ ➭ ➻ ➯ ➭ ➱ ➴ ➺ ➮ ➯➮ ➴ ➾ ➪ ➺ ➯ ➬ ➯ ➺ ➷ ➘➴ ➪ ➯ ➲ ➭ ➵ ❿ ➅ ❿ ➊ ❻ ❸ ➂ ➅ ❾ ❾ ➎ ➂ ❷ ➒ ➊ ↔ ❻ ❹➊ ➒ ❿ ➧➨ ❽ ➦ ➢➤➥ ➡ ➠➡ ➟ ➂ ➅ ➒ ➄ Ö ❷ ➂ ➊ ➇ ➂ ➓ ➒ ✃ ➯ ☞✌ é í ì æ Þ æ ë Øê è ç Ø ç æ å Ø Þ ä î æ ã ✂ ✝ ✠ ✂ ✝ ✟ ✝✞ ✂ �✁ òó ü ø ÷ ö ñ õ ó ô Ù â ➼ ❮ ➯➾ ➼ ➶ ❮ ❮ ❮ ❒ ❮ ❰ ❮ ➯ ➭ ➳ ➶ ➾ ❐ ➯ ➽ Ï á Ü à Þ Þß Ú Ù Ø Ý × Ð Û Ú Ù Ø × Ó Ò Ñ ✮ ✤ ➂ ✚ ✛ ✯ ❃ ✗ ✘ ✤ ✛ ✘✙ ✢ ✗ ✔ ✰ ✗ ✘ ✤ ★ ❂ ✸ ✢ ✚ ✢ ✣ ✮ ✘✙ ✤ ✘ ✽ ✸ ✣ ✮ ✸ ✮ ✢ ✥ ✧ ✛ ✯ ✔ ✯ ✢ ✽ ✗ ✼ ✥ ✣ ✹ ❃ ✘ ✚ ✗ ✸ ✮ ✘ ✬ ✗ ✣ ✗ ✬ ✘ ✤ ✢ ✫ ✣ ✩ ✣ ✫ ✛ ✤ ✸ ✛ ✣ ✤ ★ ✢ ✫ ✣ ✣ ✤ ✵ ✛ ★ ★ ✴ ✚ ✸ ✮ ✘ ✦ ✯ ✗ ✤ ✴ ✵ ✸ ✚ ★ ✢ ✫ ✴ ✽ ✗ ✢✣ ✮ ✱ ✘ ✽ ✚ ✸ ✮ ✸ ✢ ❀ ✫ ✤ ✣ ✩ ✗ ✘ ✤ ★ ✕ ✯ ✣ ✚ ✧ ✮ ✘ ✪ ✣ ★ ✛ ✘ ✢ ✤ ✛ ✚ ✙ ✘ ✜ ✔ ✜ ✛ ✯ ✘ ✘ ✛ ✗ ✢ ✜ ✬ ✘ ✪ ✩ ✪ ✘ ✘ ✤ ★ ✿ ✙ ✘ ✤ ✛ ✘✙ ✔ ✢ ✤ ✸ ✮ ✢ ✗ ✘ ✤ ✛ ✙ ✜ ✘ ✤ ✸ ✩ ✣ ✸ ✘ ✮ ✙ ✛ ✦ ✧ ✘ ✩ ✘ ✤ ✢ ✕ ✸ ✬ ✘ ✗ ✾ ✱ ✘ ✽ ✚ ✤ ✣ ✣ ✮ ✦ ✘ ✢ ✯ ✢ ✛ ✧ ✼ ✘ ✚ ✢ ✤ ✷ ✧ ✷ ✧ ✧ ✩ ✙ ✪ ✪ ✯ ✢ ✛ ✧ ✮ ✘ ✤ ✛ ✯ ✢ ✘ ✩ ✘ ✫ ✘ ✥ ✢✣ ✩ ✢ ✤ ✔ ✤ ✛ ✭ ✘ ✩ ✘ ❁ ✮ ✦ ✢ ✤ ✛ ✫ ✘ ✥ ✢✣ ✘ ✚ ✱ ✫ ✔ ✢ ✯ ✢ ✣ ✩ ✣ ✛ ✛ ✤ ✸ ✢ ❂ ✛ ✯ ✣ ✤ ✶ ✢ ✯ ✗ ✜ ✴ ✜ ✣ ★ ✢ ✛ ✷ ❀ ✤ ✔ ✤ ✛ ✚ ✿ ✔ ✢ ✼ ✘ ✛ ✤ ✻ ✪ ✤ ✣ ✚ ✫ ✛ ✵ ✣ ✘ ✚ ✭ ✢ ✫ ✸ ✜ ✧ ❾ ➅ ❹ ✘ ✓ ✚ ✔✵ ✥ ✖ ✚ ✍ ✔ ✏ ✱ ✰ ✑ ✪ ✗ ✍ ✦ ✮ ✔ ✑ ✽ ❆ ✿ ✽ ✼❅ ✼ ❂ ❁ ✼ ✪ ✸ ✑ ✏ ✫ ✛ ✚ ✑ ✗ ✛ ❈ ✫ ✧ ✘ ✴ ✑ ✪ ✔ ✏ ✪ ✚ ✒ ✖ ✥ ✖ ✚ ✍ ✘ ✑ ✛ ✤ ✔ ✑ ✏ ✧ ✘ ✗ ✖ ✒ ✜ ✫ ★ ✒ ✖ ✚ ✒ ✙ ✑ ✏ ❇ ✿ ✰ ➌ ⑤ q④ r ♦ ✈ rs ✉ rs ① ♥ q ♥ ♣ ♦ ♠ ♥ ④ q⑥⑦ ❥ ❿➀ ➅ ❸ ❹ ➄ ➀ ➃ ❾ ❾ ✈ ❹ ❻ ❺ ❹ ⑩ ⑨ ⑧ ♠ ❲ ✼❊ ■ ❚ ❘ ▼ ◆❖P ▲▼ ❑ ❏ ❁ ❨ ❇ ❁ ✽ ❍ ● ✽ ❋ ❳ ❳ ❪ ❫ ❲ ✐ ❤ ❢❣ ❲ ❝ ❱ ❵ ❲ ❱ ❲ ❪ ❨ ❴ ❲ ❬ ❩ ✔ ✑ ❷ � û û ✂ þ þ ✆ û ☎ û ✄ û ✂ þ þ ý ù � ✝ ë ✗ ✖ ✚ ✍ ✘ ✔ ✙ ✘ ✒ ✞ ✑ ✏ ☞ ✠☛ ✡ ✠ ✟ ô ì ✛ Ü ã ãä ß Þ Ý â á à æ ß Þ Ý Ü Ù Ø × å ç ó î ò ñ ë ã ë ð Ýï í è Ý ì ë ê Ý ã é Þ ✚ ✑ ✪ ✍ ✖ ✚ ✒ ✰ ✚ ✪ ✗ ✦ ✒ ✗ ✑ ✛ ✤ ✚ ✏ ✧ ✰ ★ ✯ ✑ ✗ ✍ ✦ ✗ ✑ ✛ ✤ ✏ ✜ ✧ ✘ ✗ ✔✖ ✓ ✒ ✔ ✍ ✏ ✒✔✤ ✛ ✏ ✩ ✧ ✔ ✑ ✖ ✑ ★ ✪ ✒ ✔ ✧ ✑ ✍ ✦ ✥ ✔ ✖ ✒ ✚ ✔✖ ✗ ✍ ✩ ✏ ✮ ✭ ✏ ✒ ✔ ✥ ✑ ✬ ✥ ✫ ✒ ✫ ➆ t ➇ ➊ ➊ ❽ ❶ ➁ ➎ ➂ ➅ ➂ ➎ ➓ ➌ ❾ ➎ ➂ ➅ ➀ ➂ ➋ ➍ ➅ ➂ ➍ ❿ ➄ ➃ ❾ ➂ ❿ ➏ ❾ ➎ ➂ ❷ ➒ ➛ ⑨ ➂ ❿ ❿ ❿ ❿ ❸ ➎ ➂ ➓ ➒ ➄ ❾ ➍ ❼ ↕ ❸ ❻ ➄ ➄ ❿ ❹ ❾ ➄ ➇ ➃ ❾ ➂ ➅ ➂ ❽ ↔ ❷ ➂ ➉ ❷ ➃ ❾ ❿ ➁ ➙ ➀ ❷ ❼ ❿ ➅ ➀ ➂ → ➙ ➂ ➅ ❾ ➎ ➎ ➂ ❽ ❼ ❻ ➆ ➅ ➂ ➁ ❹ ➅ ❷ ↔ ➅ ❷❸ ➎ ➁ ➂ ➂ ➃ ❽ ↔ ➎ ➂ ❻ ➁ ❷ ❸ ➁ ↔ ❷ ❿ ❾ ➂ ❽ ❶ ❹ ❸ ❶ ➂ ❿ ❷❸ ➎ ❾ ↔ ➊ ➀ ➅ ❾ ❸ ➓ ❷ ❸ ❹ ➜ ➏ ➊ ➂ ➌ ➃ ❾ ➀ ↔ ❷ ❿ ➀ ❿ ➄ ➆ ❻ ❸ ❷ ❷❸ ❾ ❼ ➃ ❷ ➉ ➀ ❿ ❹ ❿ ❽ ➒ ➔ ❸ ➄ ❻ ❽ ❿➎ ➊ ➊ ❹ ❽ ❶ ➁ ❿ ➁ ❽ ➀ ❽ ❹ ➆ ❿ ❹ ❿ ➍ ➒ ➓ ❼ ➄ ❻ ❸ ⑩ ❹ ❹ ❹➊ ❿ ➅ ➈ ❿ ❸ ❿ ➑ ➁ ➄ ➆ ➁ ❷ ❸ ❶ ❶ ❻ ➎ ➎ ❽ ➂ ➊ ➂ ➅ ❼ ❻ ❸ ❻ ❿ ❼ ❸ ❹ ➁ ➎ ❷ ❸ ➂ ❹➊ ❿ ❸ ➅ ❼ ➆ ❸ ❷ ➒ ➄ ❷ ➉ ❽ ❹ ➅ ➋ ❸ ↔ ➅ ➅ ❸ ❷ ➒ ➄ ➓ ➓ ❿ ➌ b P. P, 146 Create a new root, Q, whose only child is P P 3 P ø❙û P ù✣ú P P - a,b)- Let Q be the parent of P.  (b+1)/2  while P has b+1 children do ✰✲✱ ✸✺✹ if P is the root then ✡✆☛ P ✡✆☛ ✶✖✷ :  (b+1)/2  ï❙ðñ P = Q; P else ✄✆☎ ÔÖÕ - ✔✖✕✗ ýÿþ n 145 ❶✎❷ b). 3 ✓✕✔ ➈✣➉ ✙✷✶ ❇✻❉ ❶✎❷ ❲❡❞ ➀➐➏ a,b)- ❛✻❜ ➳✻➵ ➔✕➂ ②✕③ ÷❀ø ✜✣✢ ✇✻① ✹✕❅ õ✍ö ÿ ✁� ➨➫➩ (2,3)- ➂❡➣ ➁✣➂ ❬✻❫ ✱✳✲ ❃✕❄ b ➁✣➂ ❭✕❪ a,b)- ➚✻➾ a ➁✣➂ ❼❙❽ ÿ ✁� ➍✣➂ ✓✕✔✖ a,b)- ➒✎→ ✾✕✿❀ ◗❙❘ Ú✉Û - ❦✻❧ a ❶✎❷❸ ❶✎❷❸ ✹✻✺ ✌✎✍ úüû ❯✎❱❲ ➝✣➞ ➧➫❒

  73. ❄ ❅ ▼ ◆ ▲ ❊ ❖ ❩ ❚ ■ ◆ ❬ ❃ ❈ ❊ ❃ ❖ ❭ P ❏ ❅ ❊ ❉ ▲ ❆ ❅ ■❈ ❇ ❑ ❈ ❯ ❊ ❳ ❅ ❑ ❅ ❊ ▲ ❆ ❈ ▼ ◆ ❏ ❅ ❊ ■❨ ❇ ❏ ❈ ● ❊ ▲ ❊ ❈ ❊ ❋ ● ❃ ❅ ❊ ❅ ❖ ❆ ■❈ ▼ ◆ ❅ ❊ ❖ ❆ ▲ ❑ ❅ ❊ ❉ ❆ ❏ ❆ ❅ ❃ ■ ❑ ❅ ❊ ❖ ❆ ▼ ❃ ❈ ❇ ❑ ❈ ▲ ❭ ❩ ❇ ❑ ● ❪ ❵ ❈ ❆ ❉ ❆ ▼ ❈ ❆ ❚ ❑ ■ ◆ ❏ ❊ ❖ ● ❃ ❅ ❊ ❉ ❆ ❖ ❅ ❖ ❑ ❑ ✱ ✜ ✤ ✥ ✦ ✧★ ✩ ✪ ✩ ✦ ✭ ✬ ✮ ✲ ✢ ✳ ✴✵ ✸✹ ✳ ✺ ✻ ✻ ✵ ✼ ✽✾ ✿ ❀ ❁ ✣ ✗ ❆❇❈ õ × Õ ã Ý Ö å Ò ×Ø â ò ó ô ö ✓✔ ÷ ø ùú û ü ÿ � ✁ ✆✝ ❅ ✍ ✑ ✒ ❅ ❉❊ ▲ P ▲ ❆ ❈ ▼ ◆ ❅ ❊❖ ❘ ❙ ❏ ❅ ❚ ❅ ■ ❊ ❇ ❯ ❃ ❈ ▼ ◆ ▼ ◆ ▲ ❯ ❆ ❆ ❊ ❃ ❃ ▲ ❅ ❊ ❆■❈ ❏ ❅ ❃ ■ ❑ ❅ ❃ ■ ❊ ❆ ❅ ❈ ▼ ◆ ❅ ❊❖ P ❏ ❃ ▼ ❃ ◗ ❈ ❑ ❇ ● Þ ❧ s ♠ r ♣ ⑥ q ❡ ✇ r ♣ ⑦ ⑧ ♣ t ④ s ♠ ♦ ⑩ ⑦ ♣ ⑥ ⑦ ❤ ❹ ① ❧ ✉ ❧ ✇ s ❤ ♠ ♦ s r ❤ ④ ♠ ⑦ ♣ ❸ ♦ r ① ♣ ④ ✐ s ♠ ② t r ♣ ⑥ ⑧ ♣⑥ ❹ t s ♣ ❤ ♣ ⑦ ❤ ❧ ♠ ♥ ♦ ❦ r ① ⑦ ⑧ ♠ ♠ ② ♠ ❷ ⑦ r ♣ ⑥ ④ ❤ r ⑨ ❼ ❢ r s ♠ s r ❤ ❺ ❤ ⑥ r ⑨ ✇ s r ④ ✐ ♠ ⑦ ② ♠ ❷ ⑦ ❢ ❻ ⑥ r ❦ ♦ s ♠ ❷ ④ ⑤ ❪ ♣q ❇ ❑ ● ❪ ❊ ❅ ❚ ❉ ❝ ❞ ❣ ❤ r ❇ s ❧ ♦ ❤ r t ✉ s ♠ ❢ ✈ ✇ r ❑ ■❈ ❡ ❆ ❊ ❱ ❆ ❉ ❅ ❈ ❇ ❆ ❜ ◆ ❏ ❅ ❅ ❅ ❆ ❈ ❆ ❑ ❅ ❊ P ❏ ❅ ❊ ❉ ▲ ❆ ① ❤ ♦ ⑥ ❧ ❤ q r ① ⑦ ⑦ s ⑦ ❷ ⑦ ♣ ♦ r s r ♣ ⑥ q ❡ ✇ s ⑩ ⑦ ❤ ④ ❦ ⑩ ❤ ② ♣⑥ s ♠ ④ ❦ q ④ ❦ ♦ ⑤ ♣ q r ♦ ⑧ s ♦ ♣ ⑧ ❤ ♠ ② ♠ ⑨ ④ ❤ ♠ ♣ à ▲ å → t s ➅ ➋ ➏➐ ➑ ➍ ➒ ➓ ➣↔ ➔ ↕ ❸ ➎ ➑ ➙ ↔➛ ➓ ➛ ➜ ➝ ➞ ➔ ➓ ➟ ② ➄ ↕ ⑧❹ ❷ ② ⑧❶ ✇ s ➀ t ⑨ ➁ ❿ ❻ ✇ t❶ t s ➂ ④ ③ ⑦ t s ⑨ s ④ ❺ ✉ ➛ ➛ s s ➓ ➟ ➛ ➜ ➯ ✇ s t ❺ ✈ ❹ t ⑩ ➞ ⑦ ✇ ➀ ⑧ ⑦ ➀ ✇ ② ③ ✉ ⑩ ✉ ➔ ➩ ➜ ➤ ↕ ➑ ➛ ➞ ➢ ↕➤ ➣ ➍ ➞ ➎ ➤ ➙ ➥ ➛ ↔ ➣➍ ➒ ➎ ➦ ➐ ➦ ➧ ➒ ➣ ➨ Û ❿ t ✉ ❞ ❍❩ ❬ ❱ ◆ ❱ ❭ ❪ ❫ ❲ ❱ ❛ ❜❝ ❝ ❳ ❡ ❛ ❢ ❣ ❤ ♠♥ ♦ ❜ ❧ ♣ ❤ ❢ ❨ ❍ ❝ ■ ❆ ❇ ❈ ● ❍ ■ ❏ ❑ ● ▲ ▼ ❍ ❏ ❲ ◆❖ ◆ P ◗ ❘ ❙ ■ ❚ ◆ ❍ ❯ ❱ ❵ q ❹ ❹ t ⑦ ③ ✉ ⑩ ❹ ⑦ t ❶ t ⑦ ❾ ❼ ④ ⑨ s t ⑨ s ② t ❽ ✈ t ❺ ✈ ③ ❽ ❜ ✇ t ✉ s ② ⑥ ✇ s ⑦ ✇ ⑧ ④ s ⑧❶❷ s ✇ ✉ t s ④ ❺ ⑨ s t ❻ ❼ ❶ t ➢ ⑩➲ ❰ ➾ ➱ ➪ ❐ ➚ ➹ ❒❮ ➹ ➘ Ï ➘ Ó Õ è Ô Õ Ò Ö×Ø ç Ø å ✃ ➱ ä ➷ Ù Ñ Ñ Ö× ➹ å è ➘➴ î × ➽ ➘ ➬ å ➬ ➘ ➴ ➪ ➹ ➘ ➘ ➮ ì à ➶ æ Ýè ×Ø é Ù Ø é Ý Ø å × ç × Õ Õ Ô Ù Ò Ýè ë Þ å ê Ò å × Ò Õá Ö Ý Ýä Û å Õ ä Ò Ù Ý á Ô Õ × æ ç Ø Õ × Ò Ý Ò Ô ➪ ◆ ◆❖ ◆ å P ◗ ❘ ❙ ■ ❚ ❍ Û ❯ ❱ ❲ ❍ ❳ ❨ å ❬ ❱ ◆ Ö Õ ❭ ❏ ➳ ❄ ❅ ❆ ❇ ❈ ● ❍ ■ ❑ Ò ● ▲ ▼ ❍ ■ ❏ ä Ý ñ ç ❱ ❍❩ Ý ➵ ➺ ❤ ✐ Õ ❥ ❜ ❡ ❜➸ ❧ ❜❝ Ô ❤ ❣ ❧ ❵ ➻ × ➼ ❱ Ò ❲ ♠ Ö ❪ ❫ Õ then P a delete Q and make the child the new 148 P’. if P has a sibling P’ with more than a children then ✛✠✜ ❦✚⑦ ♥✠♦ P ✘✚✙ ❯❍❛ Ð➊í P, : a Ð➈ÑÒ ❦✚❧♠ S ❙✚❊ child, 3 ✕☎✖ ✐❍❥ P ✎✠✏ P, P one S ❡❲❢ Terminate the algorithm. Q ☞☎✌ ✶☎✷ a,b)- ✡✠☛ P, P P❲❱ P P ✞✠✟ P but P if Q is the root, then ✂☎✄ ✯✚✰ while P has a-1 children do has â➊ã P P Q root. ✐❍③ Q To S ý➊þ ✫✠✬ âðï P P Q else P = Q; ❋❍● æ➈× ❊❴❫ S P ÚßÞ if Ü➈Ý P P, P Ù➈Û ❶❄⑥ ❂❄❃ else Ù➊Ú ❉❋❊ - ➽➊➾➚ 147 a,b)-Insert()? →➭➫ ⑨⑤⑩ ➜➡➠ 3 b? - ❸①❹ a ❤✆❧ ➅➊➉ ✐❦❥ ➆➈➇ a,b)- (3,5)- ⑨⑤⑩ ➃①❹ ➔✖→ ③⑤④ ❉❋❊ ✈①✇ - ❴✆❵ ➌✖➍➎ r✖s

  74. ❽ ô Ñ ➱ ❮ ø Ô Ï Ñ ❐ ø ➴ ó ➴ ò ø ÔÕ ✃ ✓ Ô ➬ ➬ ➴ ü ÔÕ ➮ Ò ➱ ➮ Ô ✓ ò ✓ Õ ÷ ❮ ➴ ò Ï ✓ ñ Ï ô ✫ ✛ ✧ ✯ ✰ ✯ ★ ✚ ✚ ✮ ✥ ✦ ✱ ✯ ✚ ✖ ✲ ✘ ✘ ✤ þ ✤ ✕ ✖ ✚✛ ✜ ✢ ✜✣ ✗ ✚ ✩✪ ✚ ✥ ✦ ✢ ✘✧ ★ ✜ ✣ ✃ ➷ ✜ õ Ô ò ü ➴ ñ ➬ Ö ➴ ❮ ➮ ➴ ñ ➮ ➬ ò ➴ ✓ ø ò Õ ø ✑ ➘ Ñ ➬ ò ✒ ➮ Ô ➴ ó ➴ ý ❾ Ñ ➱ × ü ✓ ❰ ✔ Õ × ü ÔÕ ➮ ➷ Ô ó ➷ Ô ó Ô Ñ ➱ ø ❰ ø ➬ ñ ø Ò ò × ➮ Ô ➱ ø ò Ô ➬ Õ ó ✓ ò ❐ Ò Ô ò ➬ ✧ ✲ ❰ ✇ ♥ r s s t r✉ ✈ ① ❥ ✉ ② ③ r ⑦ ② ✇ ⑤⑧ ♠ ❧ ⑤ ❨ ❜ ❵ ❯ ❬ ❜ ❲ ❳ ❩ ❪ ❴ ❙ ❯ ❫ ❣ ❬ ❤✐ ❫ r r✉ ❵ ⑧ ⑤ ② ⑥ ❺ ✇ ① ✈ ⑩ ⑦ ❶ ⑧ s ❻ ② ⑦ ❺ ❸ ❷ ❹ ✈ s ⑧ ⑤ ✈ r ⑦ ❶ ⑧ s ❸ ❷ ⑤ s ⑥ ⑦ ✈ ⑧ ✉ ❯ ❯❝ ✱ ✯ ✯ ✜ ✪ ✧ ✚ ★ ✸ ✘ ✩ ✛ ★ ✱ ✢ ✘ ✹ ✺ ★ ✦ ✳ ✼ ✘ ✯ ✲ ✩ ✜ ✬ ✳ ✩ ✧ ✷ ✖ ✴ ✘ ★ ✚ ✧ ✵ ✶ ✸ ✽✾✿ ❵ ❵ ❱ ❪ ❯ ❬ ❨ ❫ ❲❴ ❯ ❙ ❬ ❱ ❛ ❳ ❲❜ ❙ ❯ ❬ ❭ ❩❬ ❀ ❍ ❄❅ ❀ ❆ ❉ ❈❊ ❃ ❄❋● ❅ ❲ ❈ ❏ ❊ ▼ ◆ ❖P◗ ❯ ❱ Ï ò ❮ Û Ø Ñ ➱ Ò Ó Ô Ü ➮ ß àá â á ã ß ➬ × ä ➮ × ➮ ❒ ❮ ❰ ➴ ➱ ➴ Ï Ð Ñ ➱ Ò Ò Þ ç Õ ➈ ❿ ➀ ➁ ➅ ➆ ➇ ➉ ❽ ➅ ➊ ➋ ➆ ➇ ➈ ❾ ï ë ß ãá æ ä ❰ ìí Ý í æ â ì ç àî æ ç ë ➮ ❮ ➌ ➔ ➑ ➇ ➒ ➌ ➆ ➓ → ➏ ➆ ➣ ↔ ➆↕ ➙ ➔ ➐ ➎ ➔ ➈ ❿ ➀ ➁ ➅ ➆ ➇ ➉ ➌ ➅ ➊ ➋ ➆ ➇ ➈ ➌➍ ➌ ➛ ➘ ➬➮ ➢ ➽ ➡ ➾ ➹ ➷ ❮ ➺ ❰ ➴ ➮ Ð Ñ Ö ➟ ➤ ➜ ➦ ➝ → ➔ ➡➢ ➤ ➡➥ ➡ ➡ ➧ ➨ ➩ ➫ ➫ ➤➳➵ ➸ ➌➍ Þ ✝ ❰ Ï Ô Ó ☞ ✌ Ñ ✍ ✌✎ ➴ ❐ ô û ❰ ❮ ö ✡ ➬ ➮ ➬ ➬ ✏ ➮ × ✃ ô Ñ ò õ ➱ ➬ ☛ ➴ õ ➷ ✂ ✆ ✂ ☎ ➶ ➹ ✂✄ ➾ � ✠ ➬þ ñ õ ➴ Ï ò ➾ ✂ ➪ ✂ ø ✂ ➶ ➪ ☎ ➚ ➾ ➬ ➴ ý ÷ Õ õ ô ➔ ➆ ➤ Ô ➓ ➫ ➞ ➔ → ó ➆ ➣ ñ → ➝ ➜ ➛ ➔ ➌ ➔ ñ ➙ ➆↕ ↔ ð ➡ ó Õ ➒ Ô ➌ ➷ ➑ ➐ ➏ ➬ ➇ ➷ Ï ➬ò ➮ ➎ ➷ Ô 150 ■❑▲ Key, Info, LC, RC. ❞❢❡ 3 Insert amortized Ïúù ■❑❏ m logn ♦q♣ n ÷➼ø LookUp(), ⑨❢❶ ✇❽❼ ✘✫✭✬ ❇✙❈ ⑨✁⑩ ➭➲➯ õ➼✃ ❙✭❧ ❞❦❥ Splay m n ❁✙❂❃ ➾ ✟✞ To amortized ✫✭✻ ➱❍✃ O(log n). Delete(). ×✠ü O(logn). ❳✁❨ ④✁⑤⑥ ➂➄➃ - ✗✙✘ ➱➼Ò ❘❚❙ ÿ ✁� 149 3 ➫➼➻ - ➾✚➶ Ó➠ÔÕ ➭➲➯ ➚❍➪ èêé ➱➼Ò Ó➠ÔÕ a,b)- (3,5)- - - å➠æ ➱✠Ï ❐➲❒ ➱❍✃ èêé ➂➄➃ (a,b)- Ý➠Þ - ➞➠➟ ➘➠➴ (b (c Ù✠Ú

  75. ❾ ☎ ú ø ✆ ÷ ✓ ò ✂ î ú õ ✓ ø ö ú ✂ ð ✖ ✄☎ ïð ✢ ✄ ✂ ☛ ï ò ÷ ð ü ✓ ü ò ö ï ú ✂ ð ✕ ✌ ✜ ✣ ñ ✫✽ ✫ ✸ ✹ ✺ ✫ ✻ ✼ ✾ ✷ ✹ ✱ ✹ ✿ ❀ ❁ ✺ ✹ ✱ ✬ ✤ ✯ ✥ ✦ ✪ ✫ ✬ ✭ ✮ ✪ ✰ ✶ ✫ ✬ ✭ ✱✲ ✱ ✳ ✴ ✵ ð ✄ ❅ ø ✄✓ ø ï ö ❿ ☎ ú ✆ ÷ í î ï ð ñ ð ø þ ø û ú ✂ û ø ✆ ✏ ✂ ✠ ú ö ø ø ï û ï ö ø ☎ ✠ ü ✄ ö ö ü ✄ ✍ ô ✂ ñ õ ✂ ☛ ú ✔ ö ø û ø ò ✔ ð ü ö ï ï ö ñ ø ñ ð ✂ ú ú ✌ ✂ ✓ ✑ õ ö ✄ ñ ð ❂ ❄ ñ ✈ ⑤ ①⑥ ⑦ ⑧ ① ② ③ ❹ ④ ➂ ① t ➄ ① ➅ ① ⑥ ③ ➁ ⑩ ⑤ ① ⑥ ❻ ✈ ♥ ❶ ❾ ⑩ ♦ ❿ ① ⑦ ✈ ✉ ⑤ t ➀ ❹ ➅ ⑥ ✉ t ➃ ③ ➄ ① ⑩ ⑥ ➃ ① ❻ ⑩ ❹ ⑦ ⑨ ❼ ⑩ ❶ ❹ ✉ ✈ ⑩ ⑦ ⑤ ✈⑤ ✉ ⑩ ❺ ♦ t ⑤ ✈ ✉ ① ➅ ① ➇ ✈ ✉ ① ✉ ① ❈ ❳ ❙ ❲ ❞ ❴ ❨ ❢ ❨ ❣ ❛ ❤ ❬ ❵ ❪ ❨ ❳ ❬ ❲ ❜ ❬❵ ❨ ◗ ❇ ❋ ● ❍■ ❏❑ ▲ ❖ ❏❘ ❳ ◆ ❳ ❨ ❨ ❳ ❫ ❩ ❨❴ ❩ ❴ ✉ ④ ✈ ⑨ ❹ ⑩ ⑤ ❻ ✉ ⑦ ① ⑨ ❼ ⑩ ♦ ⑩ ✇ ⑧ ❽ ❺ ✉ ✐ ✉ ❝ ❳ ❫❥ ❛ ❝ ❤ ❞ ♠ ✈ ⑧ ✈ ✉ ④ ✇ ✈⑤ ✉ ①⑥ ⑦ ò ø ð Ð × Ó × Õ Ó Ò ➷ ✃ Ù ❐ Ð ➬ ➷ ❮ Û Ô Ò Ø ➷ ➱ Ï ❒ Õ ➘ ➮ Ï Ò Ö× ❒ Ô ✃ ❐ × Õ ❒ ❐ Ü Õ ❒ ä Þ ➾ ß ➻ ➸ à á å ï æçè é ì í ñ ô ì Ý Ø Ò ❐ ➷➱ Ô ➘ Ò ❒ ❮ Ï ➘ Ñ Ñ Ò ❒ Ï Ó Ò × ➬ Ô Ô ø ↔ ➓ ➍ ➇ ➔ → ➣ ➇ ↕ ➒ ➇➙ ➛ → ➍ → ➜ ➝ ➈ ➑ ➣ ➊ ➀ ➁ ➂ ➆ ➇ ➈ ➉ ➆ ➐ ➋ ➌ ➇ ➈ ➉ ➍➎ ➍ ➏ ➞ → Ó Ò Ð Ñ ➘ Ñ ❐ ❒ Ï ➷ ❰ ❒ ➮ ➷ Ò ✃ ❰ ➘ ➘Ï ❮ ➟ ➼➽➾ ➦ ➧ ➨ ➳ ➵➸ ➳ ➼ ➸ ➷➱ ➻ ➵➸ ➯ ➚ ➪ ➶ ➴➷ ➬ ñ ❒ ✂ ð ÷ ✂ ñ ✌ û ✆ ☞ ü ø ú ö í ✞ ✂ ✄☎ ò ñ � ü ð ï ☛ ø ÿ ✁ û ï ò ö ✂ ð ï ó ✂ õ ✁ ü ☞ ò ñ ÿ ✡ õ ✂ ð ☛ ïð ú ø í ð í ñ ò ÷ ø ✂ ö ✟ ✟ ÷ ó ✄ ✠ ú ✄ ü ú ✄☎ î ö ñ ø ð ï ò ú ï û ✂ ø ✟ ô ò ÷ ✄ ó ÷ û ú ú î í ø ú ù ð ï ø ÷ ø ☛ õ ð û ö õ ü û ñ ø ú ï ú ø ÷ ð ø í þ ú û ú ø õ ü ö ñ ð ☛ ✞ ö ý ✎ ✍ ÷ ø ï 152 ❝❡❞ ⑨❡⑩ ②✒③ ❥❧❦ 3 ✇❯⑩ P✝◆ Splay ♥♣♦ ②✒③ ▼✝◆ ✇❯① ❭✒❪ ❩❯❬ s✛t ➂❡➆ ❱✛❲ q❯r ❉✒❊ T ❸❡❹ ❙❯❚ LookUp(K,T ♥♣♦ q❷❶ ❆✝❇ Splay(K,T Splay(K,T Insert(K ❃✒❄ ✧✩★ - Info ➃✒③ 151 . Move-To-Front) ➷✭Ú 3 ✖✛✚ ✗✙✘ ñ❢÷ Splay ñ❢÷ õ✁ö ➤❢➥ - ➠➢➡ þ ✒✑ Splay ò❦ó î❑ïð ê❚ë ✃✁❐❒ õ✁ö Splay(K,T), K - ➺❢➻ ➶❢➮ â✟ã î❑ïð Splay ñ ✝✆ ñ❢÷ ➭✟➯➲ ➃➅➄ - ➹➢➘ ✞✒✑ ➩❚➫

  76. ➈ ➻ ➬ ➹ ð ❐ ➭ ➾ ➭ ➾ ➭ ➺ ➳ Ï ➫ ➽ ➘ ➽ ➺ ➳ ➯ ➱ ➽ Ú Ï ➺ ➻ ➼ ➭ ➽ ➽➹ Ø ➸ ❰ ➭ ➳ ❐ ➯ ➭ ➳ Ú ➳ ❰ ➷ ➯ ➱ ❮ ❐ ➽ ➸ ➯ ❐ ❐ ➼ ➭ ➳ ➶ ñ ➽ Õ ➭ î ➚ ➳ ➫ ➳ ➯ ➵ ➳➚ ➸ ➯ ➭ ➳ ➸ ➹ ➫ ➳ ➺ ➮ ➭ ➘ ➺ ➳➚ ➽ ➶ ➘ ❐ ➨ ➐➤ ➥ ➟ ➣ ➟ ➦ ➧ ➠ ➡ ➟ ü û Þ ➉ á à ÷ ➢ ➐ ã ↕ → ➐ ➑ ➒ ➣↔ ➣ ü ý ➙ ➠ ➛ ➜ ➑ ➝ ➣ ➐ ➞ ➟ ä å ➯ ➻ ❐ ➳➚ ➸ ➯ ➯ ➬ ➵ ð ➲ ➽ ➺ ➾ ➴ ➹ ➽ ➫ ➲ ➴ ➯ ã æ æ õ é å à êë ì ×ô ➘ ➭ î ï ➘ ➽ ➹ ➽ ➯ ➫ ò ➭ ➏ ❐ ➘ ❐ ➽ ✃ ➱ ➯ ➶ ➽ ➽ ➹ ➴ ➹ ➽ ➽ ➺ ➾ ➭ ➯ ➹ ➺ ➷ ➳ × ➸ ➯ ➫ ➲ ➯ ➘ ➶ ➘ Ú ➱ ➽ ➺ ➻ ➴ ➺ ➯ ➽ ➽ ➺ ➭ ➶ ❐ ➫ ➯ ➭ ➽➘ ❐ ❰ ñ ➻ ➶ Ø ➽ Õ Û ➘ ➽ ➻ ➽ ➘ ➯ ➷ ➘ ➮ ➭ ➳➚ ➯ î ó ➭ ➱ ➮ ➭ ❰ ➹ ➸ Ï ➳ ➳➚ ñ ➫ ð ➮ ➭ ➳ ➸ ➳ ➹ ➳ ➺ ➭ ➭ ➘ Õ Ù ➹ ➭ ➳ Ú ➚ ➾ ➚ ➮ ➽ ➺ ➳ ➭ ➯ ➲ ➯ ➚ ➭ ➘ ➽ × ➫ ➻ ñ ➳ × ❐ ➯ ➺ ➻ ➳ Û ➘ ❐ ➭ ➳ ➹ ➯ î ➘ ➭ ➽Ü ➳ ➻ ➯ ➭ ➘ ➫ ➮ ➽ ➯ ✃ ➷ ➯ ➘ ❐ ➭ ➳ ➷ ➴ ➹ ➻ ➫ ➳ Ï ➱ ➽ ñ ➻ ➶ ❐ ➻ ➔ ➓ ➘ ➯ ➶ ➹ ➽ ➺ ➯ ➻ ➯ ➳ ➭ ➽ ➘ ❐ ➭ ➳ ➷ ➴ ➻ ➺ ➷ ➱ ➯ ➻ ➳ ➽ ➺ ➾ ➫ ➮ ➯ ➸ ➫ ➲ ➯ ➘ ➷ ➶ ➺ ➳➚ ➹ ➻ ➵ ➯ ➶ ✎ ➭ ➳ Ñ Ò ➷ ➹ ➻ ☎ ➯ ➘ ➷ ➯ Ó ➘ ➾ ➬ ➚ ➳ ➽ ❒ ➸ ➯ ➭ ➮ ➽ ➺ ➾ ➚ ➳ ➭ ❮ ➭ ❰ ➲ ➯ ➘ ➭ ➭ ➲ ➺ ✛ ➙ ➛ ➜ ➑ ➝ ➣ ➐ ➞ ✥ ➟ ➠ ➐ ➡ ➢ ➐➤ ➥ ➟ ↕ ✓ ➟ ➒ ➊ ➋ ✦ ➌ ✖ ➏ ➐ ➑ ➓ ✤ ➏ ➔ → ➐ ➑ ➒ ➣↔ ➣ ➣ ➦ ➽ ➹ ➺ ➳➚ ➪ ✔ ➯ ✓ ➽➘ ➴ ➽ ✔ ➫ ➲ ➯ ➘ ➷ ➶ ➸ ➯➬ ➾ ➽ ➧ ✢ ➨ ➠ ➟ ✤ ✔ ✢ ✣ ✕ ✜ ➭ ➭ ➯ ✛✜ ✙ ➯ ➺ ➻ ➼ ➚ ➳ ➒ ➭ Õ Û ➘ ➽Ü ➳ ➻ ➯ ➘ ➯ ➫ ✞ ➯ ➫ ➭ ➯ ➲ ➳ ➘ Ø ➯ ➱ ➳ ➽ ➲ ➬ ➾ ➻ ➘ ➸ ➶ ➳ × ➭ ➯ ➻ ➸ ➽ ➭ ✝ ➭ ➭ ➈ ➽ × Õ ✂ ✁ � ÿ ➉ ➶ ➊ ➋ þ ➌ ü ➏ ➐ ➑ Ø ➻ ❰ ➚ ➹ ✁ ❐ ➭ ➽ ➷ ➵ ➳ ➺ ❰× ➳ ➷ ➴ ➹ ➷ ➻ ➽ ➭ ➷ ➯ ➾ ➺ ➹ ➷ ➯ Ó ➘ ➾ ➚ ➳ ❐ ➷ ➴ ➹ ➷ ➻ ➳ ➭ ❰× ➽ Ö ➭ ➺ ➴ ➹ ➷ ➻ ➳ ✍ Ñ ➳ ➳ ✌ ➱ ➯ ➭ ➳ ✡ ➯ ➭ ➳ ➻ ➶ Ø ➯ ➻ ➶ Ø ➽ ➷ ➯ ➹ ➺ ➻ ➽ ➘ ➴ ✃ ➯ ➘ ➭ ➳ Ú ❰ ➹ Ù ✟ × ➭ ➳ ➚ ✂ ☞ ☎ ✠ ➩ ➫ ➭ ➯ ➲ ➳ ➯ ➫ Parent(P) 154 Splay 3 P. ✣★✧ â✝í ✄✆☎ ➾✒➸ splay ç✒è P ➶❡➳ ➵✒➸ P. ➾✒➸ Splay ✘✚✙ ù❡ú ✕✗✖ â✝ã ö✛÷ø P ✏✒✑ P ß✒à ➶❡➹ ➲❯➳ ✡✆☛ ➍✩➎ - 1. Splay(+ ∞ , T1), 153 3 ➴❡✃ Splay(K,T). Concat T Delete(K,T) T1. ➴❡✃ ➽ÐÏ ➵✒➸ ➵✒➸ ➵✒➸ ➲❯➳ ➵✒➸ ➩✛➫ Concat(T1,T2): Delete(K,T): ➮ÝÏ ➶❡➹ Ô❷Õ ∞ ➍✩➎ -

  77. ✩ ➴ ❰ ➷ ❒ ✃❐ ➚ ➴ ➱ ➷ ➹ ➹ ➮ ➘ ➚ ➪ ➽ ➭ ➯ ➼ ➻ ➺ ➭ ✃Ï ➱ ➭ Ø Ù Þ Ô á Þ Ô Ö × ß Ü Þà ❮ ß Ô Þ ÓÔ Ý Õ ÙÚ Ö Ô Õ ➢ ➸ ä ➃ ➛ ↔ ✪ → ➔ ➓ ➉ ➈ ➅ ➐ ➝ ➈ ➅ ➆ ➉➊ ➈ ➇ ➆ ➅ ➄ ➃ ➜ ➞ ➜➵ ➨ ➳ ➲ ➜ ➯ ➭ ➫ ➜ ➢ ➩ ➝ ➧ ➟ ➦ ➥ ➢ ➢➤ ➞ ➝ ➜ ➡ ➠ ➛ ãä Ù ➁ ù ü ì é ÷ ñ õ é ÷ ð ì ÷ é ✄ ð ÷ ✂ ñ í ✁ � ÿ è ò þ é ì é í ì ð ð ÷ ð ù ì ✄ é ð ÷ ✂ ñ í ✁ � ☎ ù þ ù é Õ é ó ò ñ ê ð ì í ï ê ê è ñ ç á Þ Ô Ö × Þ å Ü á é ì ò é ê ó ù í û êü é ÷ è êö ê ô û ú é ÷ù ê ó ø ë ë ÷ ➂ ➣ ➀ ❫ ❜ ❛ ❵ ❴ ❛ ❵ ❜ ❛ ❴❵ ❪ ❣ ❭ ❬ ❩ ❲ ▼ ❙ ◗❘ ❋ ▲ ❛ ❤ ▼ ❣ ❛ ❵ ❛ ♣ ❞ ❣ ❞ ❞ ❛ ❛ ❜ ❵ ❥ ➏ ✐ ❦ ❛ ❥ ❵ ❞ ❞ P ■ ❛ ✲ ✽ ✼ ✻ ✺ ✸ ✹ ✸ ✴ ✳ ✷ ✾ ✶ ✱ ✵ ✴ ✳ ✲ ✱ ✭ ✬ ✫ ✳ ✸ ▲ ❆ ■ ❑ ❋ ● ❉ ❀ ❁ ❈ ❇ ❀ ✲ ✸ ❀ ❅ ✲❄ ❃ ❂ ✲ ❁ ❀ ✿ ❣ ❞ ⑥ ③ ④ ⑦ ⑤ t ❶ ❹ ① ✉ t ④ s ④ ✈ ⑩ ❺ ❿ ❸❹ t ⑦ ⑤ ✉ ⑨ ⑩ ❶ ✇ ⑨ t ⑨ ✇ ⑤ ✇ ❸❹ ➀ ➅ ➊ ➎ ➈ ➍ ➅ ➌ ➉➋ ➈ ➂ ➉ ① ➆ ➆ ➆ ✉ ③ ✉ ➁ ➄➅ ❷ t ➉➊ ➆ P 156 Splay: 3 ü ✝✆ ➶❏❮ Û✆Ü Ý♦æ P, Parent(P õ⑧ö Ð✗Ñ ➬✚➮ ×✚Ø ➶❏➹ ö❾ý Ý✆â ➾✚➚ í✚î Ò✆ÓÔ ë✗ì ↕✰➙ 3. - 155 ❡✚✐ Splay: ➋➒➑ 3 ❧♦q ➇✚➈ ❍❏❚ ❧✚♠ ➂✆➃ ❡✆❢ ❝✆❞ Parent(P ❳✗❨ ◆✚❖ ⑥⑧⑦ ❡✚✐ ❯✆❱ ❧♦♥ ❸❾❽ ❻❏❼ ❧✚♠ ❍❏■ ①✚② ✈✗✇ P ❊✚❋ r✒s ✮✰✯ 2. -

  78. ✞ ✯ ✵ ✴ ✳ ✲ ✟ ✰ ✣ ✶ ✤ ✫ ✪ ✩ ✣ ✖ ✣ ✭ ✹✺ ✏✧ ❉ ❍ ■ ❉ ❍ ● ❋ ❊ ❈ ✻ ❇ ❆ ❄❅ ❃ ❂ ❀❁ ✿ ★ ✱ ✦ ✚ ✒ ✏ ✖ ✗ ✕ ✖ ✔ ✎ ✓ ✒ ✑ ✘ ✙ ✛ ✥ ✖ ✏ ✤ ✣ ✢ ✏ ✜ ✑ ✠ ✡ ☛ ✎ ✏ ✑ KATAKE 157 3 D ✼✾✽ ✷✮✸ ✬✮✭ Splaying ☞✍✌ -

  79. ❏ ö ❰ ü ❮ Ï ç Ô ñ ❰ ❒ ❮ ❰ ü ❮ ❑ ô Ó Ô Õ Ò ❐ ý ô ❐ é ❒ ✃ ê Õ Ò ü û Ô ❖ Ô Ò ñ ❰ ÔÕ ø Õ Ô ▼ ❒ ô Ò Ó Õ ö Ð Ï Ó Õ ü Ó ï Õ Ò û ò Õ Ô ê ❮ ❒ Ó û ◆ ❰ ❑ ❮ ö Ò ñ æ ü ö ô ❒ ❮ ü æ ö ÔÕ Ó Ô ÷ ❮ ❐ ❰ ❰ û ❒ ô ❒ Ï Ò Ô ❰ Ð Õ ø Õ é ÷ ñ ö ❰ Ò Õ Ô ❰ Ô ö Ô Ó Ô ❰ ❒ Ï è ÷ Õ ü ❐ Ò Ô ❰ Ò Ó ó Ò ✃ ❰ ó Õ ø Õ Ò ❒ Ò ✡ ✁ ✂ ✄☎ ✆ ✝ ÿ ✞ ✗ � ✁ ☛ ✄ ✟ ☎ � ✁ ✥ ✩ ✭ ✬ ✫ ✪ ✩ ✤ ✛ ✣ ✢ ✜ ✘ ✝ ✆ ✞ � ✯ ✌ ✞ ✂ ☎ ✄ ☞ ✂ ✆ ✠ ✕ ☞ ✄ ✟ ✁ ✆ ✄ ✠ ✄ ✚ ✒ ☎ ✂ ✄ ☛ ✙ ✖ ❑ ✝ ✗ ☛ ✆ ✠ ✮ ✪ ❐ ï ô ❒ ❮ ü æ ü ❮ Ï ç Ô ñ ❰ Ò Õ Ó ❏ ö Ð Õ ø Õ Ò ÷ ❒ Ô û ø Ô Õ ❑ ❰ ö ■ ✫ ✪ ✬ ✰✱ ✰ ✲ ✳ ✴ ✵ ✫ ✶ ✰ ✪ ✷ ✸ ✹ ✺ ❍ ❀ ● ❋ ❆❇ ❄❅ ✸ ✹ ✿ ✻ ✾ ✸ ✰ ✸ ✽ ✪✼ Ô é ✆ ❜ ✞ ✆ ✝ ÿ ✞ ☛ ❬ ❭ ❪ ❫ ❝ ❵❜ ❞ ❡ ❢ ☛ ❜ ❥ ❧ ❦ ❵ ❢ ❞ ❵ ❜ ❝ ✐ ❴ ❜ ❡ ❣ ÿ ✆ ❦ � ☛ ✗ ☛ ❩ ÿ ✏ � ✂ ☛ ✛ ✞ ÿ ✒ ✌ ✌ ✂ ☛ ❳ ☎ ✂ ✞ ✛ ✏ ✆ ✞ ✔ ✓ ☞ ☛ ✞ ÿ ❵ ❢ ✠ ➀ ❿ ❾ ➊ ❿ ➋ ➌ ➂ ➍ ❺ ❶ ❿ ❺ ❶ ❸ ❺ ➀ ❼ ➇ ➏ ➍ ➂ ➐ ❾ ❿ ❿ ❶➏ ➄ ➎ ❼ ❻ ❾ ❺ ❸ ♦♣ ❻ q ❜ s ❜ ✇ ①② ③ ⑦ ⑧ ⑨ ① ❻ ❼ ❽ ❸❾ ➉ ➁ ❶➅ ➄ ➃ ➂ ❿ ❺ ❷ ❺ ❾ ❿ ❺ ➀ ❸ ❿ ✆ ✖ ❒ Ü ù ð Ú ❘❙ Ü ❚ ë ÙÚ ß Ý ❯ â Þ á ð Ï à ❒ ❰ Ð ❱ ä ã áâ Ú Ý Ú à ÛÞß Ú Ý è ❒ ❰ ò û ❰ P Ï Ò P◗ ◗ ü Ò ÷ ❒ ❑ Õ ö ï ô ❰ Ô Ô Ó Ô ö Ô ❰ ❰ ❒ æ Ó Ð Ó Ò Ï ❮ ö ✄ ✌ ✖ ✠ ✆ ☛ ✗ ✝ ❨ ✆ ✂ ☛ ✠ ❳ ✡ ✆ ✞ ÿ ✂ ☎ ❩ ✓ ✄ ✌ ✒ ✞ ☛ ✞ ✝ ✞ ✓ ✟ ✌ ✞ ❳ Ô ï Ó æ ü Õ ❐ ô ö ❰ ü Ô Õ ø è ÷ õ ö Ô ❲ Ó ö Ô ❰ ñ ❐ ❒❰ ü ÷ ❰ ó Õ ø Õ Ò ÿ ✘ ✁ ⑩ ❾ ➃ ➂ ④ ② ⑧ ❶ ❿ ⑥ ❿ ❸ ④ ❻ ❷ ⑥ ⑧ ❿ ⑥ ⑩ ② ④ ② ❹ ④ ② ➋ ② ➉ ⑩ ❿ ❹ ❸ ❿ ❸ ⑩ ❿ ❾ ❹ ➄ ❿ ❷ ④ ⑤ ➎ ② ⑨ ⑩ ❸ ➏ ➃ ❾ ➁ ❾ ❻ ⑧ ❸ ⑩ ❹ ❾ ❸ ❸ ➋ ❾ ⑩ ➄ ① ⑩ ⑧ ❹ ② ⑤ ➆ ✉➒➓ ➔ ✔ ➣ ② ⑨ ❹ ➄ ➈ ❸ ➊ ⑧ ② ❸ ④ ❶ ➃ ❾ ➁ ❾ ❻ ➙ ➛ ➜ ➝ ❾ ② ➃ ❿ ❿ ➄ ⑧ ❿ ② ➉ ⑩ ❸ ⑨ ➏ ❸ ⑤ ➋ ➄ ② ① ❾ ❸ ➋ ❾ ➏ ❹ ⑥ ❿ ❾ ➃② ⑤ ⑨ ④ ② ❻ ① ➞ ⑥ ❵ ❢ ❣ ❤ ❛ ❵ ❧♠♥ ♠ ♦ t ✇ ④ ② ⑧ ⑧ ❵ ⑨⑩ ❶ ⑩ ④ ② ⑨ ❹ ❺ ❻ ⑩ ② ⑨ ⑩ ⑨ ❳ ❡ ❼ ❳❨ ▲ ▼ ◆ ◗ ❘ ❙ ❚ ❯ ◗ ❱ ❲ ❘ ❙ ❚ ❳ ❘❞ ❩ ❬ ❭ ❪ ❙ ❫ ❳ ❘ ❴ ❵ ❛ ❘ ❜ ❝ ④ ⑩ ❸ ⑤ ➄ ⑩ ➄ ➃ ❿ ➄ ❹ ❶ ⑩ ② ⑧ ⑨➊ ➃ ➋ ⑧ ➀ ➄ ❾ ➀ ④ ❻ ⑩ ❷ ⑧ ❿ ❸ ➃ ❾ ➁ ❾ ➁ ➈ ❿ ❾ ② ⑨➀ ❿ ⑥ ➁ ② ❹ ➂ ❿ ❷ ⑩ ➃ ❾ ➁ ➇ ❿ ❾ ⑨ ① ② ④ ② ❾ ➂ ⑧ ➄ ❾ ➀ ❸ ⑨ ➜ → ➡➢ ø ñ ❰ Ô Õ ❒ò ❒ ê Ó õ ❰ö ÷ Ò Õ Õ ð ú Õ Ô ö Ð é Ò ✃ Ó ❒ ö ÷ Ò Ò ❮ ø Ú Ôê Ö â ë Ù ì ß Ý í â í Ü Û Ý ❒ Ú Û Þß à Ú Ý áâ à ã äå ➮ î ✃ ❐ Õ ñ Ï ✄☎ ✞ ✟ ✂ ✄ ✠ ✄✡ ☛ ☞ ✌ ✆ ✝ ✡ ☎ ☞ ✝ ✆ ✂ ✒ ☞ ✄ ✓ ☎ ✒ ÿ ✄ ✁✂ ✄☎ ✄ ÿ ✆ Ô æ ❐ ❒ é æ ö Ò û Ò Ó Ô Õ ü Ó é ✄☎ ý ü æ Ó ❒ ❮ ❐ ñ ❰ Ô ö Ô � ✁✂ Õ ï Ô ➴ ➫ ➴ Õ ➳ ➴ ➧ ➢➷ ➫ ➸ ➡ ➡ ➨ ➬ ➧➨ ➳ ➫ ➥ ➭ ➡ ➯ ➸ ➭ ➡ ➠ ➡ ❐ ➩ ➘ ❒❰ ➠ ➤ ➥ ➩ ➫➭ ➡ ➨ ➯ ➧ ➢ ➧➨ ➩ ➫ ➡ ➫ ➢ ➵ ➸ ➳ ➺➻ ➠ ➡➢ ➼ ➽ ➪ ➯ ➧ ➠ ❒❮ ➡ Ò ✃ ÛÞß Ýáâ Ó ÔÕ Ú Ý Ï äå æ Ú Ú Û ã Ú Ü à à Û ç é Ó ❒ Ò ❒ Ï è Ð K j , K i m 160 ç➹ô h(K) = j 3 ❴❛❤ collisions) ï⑦❰ j ➆➈➇ t✈✉ hash function modulo ❴❛❵❜ ❈❊❉ ❫✎r m h(k) = k mod m h(K j ) = h(K i ). LookUp ❁❃❂ ♠❛♥ ❷❹❸❺ ④⑥⑤ ✦★✧ m - þ➹ÿ û ▲❑ û ▲❑ ❡❛❥❦ ⑩⑥❶ LookUp() hash h: K → 159 ó⑦❰✝ù ➟③➠ Ø✮ÙÚ ❻➍➌ n-1 Ö⑦× ❶⑦❽❾ 3 ①③➄ ✞✑✏ hash table 0 Insert() ✍✎✡ ⑤⑦❸ ó✮ô ①③➄ ①③❸ Ñ➹Ò ①③➄ ➅⑦⑩✝➆ ✐✾❥❦ ➲✾➳ K j ①③② ①③❷ Delete()). ①③➄ ➦③➧ ➉⑦⑩ h(K j ). ➮➱➍✃ ➉⑦⑩ ⑤⑦⑥ {0, …, m- j ✉✮✈ function ❖✍P ①③② ①③➄ ➤➹➶ T[0..m- h ➦③➧➨ ➾✮➚ ①③② - ↔⑦↕ ♣rqs þ➹ÿ ➐r➑

  80. ➑ ë ñ á ï í á å à ì õ ê æ ê ëå ö ì é ê á ➑ ➒ ➓ ➔ → ↕ ➙ ➛ ➜ ➝ ë á ➞ ã å ï ó à ë ñ ➒ ð ë ã ë ì è á ç ä å æ ç á å ï î ô á é ê ëå ↕ ➟ è Ð ➶ ➹ ➶ ➘ ➴ ➷ ➬ ➮ ➶ ➱ ✃❐ Ï➮ ➾ ➱ ➪ ➷Ñ Ò Ó ÔÕ Ö × Ö ØÙ Ü Ý × Ù Ø ➾ ➼ ➙ ➩ ➛ ➜ ➠➡ ➠ ➢ ➤ ➥ ➦ ➛ ➧ ➠ ➙ ➨ ➫ ➻ ➙ ➭ ➯ ➙➲ ➳ ➩ ➠ ➩ ➵ ➸ ➺ ➫ ➩ ñ ó â ➪ ➙➲ ➳ ➩ ➠ ➩ ➵ ➸ ➺ ➫ ➩ ➻ ➼ ➾ ➭ ➶ ➹ ➶ ➘ ò ➷ ➶➱ ✃❐ Ï ➮ ➾ Ð ➯ ➙ ➷Ñ ➛ ➓ ➔ → ↕ ➙ ➛ ➜ ➝ ↕ ➞ ➟ ➙ ➜ ➫ ➠➡ ➠ ➢ ➤ ➥ ➦ ➛ ➧ ➠ ➙ ➨ ➩ ➱ ➴ è å æ î í í è ã á ä ë ë á å æ â ì á ç é ê â Ø Ù è Ò à è Ó å ñ á é ð ê × Ö ØÙ á à Ü Ý × å 162 3 ❒❃❮❰ Ú❃Û ➽➚➾ ➣★↔ - 161 ì▲ï 3 h(K) = j. ❒❃❮❰ ➬❹➮ æ▲ç Ú❃Û ➽➚➾ à❹áâ ➣★↔ - Þ▲ß ÔÕ❃Ö T[j

  81. ÷ ➃ ➓ ➂ ➆ ↕ ➑ ➉ ➄ ➙ ➆ ➂ ➁ ➂ ➝ ➜ ↔ ❿ ➇ ➈ ➎ ➃ ➌ ❿ ➐ ❿ ➐ ➙ ❿ ➂ ➐ ➓ ➂ ➆ ↕ ➑ ➉ ➄ ➆ ➌ ➂ ➃ ➁ ➛➜ ↔ ❿ ➇ ➈ ➎ ➃ ➂ ➓ ➃ ❯ ● ❉ ▲ ❭ ❑ ❞ ▼ ❍ ❋ ❍ ❳ ❯ ● ❘ ❉ ▲ ❉ ❭ ❍ ▲ ❖ ❭ ❣ ❍ ➂ ➞ ➆ ↕ ➑ ➉ ➄ ➙ ➆ ➂ ➃ ➁ ➜ ▼ ➟ ▲ ❖ ❱ ❍ ❯ ❘ ❋ ❉ P ■ ➌ ➎ ▲ ③ t ✉ q ✈ ✇ r s t ①② ① ④ r ⑤ ⑥ s ⑦ ① r ⑧ ⑨ ⑩ r s q ❷ ❩ ❚ ❍ ❋ P ● ❑▲ ▼ ❘ ❭ ❣ ❡ ♠ ❉ ❢ ❜ ❙ ❤ ❉ ❩ ✐ ❥ ❦ ø ❶ r❸ ➈ ➆ ➏ ➁ ➐ ➑ ➒ ➓ ➇ ➈ ➅ ➓ ➅➆ ➆ ➔ → ➐ ➓ ➉ ➆ ➅ ↔ ❿ ➇ ➃ ➁ ❹ ➃ ⑨ ① ⑨ ❺ ❻ ❼ ⑩ ⑨ ❽ ❿➂ ➁ ➎ ❿ ➇ ❿ ➂ ➆ ➋ ➁ ➌ ➇ ➍ ➂ ❍ ❘ ❞ í åæ ä æé ê ä æ ë ê è ì é ➧ î åñ ä ñ å é ó ñ è ô ➧ ➧â õ Ð Û Ð Ü Ý ➧ Ý Ù Þ ➧ß × Ò Ï Ý à ➧ × á ❰ Ý Ü Ð ➧ Û õ é Õ í ä ò ñ ú õ ð ê ð å ÷ ð ø ú ù ä û åñ ö æ ñü ÷ î ù ð õ è é ä ê å õ ð ö ÷ ö åñ ð æ ï ð ä ø ó ð õ ñ ù ê ð Ò ➧ ❭ ➠➡ ❡ ❉ ❢ ❜ ❙ ❤ ❡ ❙ ❤ ❉ ➢ ● ➤ ➥ ➡ ➢ ➦ ➧ ➡ ➯➲ ➳ ➭ ❍ ❖ ➸ ■ ❣ ❨ ❘ ❯ ❨ ❞ ❫ ❍ ❉ P ▼ ❊ ❉ ❯ ❫ ❉ ❣ ● ❉ ▲ ❉ ❳ ❘ ➵ ➺ Ú Ï ➪ ➲ ✃ ➭ ➯ ➵ ➮ ❐ ❒ ❮❰ ❮ ➬ ❰ Ð Ò ÐÕ ❰ Ö Õ ➧ ❮ Ù ➮ ➸ ➻ ➶ ➼ ➻ ➳ ➚ ➵ ➽ ➪ ➲ ➾ ➲ ➲ ➷ ➵ ➹ ➘ ➲➭ ➹ ➪ ➲ ➭ ➲ ➴ ➯ ◆ ❧ ❊ ✯ ✩ ❂ ✮ ✥ ✭ ✣ ❆ ❂ ✛ ✿ ✣ ✦ ✭ ❀ ❇ ❈ ❉ ❊ ❉❋ ● ❍ ■ ✿ ✩✪❅ ❉❋ ✛ ✣ ✢ ✙ ✦ ❃ ✦ ✿ ✥ ❃ ✬ ✥ ✮ ✮ ✩ ❂ ✘ ✙ ❄ ✮ ✩ ❂ ✛ ● ❏ ✩❂ ❭ ● ❑❯ ❲ P ■ ▼ ❉ ❨ ❱ ❬ ❪ ■ ❏ ❍ ❫ ❘ ❯ ❱ ❍ ❨ ❘ ❛ ❘ ❍ ❋ ◆ ❉ ● ❑▲ ■ ▼ ❋ ❉ ● ❑▲ ❊ ◗ ❋❚ ❉ P ❑ P ● ❘ ❙ ❈ P ● ❘ ✾ ✾ ❭ ✌ ✌ ✍ ÿ ✎ ✏ ÿ ✑ ✒ ✌ ✆ ✓ ÿ ✔ ✕ ✍ ✌ ✖ ✗ ✘ ✙ ✛ ✥ ☞ ✆ ✦ ☎ ù ú û þ ÿ � ✁ ✂ þ ✄ ÿ ☛ � ✁ ✆ ✝ ✆ ✞ ✟ ✠ ✡ � ✛ ✧ ✬ ✘ ✘ ✙ ✢ ✣ ✛ ✥ ✛ ✮ ✩ ✾ ✿ ✮ ✾ ✣ ✧ ✘ ✙ ✩ ✧ ✦ ✪ ✫ ✥✾ ✣ ★ ✯ ✘ ✩✪ ✩ ✫ ✬ ✙ ✩✭ ✮ ✥ ✭ ✩ ✪ ✫ ✰ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✴ ✽ ❬ ❉ ❪ ◆ ❋ ▼ ❏ ❊ ❘ ❲ ❲ ❊ ❞ ❚ ❱ ❍ ❋ P ● ❑ ▲ ▼ ❘ ❭ ❣ ❍ ❨ ❡ ● ❉ ◆ ❋ ❫ ❭ ❖ ❯ ❏❋ ❉ ❑ ❍ ▲ ❍ ❨ ● ❪ ❱ ❍ P ❑ P ❩ ❉ ❘ ❨ ❨ ● ❪ ❱ ❍ P ❑ P ❍ ❱ ❑▲ ❍ ❋ ▼ ■ ❊ ❘ ❲ ▼ ❍ ▲ ❍ ● ❢ ❍ ❬ ❉ ▲ ❉ ❭ ❍ ▲ ❖ ❭ ▲ ❉ ❘ ❯ ❉ ◆ ❋ ❫ ❭ ❖ ❯ ❏❋ ❯ ■ ▲ ❝ ❭ ❖ ❯ ❨ ❱ ❋ ▼ ❋ ▲ ❋ P ❍ ❨ ❞ ❫ ❍ ❉ ❱ ❫ ◆ P ❨ ❍ ❫ ❘ ❯ ❊ ■ ▲ ❉ ❉ ❉ ❉ ❜ ❬ ❘ ❭ ❪ P ❘ ❯ ❲ ❍ ❩ ❉❋ ■ ❉ ❭ ▲ ❉ ❬ ❢ ❉ ❍ ❍ ▲ ❖ ❭ ❉ ❍ ❉ ❡ ▼ ■ ▲ 164 3 Ù➀Þ m? - probes ➢➩➨ /2 ❱✤❲ Probes ) ≈ 2 + ×✤Ø ➄✤➅➆ ➄✤➅➆ ➄✤➅➆ LookUp S( ❱✤❲ ➈➊➉ Ó➀Ô ➽✤➾ ➓❁➣ ➴❁➱ ➄✤➅➆ Ð❁Ñ S ë✤ò ♥♣♦ ç➊è ) - ç➊í S( ❾➀❿➁ Ù➀à ➫➀➭ ã➀ä ï➀ð 163 ✣❁❀ ✭✲✱ 3 probes probes probe Probes ✚✜✛ n/m, load factor ❖✜P LookUp LookUp ✢✤✣ ✚✜✛ ❴❵❋ ❊❁❩ probe ❯❁❳ 1 Probe ü★ý - ❱✤❲ m ✳✵✴ 1 U U S n

  82. ý ②③ ⑧ ❶ ❹ ➅ ① ❽ ➄ ❺ ② ① ➆ ⑨ ❹ ➁ ❺ ⑨ ② ➀ ❷ ⑦ ⑨ ② ② ③ ❷ ❿ ⑨ ③ ❹ ⑤ ❹ ❽ ③ ⑩ ⑧ ❼➄ ➁ ❶ ⑨ ⑩ ⑧ ⑩ ➀ ③ ⑨ ③ ➀ ❺ ➁ ❶ ➈ ➃ ❹ ⑨ ❹ ② ➂ ❺ ❽ ③ ⑨ ❹ ⑤ ❸ ③ ❺ ⑨ ❹ ➅ ➄ ⑨ ③ ② ⑧ ⑨ ⑧ ❷ ⑤ ③ ⑥⑦ ⑧ ⑨ ⑩ ❶ ❷ ① ❹ ❶ ❺ ❻ ➄ ❼ ⑧ ❽ ① ❺ ③ ❻ ❻ ❿ ⑨ ③ ②③ ❿ ➄ ❚ P ◆ ❩ ❪ ❍ ❪ ❳ ❬ ❲ ❳ P ❳ ❏ ❭ ■ ❥ ❩ ❪ ❭ ◗ ❴ ❏ ❚ ❭ ❥ ■ ■ ❯ ◗ P ❏ ❪ ❯ ❚ ❴ ❍ ❩ ❏ þ ❏ ❪ ❯ ❭ ◗ ❴ ❏ ❯ ❱ ❍ ❙ ❪ ② ⑥ ❿ ❸ ❺ ③ ⑧ ❽ ① ③ ❽ ⑧ ⑦ ❿ ⑤ ❺ ➁ ② ❺ ② ➀ ❻ ❺ ② ① ⑩➀ ❻ P ⑩ ♦ ❨ ■ ❯ ◗ ❵ ✇ ⑥⑦ ⑧ ⑨ ❶ ❻ ❷ ① ❹ ❶ ❺ ③ ⑧ ❽ ① ❺ ③ ❹ ❽ ◗ ➄ ① ❺ ❽ ➔ ❺ ② ④ ⑨ ② ③ ⑨ ⑤ ❹ ➀ ❺ ❻ ❿ ➁ ② ❺ ❽ ➄ ❹ ③ ② ⑩ ② ➀ ➓ ➀ ③ ② ❸ ❺ ⑥ ➁ ⑨ ② ② ➀ ➓ ① ❹ ❼ ❻ ❺ ② ❿ ❺ ③ ⑨ ③ ➟ ➭ ➟ ➥ ➫ ➢ ➟ ↕ ➛➳ ➠ ➥ ➸ ➥ ➧ ➥ ➝ ➥ ➺ ➭ ↕ ➧ ➜ ➧ ➞ ➠ ❹ ➥ →➣ ➙➛ ➜ ➛➝ ➙ ➙ ➡ ➢ ➤ ➝ ➠ ➟ ➧ ➥ ➢ ➨ ➩ ➙ ➜ ➧➫ ➤ ➝ ➥ ⑨ ❶ ① ➇ ❽ ➄ ③ ➃ ❺ ❶ ❽ ⑧ ❺ ② ➏ ➁ ➐ ➁ ③ ⑨ ❿ ❶ ❺ ⑨ ② ⑦ ❺ ❹ ➁ ③ ② ❹ ⑨ ❹ ➄ ⑤ ❽ ➄ ③ ➀ ➀ ⑤ ③ ② ❹ ⑤ ❽ ➄ ③ ➀ ③ ⑦ ❶ ⑤ ➑ ③ ➀ ❺ ⑤ ❽ ❻ ❼ ⑧ ⑩ ⑧ ❼ ➃ ➁ ⑩ ❹ ➅ ⑧ ❺ ⑥ ➄ ➇ ➒ ❺ ② ➁ ➄ ⑨ ⑨ ❼ ❺ ⑦ ❸ ❷ ⑧ ❺ ② ⑦ ⑨ ❹ ⑤ ③ ❽ ➄ ③ ➀ ③ ❶ ④ ➄ ❹ ➃ ② ❨ ❭ ❫ ✏ ✆ ✝ ✞ ☛ ☞ ☛ ✌ ✍ ✎ ✝ ✠ ✑ ☛ ✆ ✒ ✓ ✔ ✆ ✕ ✖ ✆ ✡ ☎ ✘ P ❯ ❏ ❯ ❭ ❳❴ ❲ ❱ ❴ ❏ ý ✟ þ ÿ � ✁ ✂ ✄ ☎ ✆ ✝ ✞ ✗ ✓ ❫ ❏ ✸ ❪ ❜ ❝ ❞❡ ❣ ❤ ✐ ❪ ❨ ✯ ■ ❯ ◗ ❯ ❭ ◗ ❴ ❳ ❲ ❳ ✫ ✷ ☛ ✤ ✓ ✙ ✚ ✛ ✔ ✓ ✜ ✢ ✦ ✧ ✤ ★ ✧ ✩ ✪ ✫ ✬ ✭ ✧✯ ✰✱ ✶✭ ■ ❪ ❏ ✗ ☛ ✆ ✒ ✓ ✔ ✆ ✕ ✖ ✆ ✘ ✝ ✓ ☛ ✓ ✙ ✚ ✛ ✔ ✓ ✜ ✢ ✑ ✏ ✤ ✟ ÿ � ✁ ✂ ✄ ☎ ✆ ✝ ✞ ☎ ✎ ✠ ✡ ✆ ✝ ✞ ☛ ☞ ☛ ✌ ✍ ✦ ✧ ❏ ❍ ❍ ▼ ◆ ❙ ◗ ◗ ❏ ❯ ❱ ❏ ❄ ❏ ❲ ❳ ❨ ◗ P ❩ ❬ ❩ ❭ ● ❉ ★ ✯ ✧ ✩ ✪ ✫ ✧✯ ✰✱ ✶ ✭ ✤ ✷ ✫✸ ❈ ✹ ✺ ✾ ✽ ✿ ✺ ❀ ❁ ❂ ❅ ❱ ✹ ❩ ❲ ❙ ❦ ❖ ❭ ❩ ❙ ❏ ❭ ❳ ❙ ❪ s ❭ ❳ P ◆ ❩ ❪ ❍ ❱ ❫ ❏ ❩ ❏ ❲ ❲ ❳ ❙ ❩ ❱ ❴ ❏ ❲ ❙ ■ ❩ ❯ ❪ ❴ ❏ ❏ ❳ ❨ ◗ P ❩ ❬ ❪ ♣ ❦ ❴ ❭ ❩ ❱ ❭ ❏ ❪ ❯ ❭ ◗ ❲ ❨ ❙ s ❭ ◗ ❍ ❪ ❏ ❨ ■ ❯ ❩ ❳ ❴ ❈ ♣ ❏ ❯ ❱ ❍ ❏ ♣ ❵ t ✉ ❢ ❲ ✈ ❳ P ◆ ❩ ❪ ❍ ❱ ❭ ❳ ❫ ❲ ❦♥ ❙ ❳ ❲ ❳ ❱ ❳ ◗ ❯ ❚ ❴ ❧ ❭ ❭ ◗ ❏ ◗ P ❏ ❭ ❏ P ❩ ❙ ❳ ❏ ❏ ❩ ❪ ❭ ❳ ❲ ❳ ❨ ❩ ❭ ◗ ❨ ❱ ❪ ■ ❴ ❏ P ◆ ❩ ❪ ❍ ❱ ❦ ❫ ❯ ❨ ❨ ❳ ❭ ❏ P ❬ ❱ ❳ ♠ ❫ ❩ ❙ ❧ ◆ ❪ ■ ❯ ◗ ❯ ❩ ❪ ❯ ❪ 166 ➞❘➟➠ 3 ⑦r❾ ➭❑➵ ❻❘❼ ❷❑➂ ⑦r❾ ↔✥↕ ✲✴✳✵ ➯❘➲ ♦❑♣rq ❸✮❹ ❷❑➂ ◆❘❚ ■❑❥ ❸✮❹ ④✥⑤ ◆❘◗ ➌➍✴➎ ①✮②③ ➻➽➼ ➉➋➊ LookUp: ❻❘❼ ✣✥✤ Cellar – Insert: ◆❘❚ ❆✴❢ ➥r➦ - ⑦❛➇ ■❑❏ 165 3 ✲✴✳✵ ✾❋❊ ◆❘❚ ✬✮✭ ❆✴❇ ❃✴❄ ◆❘◗ ❏❛❵ ❖❑P ✣✥✤ - ✻✼✴✽ ■❑❏▲

  83. ➾ ✘ ✍ ✓ ✜ ✒ ✥ ✘ ☞ ✛ ✜ ✟ ✚ ✯ ✍ ✢ ✎ ✍ ✟ ✼ ✒ ✛ ☞ ✘ ✛ ✮ ✡ ☞ ☛ ☞ ✙ ☞ ✻ ✗ ✘ ✒ ✡ ☛ ✍ ☛ ✎ ✚ ✴ ✛ ☞ ✓ ☞ ✚ ✢ ✒ ✟ ✍ ✏ ✺ ✚ ✓ ✭ ✙ ✔ ✦ ✜ ✑ ✒ ✛ ☞ ✗ ✗ ✟ ✟ ✑ ✟ ✍ ✘ ☞ ✚ ✘ ✟ ✰ ☞ ✢ ✚ ✘ ✒ ✗ ✭ ✒ ✡ ☛ ☞ ✍ ✎ ☞✍ ✡ ✙ ✏ ✮ ☞ ✘ ✟ ✍ ✙ ✘ ✔ ✗ ✏ ✪ ✟ ☞ ☞ ✍ ✎ ✚ ✢ ✰ ✗ ✟ ✴ ✛ ✓ ✡ ☞ ✚ ✜ ✍ ✟ ✡☞ ✗ ✰ ✓ ✭ ☛ ✒ ✔ ✍ ✘ ✔ ✙ ✔ ✥ ✢ ✜ ✭ ✤ ✛ ☞ ✘ ✚ ✘ ☞ ✗ ➚ ✴ ✛ ☞ ✓ ☞ ✚ ✙ ✙ ✟ ✑ ✢ ✘ ✒ ✡ ✎ ✚ ✎ ☞ ✗ ★ ✏ ✙ ✛ ✤ ☞✍ ✙ ✘ ✒✗ ✑ ✚ ✗ ✟ ✡ ✔ ✹ ✘ ✔ ✒✗ ✑ ✚ ✗ ✟ ✡ ✟ ✪ ✸ ✘ ✎ ✘ ☞ ✖ ✘ ☞ ✛ ✔ ✑ ☞ ✛ ✚ ✑ ✍ ✗ ✧ P ❊❋ ● ❍ ■ ❏ ❉ ❑ ◆ ❖ ◆ ❩❬❭ ❉ ❪ ❫ ❜ ❡ ❱ ❘ ✐ ❥ ❱ ❥ ❊ ìí ❩ â ê ä ë ✿ â ò ❀ ñ â ❁ ï ë á ä ❂ ô ❀ ô æ ❃ ❄ ❅ ì ❣ ❦ á ❳ r s ❳ ❦ ❤ ❯ ♠ ❣ ❩ s ❤ ❣ ❧ ♠ ✐ ❱ ✐ t ✐ ❩ ❱ ❩ ♥ ❥ ❤ ❢ ♠ ❣ ❥❧ ❥ ❬ ❤ ❥ ❱ ❘ ❤ ❳ ✐ ❳ ❩ ❬ ♥ ♦ ♣ ❬ ❳ ❦ ❩❯ ❩ q â æ ✟ ✹ ☞✍ ✜ ✭ ✗ ✒ ★ ✗ ✟ ✜ ☞✍ ✓ ✒ ☞ ✚ ✒ ✤ ✛ ✒ ✥ ✟ ✗ ✟ ✡ ✛ ✰ ✡ ✡ ✟ ✘ ✟ ✗ ✒ ✜ ✰ ✘ ✹✗ ✎ ✑ ✍ ✙ ✘ ✒✗ ✑ ✚ ✗ ✟ ✡ ✟ ✢ ✧ ✘ ✾ Ö ❰ ➮ Õ Ö × ➮ Ø Ù ➮Ú Û ❰ ➱ Ö Ü Ý Þ × Ö ß à ã â ä Ô Ó ✔ ➱ ✙ ✔ ✥ ➾ ➚ ➪ ➶ ➹ ➬ ➮ ✃ Ò ❐ ➬ ❒ ❮ ➮ ➱ ✃ ❰Ï ❰ Ð Ñ ✰ ✟ ✟ ✧ ☞ ✗ ✙ ✤ ✔ ✜ ✟ ✘ ✚ ✒✗ Õ ✘ ✟ ✍ ✟ ➮ ✙ ✚ ✎ ☞ ✎ ✦ ❰ Ö Ù ✑ Ø ✒ ✍ ✒ ✖ ➮ × ✗ ✥ ✘ ✟ ✍ ✎ ☞ ✚ ✡ ✘ ☞ ✥ Ô ✜ ✑ ✡ ✒ ☛ ✒ ★ ✓ ✚ ✟ ✟ ✯ ✟ ✓ ✭ ✙ ☞ ✍ ✥ ✦ ✟ ✡ ✟ ✍ ✬ ✗ ✍ ✟ ✡ ✏ ✓ ☞ ➱ ✡ Ó ✎ ✜ ✚ ☞ ☛ ✭ ✮ ✤ ☞ ✘ ✟ ✍ ✒ ✍ ☛ ü ÿ � ú ✁ ý ✂ ✄ ÷ ☎ ✆ ý ✝ ú Ö × ✡ Þ ✎ ✍ ✏ þ ø ✑ ë è ñ ç æ ä ê å â ò ä ß èó ô â õ ã ù ø ú à ✟ ✒ ✙ Û ✛ ✜ ✟ ✘ ✍ ✙✜ ✒ Ö ✟ ➮Ú ✡ ☛ ☛ ✏ ✎ ☞ ✗ ✤ ✛ ✔ ☞ ✟ Ý ❰ ✡ ☞ ✖ ✒✗ ✘ Ü ✟ ✍ Ö ✙ ✘ ✘ ✒✗ ✑ ✚ ✗ ✟ ✡ ✟ ✡ ✟ ìí ✒ ✒ ✮ ✭ ✙ ✔ ✑ ✒ ★ ☞ ☛ ✭ ✤ ✰ ☞ ✘ ✟ ✍ ✮ ✍ ✟ ✡ ✏ ✓ ✘ ✒ ❐ ★ ✒ ✘ ✒ ✡ ➬ ✍ ✎ ✚ ✪ ✙ ✃ ➱ ✳ ✦ ➮ ➬ ✳ ➹ ✒ ✑ ✍ ✚ ✗ ✥ ✓ ✭ ✙ ☞✍ ✥ ✙ ✘ ✔ ✟ ✡ ✡ ✒ ☛ ✒ ★ ✓ ✚ ✟ ✟ ✭ ✍ ✒ ✷ ✡ ☛ ☞ ✍ ✎ ✚ ✢ ➶ ➪ ✏ ✤ ✲ ☞ ✘ ✍ ✥ ✎ ✍ ✟ ✎ ✒ ✎ ✍ ✒ ✚ ✪ ✳ ✟ ✡ ✒ ☛ ✒ ★ ✓ ✟ Ð ✡ ✟ ✓ ❰ ✒ ✛ ✚ ❰Ï ✘ ✟ ✩ Ñ ✴ ✔ ✓ ✚ ✟ ✟ ✗ ✟ ✧ ✰ ✘ ✙ ✢ ✔ ✥ Ò ✲ ✭ ✘ ✟ ✙ ✔ ✥ ✘ ✍ ✏ ✘ ✗ ✟ ☞ ➱ ✲ ✟ ✛ ✘ ✏ ➮ ☞ ❮ ❒ ✟ ✍ ✡ ✟ ✍ ✟ ✑ ✯ ✙ ✚ ✛ ✒ ☞✍ ✡ ✏ ✑ ✒ ✍ ✒ ★ ✡ äë ✟ ✗ ✃ ✟ ✡ ✟ ✍ ✜ ✑ 168 3 H(K,p+1) = (H(K,p)+1) mod m. ❢✌❣ ❍▼▲ á✥â h(K); ❲❨❳ ❝❨❞ ❢✌❣❤ ➘➷➴ - ❚✌❯❱ ◗❙❘ ❴❛❵ ❆❈❇ probe 167 ✩✫✪ 3 ✧✕☞ ✩✫✪ ✱✌☞ î✴ïð ✽✵✗ ☛✌★ ✯✵✗ ☛✌☞✍ LookUp. ☛✌☞✍ é✮ê ✓✕✔ ✧✕☞ p = 0, 1 … ✖✣✢ ☛✌☞✍ ✯✵✗ p- ☛✌☞ û❋ü LookUp: ☛✌☞✍ sequence á✥â Insert: ✩✌✶ K,p ➘➷➴ - ö÷✴ø ✞✠✟

  84. ✉ ❭ ❯ ❱ ❩ ❬ ❭ ❪ ❫ ❩ ❴ ❵ ❬ ❪ ❙ ❛❜ ❛ ❝ ❞ ❡ ❢ ❭ ❣ ❛ ❬ ❤ ✐ ❚ ❘ ❬ ✹ ì ✵ ✲ ✼ ✲ ✴ ✶ ✲ ä é ✻ í ê ◗ ✷ ì ✹ ✳ ✼ ✴ ✶ ✲ ✾ ▲ ▼ P ❥ ❦ ✿ ➆ ❹ ❼ ④ ❼❽ ➀ ➄➅ ➆ ➅ ➄➇➈ ➉ ➅ ❿ ❷ ➊ ➉ ➂ ➃ ➋ P ❏➓➔ → → ➣ ↔ ↕ ② ⑦ ❧ ✈ ❬♠ ♥ ✐ ❛ ✐ ♦ ♣ q ❥ ✐ r s ✉ ✇ ✇ t ✉ ③ ✇④⑤ ✉ ⑥ ✉ ⑨ ✉ ⑩ t ✴ ç ➛ ✘❅ ✤ ✓ ✚ ❂ ✤ ❃ ✜ ❄ ✦ ✮ ★ ❆ ✈ ✞ � ý ✆ ü ý ü õ ✂ ò ý ñ ✔ ✛ ✆ í ✴ ✶ ✿ í ✵ ✵ ç è í ✼ ç ì é ✮ ✷ ä ✹ ✽ è ✲ æ ✲ ✾ ❀ ✒ ✢ ò ü é ó ü ó ø ✝ ü ✠ ù ñ ✄ ý ñ ✄ ò � ù ø þ ✆ ✠ ò ✏ ❋ ✶ ✲ ✹ ✷ õ ö ✞ ñ ❇ ❈ ❉ ✡ ý ñ ø ó ✝ ü ✠ ù ✄ ü ✟ ✂ ò ó ò ❊ ✞ ü ✂ ✁ ñ ø ó ➙ ➛ ✲ ➥ ➜ ↕ ➨ ➩ ➡ ➥ ➼ ➝ ➙ ➜ ➡ ➻ ➜ ↕ ➙ ➢ ➺ ➥ ➤ ➡ ➞ ➳ ➸ ➤ ➢ ➙ ➦ ➡ ➜ ➥➧ ➟ ➝ ➥ ➜ ➟ ➥ ➪ ➵ ➦ ➜ ➙ ↕ ➬ ➸ ➡ ➨ ➵ ➶ P➹ ➘ ➴ ◗ ➷ ➤ ➡ ➤ ➢ ➸ ➟ ➙ ➭ ➸ ↔ ➥ ➧ ➛ ➨ ➡ ➥ ➟ ➙ ✃ ➨ ➩ ➟ ➙ ➜ ➫ ↕ ❐ ➡ ➥ ➵ ➾ ➜ ➱ ➟ ➩ ➙ ➝ ➥ ➭ ➥ ➧ ➳ ➩ ➙ ➳ ➙ ➨ ➟ ➮ ➙ ➜ ➢ ➩ ➦ ➜ ➙ ➯ ➫ ➬ ➵ ➢ ➽ ➙ ➜➝ ➦ ➙ ➡ ➜ ➤ ➛ ➙ ➟ ➝ ➭ ➨ ➜ ➩ ↕ ➣ ➙ ➝ ➥ ➭ ➥ ➧ ➳ ➩ ➙ ➙ ➟ ➛ ➨ ➠ ➛ ➞ ➙➟ ➙ ➠ ➢ ➤ ➢ ➜ ➤ ➥ ➟ ➙ ➨ ➝ ➩ ➛ ➨ ➦ ➜ ➫ ➭ ➯ ➝ ➙ ➡ ➙ ➙ ➢ ➟ ➝ ➳ ➸ ➤ ➨ ➜ ➵ ➡ ➥➧ ➫ ➩ ➟ ➙ ➼ ➜ ➝ ➙ ➜ ➦ ➨ ➟ ➫ ↕ ➸ ➫ ➨ ➜ ➵ ➡ ➤ ➝ ➢➵ ➫ ↕ ➸ ➫ ➨ ➜ ➺ ➡ ➨ ➭ ➜ ➻ ➵ ➼ ➟ ➙ ➫ ➨ ↕ ➜ ➨ ➜ ➯ ➭ ➨ é ❁ æ òó ➧ ➦ Þ á Þâ æ é êë ì íî ï ô ÕÙÚ ó ò õö ú û õ ü ý þ ó üÿ ➵ ➥ õ ➹ ➷ ➱ ✃ ➻ ❐ ➼ ➚ ❰ ➷ ➴ Ñ ➚ Ø Ô ➵ ➧ ➦ ➯ ➳ ➨ ➩ Õ ➳ Ö ➥× � ü ➷ ý ñ ✲ ÷ ü ✠ ö ✞ ø ø ✞ ✂ õ ö � ✟ ✂ ò ö ü ✞ ò ✂ ñ ô ✞ ø ò ó ô ü ✁ ù ò ñ ü ✂ õ õ ô ó ü ✁ ✆ ý ü ü ✂ ✁ ✝ ✄ õ ✂ ø þ ✞ ✟ ✄ ✆ ✃ ➾ ý ➉ ➃ ⑦➄ ➅ ➀ ❸ ➀ ➆ ➇ ➈ ➁ ➀ ➊ ⑦ ➍ ➌ ➎ ➋ ➌ ➒ ➎➓➔ ➌ → ➌ ↕ ➂ ➁ ➙ ⑦ ✇ ① ② ⑥ ⑦ ⑧ ⑨ ⑩ ⑥ ❶ ❷ ⑧ ➀ ⑨ ❸❹ ❸ ❺ ❻ ❼ ❽ ⑧ ❾ ❸ ⑦ ❿ ➌ ➋ ➘ ➶ ➚➮ ➷ ➻❐ ❐ ❒ ➻ ✃ ➚ ➼ ➚ ❐ ➹ ➽ ❐ ❒ ❮ ➽ ➹ ➻ ✃ ➚ ➼ ➻ ❒ ➻ ➾ ➬ ➎ ➭ ➣ ➜ ➐ ➝ ➡ ➓ ➡➢ ➧ ➧➨ ➩ ➫ ➯ ➷ ➦ ➲ ➵ ➸ ➵ ➼ ➽ ➪ ➶ ➹ ➶➘ ➴ ☞ ü þ ✟ é í ì ä è ✳ ê é í ì í æ ✲ ✲ ✲ æ ✴ ê ✵ ✲ é ✶ ì ✵ ✷ ✸ ✴ ✹ ✶ è ✱ é ✛ ✢ ✣ ✕ ✤✥ ✦ ✥ ✧ ✓ ✚ ✕ ★ ✤ ✓ ✮ ✜ ✩ ✪ ★ ✫ ✥ ★ ✚✥ ✚ ✬ ✭ ✜ ★ ì ê ✚ ✶ í ì í æ ✲ é ✲ æ ✴ ê ✵ ✲ é ì ê ✵ ✷ ✸ ✴ ✹ ✶ ì é ê ✴ ✶ ✵ ✶ é ✳ ✴ ✷ ✶ ✵ ✶ ✲ æ ✴ è ✻ ✼ ç ì í é ä è ✹ ✽ è ✲ æ ✲ ✾ ✱ ✲ è í ì ä ✓✜ ✕ ✛ ñ ü ✂ ✁ ø ✝ ✄ ✚ ø ✌ ó ý ✄ ✌ ó ò ✌ ✎ ✄ ý ✄ ñ ø þ õ ý ø ✟ ó ù ✞ ø ✌ ý ü ✂ ✁ ✡ ü ó ✍ ô ✞ ü ✁ ñ ô ñ ø ✂ ü ✁ ý þ ✟ ù ü ✟ ø ø ñ þ ✁ õ ö þ ✏ ✆ ✑ ô ✒ ✓ 170 [(m/d)h 2 (K)] mod m = [m(h 2 (K)/d)] mod m = 0, 3 K. , p+1) = (H(K, p) + h 2 (K)) mod m m: ➙➚➾ h 2 (K) = 1, ∀ ➦❸➧ ❺❍❻ ①✙② ❶❸❷ h 2 (K ➞✖➡ ➞✖➡ m t✖✉ ,0) = h(K); ➯☛➲ . m/d)- ⑦✖⑧ m h 2 (K) > 0 ▼➒➑ d ❲❨❳ ➸❑➽ - ➁➂✙➃ ➏☎➐ h 2 (K ➌➎➍ H( H( ❾❑❿ 169 primary clustering) 3 ç ☎✺ ç ☎✺ ✠☛✡ probes ÷✌ø ➘ÓÒ ➞➠➟ cluster ➱✵✃ ➏➑➐ c ★✰✯ ➛✕➜ ç ❍● Ï✵Ð ➫▼➳ ÷✌øù ÷✌øù ç✌è ➋▼➌ ã✵äå ➣▼↔ ✗✙✘ ➾✕➚ ÷ ☎✄ ß❙à ③⑤④ ✔✖✕ ◆☎❖ - ➱✵✃ ➺✠➻ Û✵ÜÝ ð▼ñ ➤❈➥➦ ■❑❏

  85. ❒ ➧ ➦ ➥ ➤ ❍ ➦ ➞ ➟ ❆ ➟ ➡ ❊ ➭ ➳ ➸ ➺ ➸ ❆ ➞ ➾ ❆ ➦ ➧ ❍ ➤ ▼ ➤ ➡ ❍ ➝ ◆ ➟ ➧ ➧ ➝ ➥ ➟ ➜ ➽ ➶ ❆➫ Ý Õ ➱ Õ Û ➱ Ø Ü Ú ❐ Ü ➱ ➬ Ö Õ Ð Þ × Ø Ö× ➼ ❰ ➶ ➹ ➶ ➺ ➻ ➘ ➱✃❐ ➬ Ò ➱ Ï ❐ ➷ ➬Ð Ó Õ ➬ ❆ ◆ ➬ ● ➥ ➤ ❍ ➤ ➥ ❆ ➠ ❉ ❆ ➡ ➟ ➡ ➟ ➤ ❖ ➩ ➡ ➦ ➨ ■ ➤ ➥ ➤ ▼ ➤ ▼ ➤ ➡ ▼ ❊ ❆ ❉ ➧ ➤ ➡ ❍ ❮ ➧ ❍ ❋ ❍ ➥ ➤ ❍ ➟ ❖ ❍ ➜ ➤ ➩ ▲ ➡ ➟ ➡ ➤ ➥ ➤ ❆ ❖ ▼ ➤ ❖ ➜ ➩ ◆ ❊ ❍ ❖ ❆ ■ ❆ ▼ ❍ ■ ➡ ❊❋ ❖ ➦ ➥ ➤ ❐ ❐ ❖ â ❮ â × ➬ Ó ➱ Ú Ü ê ➱ Ô Ð Ø ❐ � Ô ➱ ó Ò í ó å þ ú ñ ý ù æ æ þ í ò ð ç å æ é â Ô ú ✂ Õ ➬ Ô ➱ ÿ ✄ ÿ Ð â ☎ Õ ➬ Ï ➬ Ý Ú à Ú Ü Õ ➱ Ð Þ Ó Õ ✁ ✂ Ö Õ Þ Ù Ú ➱ ➬ ➱ Ö Õ Ü ý ö Ü Ï Õ Ü ➱ Ô Ü Õ Ù ➬ Õ ➱ Ó Ñ Ô â ã ✃❐ Ø Û ➱ æ Ú Ø ß × ❐ à á Ù ➱ Ú ➬ ➱ Ï ❒ Ú ➱ Ü ❐ à çèé ð è å æ õ ö ÷ õ í øù ç öúû ü î í î ö õ ø ó õ è ñ ê ë å î å æ í ò ëíô ð ç ë ç ë ç ó ç ➦ ➞ ❊➥ ✦ ✤ ✩ ✪ ✩ ✤✫ ✬ ✥ ✦ ✭ ✩ ✭ ✮ ✪ ✥ ✪ ✢ ✢ ✢ ✍ þ ✟ � ÿ ☛ ☞ ✌ ✎ ✥ ✎ ✏ ✔ ✕ ✔✖ ✚✛ ✤ ✮ ✦ ✝ ❍ ❀ ➡ ✸ ❁ ❊❋ ● ❉ ❊■ ✵ ❏ ▼ ❙ ❚ ❯ ❱ ❲ ✲✼ ✷ ✯ ✱ ✪ ✥ ✰ ✩ ✯ ★ ✧ ✲ ✾ ✳ ✷✸ ✹ ✷ ✵ ✲✼ ✽ ☛ ✁✡ ❫❨ ã Õ à Ú Ô á â Ô Þ ä å Ôæ ç â Ú â ß Ý é × ❰ Ï Ð Ó Ô Õ Ö Ó Ü Ø Ù Ô Õ Ö ÚÛ Ú è ê þ ÿ ó ÷ ø ÿ � ✁ ✂ ✄☎ ú ✟ ✠ ÿ ✠ ✁ þ ✟ ñü ì ã ñõ â î ñ ò ó ì ô ø ó ù ì ú ì û ð í ❚ ✲ s ✈ ⑥ ⑤ ④ ② ②③ ✉ ➍ t ➎ s ① ✇ r ➏ t ➎➐ ✉ t s r ➆➑ ➒ ➎ ➁ ➏ ♥ ➊ ♠ ⑦ ⑧ ❦ ❼ ➇ ➂➈ ➄ ➃ ➂ ➁ ➀ ❿ ➋ ➂ ⑩ ❶ ❽ ❻ ② ⑩ ➌ ② ⑩ ❺ s❹ ❸ ❷ s ❶ ⑩ ⑨ ❴ ❧ ➀ ❥ ❆ ➜ ❆ ➃ ❜ ❜❝ ➜ ➟ ❖ ❊ ❋ ❆ ➠ ❨ ➡ ❡ ❯ ❚ ● ❪ ➢ ❭ ❆ ➡ ➤ ❛ ❴ ❵ ❍ ➞ ❴ ❞ ❡ ❪ ❱ ➌ ❡ ↕ ❬ ❨ ❴ ➙ ❱ ❭ ➂ ➛ ❢ ➒ ❱ ● ❴ ❪ ❴ ❨ 172 Ñ✶Ò 3 i ❒✶ÿ m ï✓ð Ù✶Ú Ù✶â ➅✓➆ ß✙Ü ì✓í ➚❩➪ Ñ✙Ô n ❾✞❿ i = i/m, i ≤ n i ➉✞➊ Ñ✶Ò ➝✙➞ load factor. ➻✞➼ ❒✶❮ ➝✙➞ ♦q♣ - →➣✓↔ ➬❘➮ 1. 2. ➯✞➲➳➵ ➓❇➔ ➴✶➷ ä❇å 171 ❑❄▲ ✑✓✒ 3 ❈✣❉ ❅❇❆ ❂❄❃ ✺✓✿ ö✖÷ W. Hooper? ✆✞✝ ✺✙✻ ❬✞❭❪ ï✖ð ✧✞★ ❳❩❨ ❡❤❘✐ Ñ❨Ò ë✙ìí - ✗✙✘ ❣❩❲ ✴✶✵ ◆P❖❘◗ T ✜✣✢ ý❑þ

  86. ✆ ➸ ➸ ➯ ➭ ➚ ➱ ⑨ ➵ ➭ ➯ ➲ ➺ ➻ ë ➮ ➭ ➸ ❐ ➯ ➳ ⑩ ➮ ➴ ➺ ➵❐ ó í ➯ à ÓØ Ò Ñ × Ý à ÓÛ á Ñ Ï Ñ ä ê × á ã æ ê é ë ì íî ñ ò ➭ ➪ Ý ➻ è ð ÷ ñ þ ë ÿ ➯ ➲ ➺ ➸ ⑧ í ➱ ➹ ⑧ ➵➸ ➫ ➯ ➵➸ ➮ ➯ ➷ ➘ ý ø ➸ ➚ ➫ ⑨ ➶ ➚ ➶ ➹ ➫ ➯ ➯ ➯ ➾ ➭ ü ➸ ➲ æ ê é ë é ø ù ú û ë × Ó ô ➯ ➪ ➸ ➫ ⑨ ➶ ➚ ➶ ➯ ➘ ➯ ➵ ➶ ➭ ➵ ➶ ➵ ➮ ➚ ➭ ⑨⑩ ➷ ➱ ⑦ ➫ ➯ ➻ ⑩ ➨ ➛➜➠ ➡ ➣ ➢ ➣ ➤➥➦ ➝ ➣ ➢ ↕➧ ➝ ➠ ➲ ➭ ✝ ➺ ➸ ➻➼ ➭ ➯ ➫ ➯ ➯ ➭ ➸ ⑨ ✃ × Ò ÓÛ Ü Ý Ñ Þ ß × Ù × Ù Ö Ó Ú Ï Ù â Þ ØÙ ã Þ Ó Õ Ñ ã Õ ÓØÙ ➸ ➫ ➯ ➯ ⑧ ⑨ ➭ ⑩ ➷ ➚ ➶ ➵❐ ➯ ➯ Õ ❒ ➚ ➭ ➼ ➮ ➵➸ ❐❮ Ð ÏÑ Ò ÓÔ Ñ ⑨ ➫ ➝ ✠ ➯ ➘ ➸ ➮ ➭ ⑨ ô ➶ ➵ ➭ ⑨ ✡ ➸ ➶ ➵ ➭ ⑨ ✠ ➲ ➯ ➸ ➮ ⑩ ➫ ➲ ➭ ➷ ➺ ✃ ➸ ➯ ➭ ⑨ ➲ ➳ ➵➸ ➺ ➴ ✠ ⑨ ➫ ➵ ➲ ➺ ➴ ✠ ✡ ☛ ➭ ⑦ ➭ ➵ ➯ ➵ ➴ ➚ ❒ ➲ ⑨ ➳ ⑨⑩ ➘ ➴ ➯ ❐ ➯ ➫ ➯ ➼ ❐ ➭ ➚ ➮ ➚ ❐ ➭ ⑨ ⑩ ✠ ✡ õ ➯ ➚ ❒ ➯ ⑨ ⑩ ➶ ➵ ➶ ➵ ➭ ➚ ➫ ➵➸ ➮ ✃ ➭ ➬ ✃ ➼ ➭ ⑨⑩ ✠ ✡ ➾ ➻ ➮ ➵ ➸ ❐ ➮ ➹ ☎ Û ã × á Ï Ñ Ú Ù Õ Ù ✄ Ý × Ñ Ñ Ï Ñ à ä × á ã á ✆ Ñ Ï Þ Õ ➭ ➮ ➸ � ➮ ➭ ➵ ❮ ✁ Ñ Ú Ó Ö Û Ü Û ✂ Þ Ù ✄ Þ ØÙ á Ô Ù Ñ ß ➚ ➲ â å ✟ ê ç î ➮ ➭ ➚ ➫ ➯ ⑨ × ➳ ⑨⑩ ➘ ➴ ➯ ➯ ➫ ➯ ❒ ➼ ➭ Ù ß × Ó Ù ☎ Ñ Þ Ø Ð Þ Ó Û Ï Ñ Ý Þ Ï Ñ Û Ò Û Ú Ø ß × ÓØ Ñ Ñ ➟ ➯➲ ➞ ♠ ❈ ❏❪ ❛ ❣ ❤ ❤ ✐ ❛❥ ❦ ❧ ❝ ❭ ♠♣ ♦ ♣ q r ❥ ♠ ❣ ♠ s ❘ ❆ ❊ ● ● ❆ ❲ ❳ ❨ ❊ ❏ ❩ ❏ ❬ ❘ ◗ P ❭ ❆ ❚ ❩ ▼ ❊ ❏ ▲ ❈ ◗ ❑ ❊ ▲ ❍ ▲ ② ❋ ❍ ❈ ◆ ❋ ➝ ❈ ❳ ❨ ❊ ❏ ❈ ❊ ❆ ❈ ❉ ❊ ❆ ❋ ❉ ❊ ❆ ● ❍ ❆ ❊ ❈ ❈ ■ ❏ ❑ ❉ ❋ ❍ ❈ ◆ ❋ t ❏ ❖ ❑ ▼ ❆ ❊ ❏ ▲ ❈ t✉ t ❚ ❳ ❬ ❈ ▼ ■ ✩ ✍ ✣ ✤ ✍✥ ✦ ✜ ✔ ✜ ✧ ★ ✢ ✜ ✜ ✭ ✮ ✯ ✰ ✱ ✵ ✭ ✮ ✯✶ ✫ ✢ ✛ ✯ ✍ ✞ ✟ ✠ ✌ ✍ ✎ ✏ ✑ ✌ ✒ ✓ ✎ ✍ ✏ ✔ ✕ ✔ ✖ ✗ ✘ ✙ ✎ ✚ ✔ ✹ ✺ ❖ ❏ ❏ ❑ ❉ ❋ ❍ ▲ ❊ ❉ ▼ ❊ ◆ ❖ ❈ P◗ ▼ ❉ ❘ ❆ ▼ ❈ ❆ ❈ ❙ ❚ ■ ●❍ ✻ ✺ ✼ ✽ ✼✾ ✳✿ ❀ ✼ ✽ ✸ ✰ ❀ ✯ ❁ ❆ ✶❂ ✳ ✸ ✼ ❈ ❉ ❊ ❆ ❋ ❉ ❊ ❈ ❖ ❏ ❖ ▼ ❈ ❆ ❈ ❙ ❚ ▼ ❘ ▲ ▼ ✇ ⑥ ◆ ● ❆ ➎ ❘ ◗ ❊ ◆ ➓ ❏ ❖ P ▼ ➏ ➒ ➄ ❉ ➌ ➑ ❾➐ ❉ ❆ ❉ ❿ ❾ ➃ ➂ ❽ ➁ ➀ ❾ ➀ ❽ ❺ ❹ ❸ ❷ ❶ ❿ ➄➅ ✉ ❾ ❍ ❪ ❾ ➍ ➌ ➋ ➄ ➄ ❑ ➊ ❿ ➉ ➈ ➇ ➆ ▼ ➌ ⑨⑩ t ● ❏ ❊ ❨ ❳ ③ ● ❖ ▲ ● P ● ❊ ❚ t✉ ❆ ❈ ▲ ❏ ❊ ❆ ▼ ➛➜ ❖ ❈ ❏ t ❍ ❋ ❉ ❈ ❖ ❈ ➔ ❉ ❑ ❏ ■ ❈ ❉ ➍ ❆ ❊ ❋ ● ❊ ▲ ❋ ❆ ➌ ❉ 174 ➹❇➫ ➹❇➫ 3 à⑤á á❱å Ordered Hashing ➾✬➚ ➳✬➽ â ✞✝ ➳✬➵➸ ✠✌☞ ï✴ð Ö➙× ➻①õ ö❵÷ ➴❇➬ ↔➙↕ ➳✬➵➸ ô❞➶ ➳✬➵➸ →✴➣ ➳✬➵➸ ❻☞❼ ô❞➶ çè✴é çè✴é ➳✬➽ - ➩❇➫ ➴❇➷ ➩❇➫ ❰✈Ï probes 173 2 (1-a n-1 ) + … ❃✬❄ 3 ) 1 + = (m/n) (H m – H m-n ), ) 1 U n-1 = 1*(1 – a n-1 ) + 2*a n-1 *(1-a n-1 )+3a n-1 - i − − i a H i = 1 + ½ + … + 1/i ≈ ln i. ♥✴♦ m probes − − 1 /( ✲✴✳ 1 probes − /( 1 i U S n ≈ (m/n) (ln m – ln (m-n)) 1 1 = n probes i 1 1 ∑ S n = (1/n) ∑ = = = n i n i ∑ ∑ = (m/n) ❬①✇ = (1/n) 2 + … = (1/a n ) ln (1/(1-a n )). ③⑤④ ❩✈❏ = (m/n) ln(m/m-n) ❩✈❏ ✪✬✫ n- = 1 + a n-1 + a n-1 probes = 1/(1 - a n-1 ) ▼❱❯ n ❡✴❢ ✷✬✸ ✡☞☛ ❜❞❝ ❫❵❴ i- - ⑦❞⑧ ❅❇❆

  87. ✍ ❄ ▼ ✷ ✵ ✶ ✹ ✼ ✽ ❩ ❑ ✼ ✴ ● ❀ ❁ ❩ ▼ ✷ ✵ ✶ ✹ ✼ ❂ ❨ ✻ ✷ ✽ ✹ ✵ ✹ ✷ ❂ ✿ ✴ ✽ ❩ ❑ ✿ ✼ ✴ ❈ ✷ ▼ ✷ ✵ ✶ ✹ ❄ ❄ ✵ ✶ ● ❄ ✷ ❊ ❄ ✷ ▼ ❁ ❉ ✷ ❄ ❂ ❁ ❊ ❅ ✿ ✿ ❅ ✷ ❄ ✴ ❂ ✼ ❊ ❈ ❅ ✷ ✵ ❈ ✴ ❄ ✼ ❈ ✷ ▼ ✷ ✵ ✶ ✎ ❄ ✴ ❄ ❁ ✼ ❀ ✼ ✴ ❂ ❄ ✴ ❂ ❁❨ ✵ ❊ ✵ ✶ ✷ ❀ ✵ ❄ ❂ ❄ ✴ ❉ ● ❀ ❂ ❄ ◆ ✷ ❀ ✳ ❳ ❲ ❱ ❚❯ ❙ ❘ ✹ ❄ ✼ P ❉ ✴ ✴ ✺ ❛ ❄ ❂ ❄ ❉ ❂ ❨ ✻ ❉ ❑ ✿ ✷ ❅ ❀ ✵ ❉ ✻ ❄ ✷ ❊ ❏ ✼ ❂ ● ✽ ❑ ❏ ❉ ❉ ❂ ✵ ❞❡ ❝ ✵ ✻ ❏ ✼ ✷ ▼ ❁ ✿ ✼ ❀ ❁ ✽ ❀ ✻ ✷ ❄ ✴ ✵ ❜ P ✷ ❄ ✴ ✵ ❂ ❩ ✷ ✵ ✼ ✵ ❬ ❑ ❑ ✿ ✴ ✷ ❀ ✵ ❄ ✷ ❊ ❏ ✼ ❂ ● ✽ ❏ ✻ ✼ ❂ ✵ ✻ ● ❂ ✵ ✻ ❄ ❂ ❄ ❵ ❴ ❱❫ ❉ ❨ ✻ ✶ ● ❂ ✵ ✻ ❄ ❂ ❄ ❵ ❴ P ✷ ✵ ❛ ❈ ✶ ❂ ❄ ❂ ✵ ▼ ❉ ❅ ❉ ✴ ✺ ❛ ❄ ❂ ❄ ❉ ❂ ✹ ✴ ✪ ✪ ✢ ✪ ✯ ✵ ✗✮ ✭ ✬ ✗ ✫ ✩ ✱ ✗ ✢ ★ ✘ ✧ ✦ ✥ ✤ ✢ ✒ ✰ ✲ ✙ ✺ ❄ ✴ ❄ ✹ ❄ ✹ ❄ ❀ ✼✿ ✴ ✾ ✫ ✽ ✼ ✻ ✺ ✗ ✴ ✳ ✏ ✑ ✪ ✢✣ ✘ ✵ ✰ ✥ ✦ ✧ ✘ ★ ✢ ✪ ✫ ✲ ✱ ✪ ✢ ✢ ✪ ✯ ✗✮ ✭ ✬ ✗ ✫ ✪ ✩ ✤ ✢✣ ✗ ✏ ✜ ✛ ✖ ✚ ✙ ✘ ✗ ✖ ✒ ✑ ✎ ✙ ✍ ✖ ✗ ✘ ✙ ✚ ✖ ✛ ✜ ✗ ✘ ✹ ✵ ✽ ✶ ❄ ✵ ❄ ◆ ❇ ❈ ✷ ▼ ✷ ✻ ✹ ❉ ● ❀ ❄ ✿ ✼ ✶ ● ❂ ✿ ❅ ✷ ❁ ● ❄ ✻ ❈ ✽ ✷ ✵ ❅ ❉ ✴ ● ❀ ❄ ▼ ✴ ❉ ✴ ✽ ❈ ❖ ✵ ❂ ❄ ❄ ✷ ❏ ✵ ❂ ✵ ✴ ✿ ❄ ✵ ❅ ✺ ❊ ✵ ❄❊ ❅ ✷ ❂ ❂ ❉ ✴ ❈ ✴ ✵ ✻ ✼ ✻ ❅ ✷ ✴ ❄ ● ❋ ✷ ❈ ✵ ❅ ❉❏ ✴ ✵❑ ❍■ ❂ ▲ ✷✹ ✴ - 3 - 3 ✓✕✔ ✓✕✔ Ordered Hashing Ordered Hashing procedure OrderedHashInsert( key K, info I, pointer P): ✶✸✷✹ ❁❃❂ ❆✞❇ if (P->size == m-1) then error; ✺❃❏ T = P->Table; ordered hashing pos = h(K); ❁❃❂ while (T[pos] != NULL) do ❁❃❂ ❄◗P if (T[pos]->Key > K) then swap(K, T[pos]->Key); swap(I, T[pos]->Info); else if (K == T[pos]->Key) then T[pos]->Info = I; return; ❚❪❭ pos = (pos + h 2 (K)) mod m; S n T[pos]->Key = K; T[pos]->Info = I; P->Size++; Un function OrderedHashingLookUp( key K, pointer P): S n . ❁❃❂ info T = P->Table; : pos = h(K); while (T[pos] != NULL && T[pos]->Key < K) do pos = (pos + h 2 (K)) mod m; S n ≈ U n ≈ (1/a n ) ln(1/(1 – a n )). if (T[pos] != NULL && T[pos]->Key == K) return T[pos]->Info; else return NIL; 175 176

  88. ❢ ✄ ☛ ✠ ☛ ✠ ✟ ✠ ☛ ✝ ☞ ✄ ✠ ✟ ✡ ✢ � ✂ ❨ ✠ ÿ ✤ ✠ ✟ ✂ ☞ ❱ ✣ ✟ ✠ ✡ ✗ ý ✠ ✌ ☎ ✎ ÿ ✝ ✡ ✟ ✣ ✂ ✝ ÿ ✠ ❱ ❲ ✣ ✢ ✣ ✟ ÿ ❳ ✤ ✕ ✂ ✝ ✣ ✢ ✂ ✣ ☛ ✌ ☞ ☞ ✁ ✢ ☞ ✖ ☞ ✣ ✤ ✑ ❚ ❬ ☞ ✎ ✢ ✡ ✄ � ✘ ✡ ✠ ✟ ✂ � ✢ ✄ � ✗ ✝ ✗ ✣ ✁ ✠ ✝ ☛ ✎ ✛ � ÿ ❱ ✖ ✖ ✔ ÿ ✤ �✁ ✂ ✕ ☞ ✤ ✕ ✁ � ✟ ✠ ✖ ✣ ☞ ✢ ✣ ✗ ✄ ✠ ✔ ✎ ✄ � ✣ ❩ ✟ ✢ ✤ ✣ ✢ ✠ ✜ ✟ ✂ � ✝ ✤ ✛ ✜ ✢ ✝ ✡ ✤ ✘ ✡ ✠ ✟ ✡ ✤ ÿ ✝ ✂ ✩ ★ ✧ ✑ ✏ ✠✡ ✌ � ✝ � ✟ ✤ ✝ ✌ ✡ ✗ ✫ ✑ ✟ ✣✤ ✛ ✜ ✢ ✣✤ ✠ ✁ ✟ ✎ ✤ ✢ ✝ ✏ ✥✦ ☎ ✟ ✝ ✢ ✡ ✝ ✆ ✌ ✠ ✗ ✄ ☞ ✝ ✟ ❣ ✝ ✪ ✯ ✡ ❯ ● ❑▲ ▼◆ ● P ❋ ❊❘ ▲ P ● ❑ ■ ▲ ❚ ✌ ✾ � ÿ ✝ ✁ � ÿ ✤ ✗ ✂ � ✛ ✜ ✄ ✡ ✠ ❊❋ ✿ ✰ ✱ ✺ ✹ ✸ ✶ ✶✷ ✲ ✰ ✱ ✵ ✴ ✯ ✳ ✲ ✱ ✻ ✼ ❆ ✰❂ ❅ ❄ ✾ ✶ ✾ ❃ ❁ ✶ ❀ ✰ ✿ ✾ ✽ ✰ ✜ ❳ ✟ ③ ❫ ⑧ ❫ ❜ ❶ ❴ ❵ ✉ ❞ ❿ ✇ ✉ ① ② ❜ ✉ ✉ ✇ ✉ ❵ ⑨ ⑦ ✈ ❺ ④ ✉ ❵ ⑨ ⑧ ✈ ❜ ❺ ① ✉ ① ⑧ ③ ⑨ ❫ ⑧ ✈ ❽ ❹ ③ ③ ✉ ⑤ ⑥ ❻ ❺✈ ❵ ❜ ❜ ❹ ⑧ ✇ ⑩ ❾ ⑦ ❜ ✉ ✈ ⑧ ⑨ ❜ ⑦ ✉✈ ⑦ ❼ ❾ ✉ ❺ ❻ ① ✉ ❾ ⑧ ❛ ❜ ❹ ❶ ⑧ ④ ❴ ❜ ⑨ ② ❞ ✉ ❺ ❶ ⑧ ❾ ✇ ❝ ⑩ ➀ ❞ ➂ ❞ ⑦ ❵ ⑧ ❼ ❻ ③ ⑧ ❻ ✈ ❺ ① ✈ ❫ ⑧ ① ✇ ⑨ ⑧ ❫ ⑨ ⑧ ✉ ❫ ③ ③ ❹ ❽ ✈ ⑧ ⑨ ✈ ③ ⑧ ❻ ❺ ❿ ❞ ❻ ⑦ ✂ ✠ ✄ � ÿ � ✡ ✣ ✛ ✝ ✌ ✝ ✂ ✠ ✖ ✖ ☛ ✢ ✝ ❜ ❫ ❪ ✤ ✕ ✂ ✌ ✟ ✟ ✂ ✠ ☎ ☛ ✡ ✡ ✣ ❣ ✠ ✢ ✟ ✣ ✗ ✝ ✁ ✟ ✝ ☞ ✝ ❱ � ✁ ✢ ✄ ☞ � ✔ ✠ ✂ ✌ ✝ ☞ ÿ ✂ ❭ ❚ ✑ ✣ ✄ ✎ ❢ ❤ ✉ ⑦ ① ⑩ ❫ ❜ ❶ ❷ ❝❸ ⑩ ③ ❜ ❹ ⑧ ⑦ ❺ ❴ ⑧ ✉ ❜ ③ ❜ ❵ ❹ ❻ ❵ ❫ ❴ ❺ ✉ ✇ ❝ ❜ ⑨ ③ ✐ qr t ♥s ♦ ❧ ♠ q ❥ ✉✈ ♣ ♥ ♦ ❤ ❣ ❧ ❵ ✇ ⑧ ⑤ ① ❜ ⑦ ✉ ❝ ❜ ✈ ✉ ❜ ④ ❜ ③ ② ① � ✂ ✝ ➵ ❒ ❐ ➶ ➽ ➶ ➘ ➹ ➵ ➚ ➚ ➼ ➸ ➷ ➮ ➳➵ ✃ Ø ➷ ➬ ✃ ❮ ✃ ➮ Ð ➲ ❮ ➬ Õ Ö ➬ ➚ ➬ Ù Ö Ñ ➋ Ò Ó ➇ ➊ ➦ Ô ➉ ➋ ➲ ❮ ➬ Õ ➬ ➻ ➼ Ï ➹ ✃ ❮ ➼ ➚ ➶ ➪ ➶ ➼ ➶ × ➚ ➼ ➸ ➑ ➘ ➻ ➹ ➚ ➵ ✄ ➼ ❮ ➚ ➶ × ➻ ➼ ❮ ➱ ➶ ➵ ➳ ➬ ➵ ➼ ❮ ➬ ➼ ➮ ❐ ❒ ❮ ➻➼ ➮ ➸ ➸ ➬ ➽ ➼ ➼ ➱ ➚ ➻ ➬ ✃ ➳ ➾ ➽ ➼ ➵ Ú ➵ ❮ ➸ ➵ ➸ ➸ ➷ ➚ ➼ ➬ ➷ ➼ ➷ Ï ➹ ➵ ➬ Õ ➹ ➡ ➔ ➼ ➊➋ ⑤ ❶ ❷ ❸ ⑥ ⑤ ❻❼ ❽ ❻❾ ➂ ➇ ➅ ➈ ➉ ➎ ⑤ ➏ ➇ ➊ ➈ ➐ ➔ →➣ ➓ ➛ ➡ ➂ ➇ ➅ ➈ t ⑩ ➦ t✉ ❤ ✐ ❥ ♠ ♥ ♦ ♣ q ♠ r s ♥ ♦ ♣ t ♥⑨ ✈ ✇ ① ② ♦ ③ t ♥ ④ ⑤ ⑥ ♥ ⑦ ⑧ ➅ ➧ → ❐ ➴ ➷ ➳ ➶ ➳ ➬ ➵ ➮ ➵ ➪ ➺ ➱ ➬ ✃ ✃ ➾ ➪ ➶❰ ➸ ➵ ➬ ➻ ➻ ➱ ➽ ➻➼ ➵ Ð ➐ ➣ ➵ ➻ ➨ ➾ ➩ ➈ ➫ ➦ ➊ ➭ ➄ ➍ ➤ ➎ ➯ ➈ ➸ ➽ ➽ ➴ ➵ ➵ ➹ ➘ ➵ ➼ ➽ ➳ ➻ ➾ ➻ ➹ ➶ ➚➪ ➳ ➵ ➬ Ý õ ✛ ✠ ✡ î â ✄ ã ✕ ÿ ò ó äô ✠ ç ✡ í î í ô õ ✟ ù ï ì ð ÷ è ✜ úû ❮ Ö ➬ ➵ ➳ Ú ➹ ➻ ➼ ✟ ✃ ❐ ❒ ✃ ✢ ✌ ➸ ➻ ➳ ÿ ➷ Ð áâ æ �✁ ✟ ✣ í ☎ ➻ ☞ ✡ ✟ ✝ ✗ ✍ ✎ ✂ ✓ ✓ ✑ ✍ ✠✡ ✂ ✌ ✔ ✡ ☛ ✌ ☎✕ ✝ ✖ ✠ ✠ ✁ ✟ ✠ ✠✘ ✝ ÿ ÿ �✁ �✂ � ✄ ☎ ☛ ✟ ✠✡ ✝ ☛ ✠✡ ✄ � ✡ ☞ ✄ ✝ �✁ ☞ ✌ ✂ ✠ ✝ ✌ ✂ ✠ ➼ ➬ ❮ ❐ ❰ ➳ ❒ Ï ➻➼ ➵ ➵ ❰ ➬ Ö ❮ ➬ ✃ ❒ ✃ ❮ ✟ ✜ ✃ ✛ ➽ ➼ ➵ ➷ ➹ ➱ ➴ ➻ ❮ ➬ ➵ ➬ Þ �✁ ➻ ➳ Ö ➮ ➪ ➾ ➳➵ ➸ ➵ ➸ ➵ ➵ ❮ ➸ ➻ ➹ ➚ ➵ ➬ ➼ ❮ ➚ ➶ × ➸ ➵ ➼ ➬ ➘ ➼ ➺ ❮ ➬ ✃ ❐ ❒ ❮ Ð ✃ Ý ☛ ➵ ➺ ➻ ➱ ✠ ✃ ✂ ❐ ➵ ❮ ➬ ➵ ➚ ➵ ➬ ➼ ☎ ➶ ✠ ➺ ý ➺ ➶ ❐ ➳ ➵ ➱ ➚ ➻ ➾ ß b 178 3 ❙✞◆ ✈➁➀ Extendible Hashing ⑥❏⑦ directory ❖◗P ❴✚❵ leaf pages ❝❡❞ ❍❏■ ❫❉❛ ♠◗♥ ❴✚❵ - ❇❉❈ ❢❦❥ ✬✮✭ bytes - LookUp() 177 ✏✒✑ j bit ã➁äå ☎✚✙ üþý 3 ❿➁➀ Deleted bit ÛÜ➪ j ❹✸❺ Þ✞à â➆ø H LookUp Ordered Hashing – j j bits ö❃÷ ➾❃Ï Deleted ➞❃➟➠ ➌➆➍ →➝➜ ➺✸➻➼ ➢➥➤ ð❪ñ ↔✸↕➙ ❒❪❮ ã➁ï ➺✸➻➼ ✆✞✝ ❦✕❧ - ➃➄➆➅ ➃➄➆➅ é➆ê ➑✸➒➓ ➲❃➳➵ ë➥ìí j

  89. ➃ ➴ ❰ ➮ ➴ ❐ ➴ ❐ ➱ ➬ ➴ ❐ ➶ ➚ ➘ ➴ ➱ ➽ ❒ ➷ ➾ Ñ ➴ Ó ❒ ➮ ❒ ➘ Ï ➱ ➶ ➱ ➴ ➹ ß à ä ➹ ➘ Ð ➷ Ñ ➶ ❰ ❐ ➱ ➶ Ó ➶ ➶ ➷ ➶ Ð ❰ Ñ ➶ Ó ç ➘ ❒ ➹ ❮ Þ ❒ ➶ ❰ ➮ ➴ ➴ Ó Ó ➽ ❰ ➮ ➴ ❐ ➴ ❰ ➶ è ➽ ➬ ➱ ❒ ❮ ➱ ❒ ➬ ➚ Ó ❒ Ý Ô ➘ ➴ ➱ ❒ ➴ ❒ ❒➮ ➘ ➮ ❒ ➽ ➚ Ú Ý ➶ ➴ ❐ ➴ ➄ Ú Þ ❒ ➘ Ò Ó ❒ ➷ æ ➘ Ô Ü ➹ ➷ ➱ ❒ ➾ Þ Ý ❒ Ó ❐ ➾ Ý ➶ ❒ ➴ ➘ à ➾ Ï ➶ ➱ ➴ ➹ ß Õ Ñ ➱ ✃ Ó ➷ ➶ Ð ❰ å ì ➘ ➶ ❰ ❐ ➱ ➶ Ó ➷ ➷ Ñ ➬➮ ➶ ➪ Ú ➘ ➶ ➮ ➶ ç Ó ➶ ❰ Ñ ➴ ➹ ➷ ➪ ❮ ➹ ➶ Ú ➱ ➹ ý ➱ Ú ➱ ➶ Ý ➶ Ð ❒ ➬➮ ➷ ➱ ✃ ➷ ➶ ❰ ➪ Ñ ➴ ➴ ➬ ➱ Ò å ➮ ❰ Ô ➽ ➷ ➶ ➹ Ó ➱ ❒ ➬ ❰ ➶ ➮ ➴ ❐ ➴ Ý ➴ Ñ Ð ➷ í ❰ û ➼ ➽ ❒ Ð ❒ Ü Ï ð ❒ ➬ ➘ ➶ ➱ ❒ ü ï ö ❒ ó ì ë îï ð ñò ó ôõ ë ë ê ö ë ö ù ôú î ❐ Ð ➹➴ ➷ ➹ Ô ➪ ➶ ➱ ➴ ➹ ➱ ➶ ❒ ü ➮ Ü ➹ ➴ ➘ ➚ ➽ ❒ ❒ ➬ Ý ❒ ❮ ➘ ➶ ➱ Ñ ➬ ➶ ❰ ❐ ➱ ✃ ➽ ❒ Ô ➽ ➹ Ñ Ï ➶ ➹ ❐ Ð ➶➹ ➹ ❒➮ ➾ ➷ ➶ ➘ Ò ❐ ❰ ❐ ➬Ó ➘ ➚ ➘ ➴ ➱ ➹ ➷ ❒ ➹ ❮ ➽ ❒ ➬ ➴ ➽ ➶ ❒ ➱ ❐ ➱ ➴ ➽ ➚ ➚ ➱ ➴ ❒ ❮ ➬ ➃ ➄ ➅ ➆ ➇ ➊ ➱ ❒ ➹ ➴ ➘ Ô Ó ❐ ➴ ➱ ❐ ➘ ➶ ➚ ➘ ➴ ➱ ➹ ➷ ➶ ➴ ➌ ➙ ➣ ➌ ↔ ➑ ➋ ↕ ➛ ➔ ➋ ➜ ➝ ➋➞ ➟ ➙ ➑ → ➓ ➠ ➍ ➅ ➆ ➇ ➊ ➋ ➌ ➎ ➑ ➊ ➏ ➐ ➋ ➌ ➍ ➑➒ ➙ ➡ ➱ ➬➮ ➽ ➪ ➶➹ ➘ ➹➴ ➷ ➾ ➳ ➚ ➱ ✃ ➷ ✃ ❐ ➴ ➼ ➻ ➢ ➨ ➛ ➙ ➦➧ ➨ ➯ ➲➳ ➸ ➯ ➧ ➦➺ ➯ ➸ ➨ ➭ ➫ ➋ Ø ➍ ➴ Ð ➴ ➶ ❐ ➶ ❰ ➮ ➱ ➶➹ ➴ ❐ Ð ➶➹ Ñ ➹ ➾ Ú ➪ Û➮ ➶ ➨ ➭ ➫ ➯ Ù ➹➴ ➷ Ð Ô ❰ Ñ ➎ ➽ ➶ ➚ ➹ ➱ ❒ ➯ ➹ ➷ ➘ ❒ ❮ Ô ➪ ➶ Ô ➱ ➮ ➴ ➷ ➬Ü ❐ ➶ ❐ ➹ ➴ ➽ ➘ ❒ ❰ Û ➮ ✃ ➱ ➹ Ò ➘ ❐ ➴ ➱ ➴ ❐ ➶ ➚ ➸ ➴ ➺ ➑ ➙ ➑ ➙ ➟ ➋➞ ➝ ➜ ➋ ➛ ➦ ↕ ➋ ↔ ➡ ➌ ➣ → ➔ ➓ ➑ ➑➒ ➍ ➌ ➋ ➐ ➏ ➊ ➠ ➙ ➢ ➲➳ ➦➧ ➨ ➯ ➥ ➙ ➤ ➸ ➨ ➧ ➛ D-d 180 áãâä áãâ 3 ➻✞➳ d < D D D. bits. ÷✚ø ➵◗➸ h D (K); Ô✚➮ Ó❡å ➩❏➫➭ directory T[h D (K)]; (depth ➈✮➉ - é◗êë 179 3 L. h(K), d <= L bit strings ➵◗➸ d bits ➩❏➫➭ L bits: 2 L ➤❉➥ ➈✮➉ Õ×Ö - h d (K ➾◗➚

  90. þ ❳ ❘ ❯ ■ ◆ ❊ ❂ ❏ ❊ ❙ ■ ◆ ❘ ❯ ■ ◆ ❋ ❁ ❀ ❃ ◗ ● ❏ ❀ ❁ ❂ ❄ ✿❀ ❅ ❥ ✿ P ❃ ❀ ❁ ❭ ✿ ● ✿ ◗ ❯ ❇ ● ❅ ❂ ❭ ◗ ❯ ✐ ● ❑ þ ÿ � ❁ ❄ ❭ ❃ ❀ ❯ ❉ ❃ ❃ ◗ ◗ ■ ❀ ❈ ❆ ✿ ❭ ■ P ❁❯ ◆ ✂ ❢❣ ❅ ❀ ❘ ❁❯ ■ ✶ ❤ ❃ ✸ ❝ ✶ ✽ ▼ ❃ ❃ ❄ ❂ ❳ ❄ ❊ ✿ ■ ◆ ❈ ❳ ❀ ❁ ❯ ◗ ❘ ● ❏ ❀ ÿ ❀ ❴ ■ ■ ✐ ❁❯ ❂ ❀ ◆ ❯ ❄ ❁❯ ● ❀ ❅ ❂ ❄ ❆ ● ■ ● ❀ ✿ P ❃ ❉ ❊ ❂ ❈ ❭ ✐ ✐ ● ❅ ❂ ✿ ❭ ❯ ✁ ✝ ❅ ❈ ❭ ❀ ❯ ❀ ❭ ❂ ❊ ❊ ❀ ❯ ✿ ❁ ❊ ■ ❱ ❴ ✿ ❖ ❃ ❀ ❳ ❇ ❳ ❴ ❭ ❂ ❯ ✿ ❅ ✿ ■ ❱ ❊ ❂ ❵ ♣ ♣ ❈ ❭ ❀ ❯ ❀ ❭ ❂ ❊ ❊ ❀ ❯ ✿ ❁ ❊ ■ ❱ ❴ ✿ ❁ ❃ ❀ ❳ ❇ ❳ ❴ ❭ ❂ ❯ ✿ ❅ ✿ ■ ❱ ❊ ❂ ❁ ❀ ✞ ✘ ✞ ✔ ✕ ✖ ✞ ✗ ✞ ✓ ✙ ✚ ✕ ✌ ✕ ✛ ✜ ✌ ✟ ✖ ✞ ✟ ✠ ✡ ✝ ☛ ☞ ✟ ✒ ✠ ✌ ✍ ✌ ✎ ✏ ✑ ✢ ✕ ❵ ■ ✱ ❧♠ ❧ ♥ ♦ P ❙ ✬ ■ ◆ ❊ ❂ ❋ ❯ ✿ ✪ ✣ ✣ ✱ ✤ ✦✧ ★ ✬✭ ✮✯ ★ ✧ ✭ ✦ ✳ ✭ ✱ ★ ✬ ✪ ◗ ❘ ◗ ✿ ❂ ❈ ❆ ❂ ❄ ❃ ❁ ◆ ◗ ❊ ✿❀ ❘ ❁ ❂ P ❄ ❄ ❏ ❊ ❅ ❂ ❅ ■ ❊ ● ❂ ▼ ● ❁ ◆ ❊ ❖ ❃ ❅ ❂ ● P ❈ ❏ P ❂ ✿ ❀ ❙ ❆❇ ❈ ✿ ❁❯ ❂ ❄ ❀ ❄ ❀ ■ ❘ ❊ ❂ ❖ ❉ ❙ ❘ ❅ ❚ ◗ ❁ ❂ ❃ ❄ ✿ ❉ ❂ ✞ ✏ ✑ ✒ ✟ ✓ ✌ ✔ ✌ ✕ ✖ ✞ ✗ ✘ ✞ ✎ ✍ ✚ ✠ � ✁ ✂ ✝ ✞ ✟ ✡ ✌ ✝ ☛ ☞ ✞ ✟ ✠ ✙ ✕ ❈ ✺ ✭ ✴ ✯ ✸ ✹✺ ✻ ❁ ✬ ❂ ❃ ❄ ❅ ❀ ❆❇ ✪ ★ ✌ ✦✧ ✕ ✛ ✜ ✢ ✖ ✕ ★ ✱ ✬✭ ✮✯ ★ ✱ ✧ ✦✳ ✭ ❁ ❀ ❅ ❊ ❃ ■ ✿ ❊ ◗ ❅ ✿ ❭ ❴ ❁ ◗ ❯ ❂ ✿ ◆ ❀ ❭ ❂ ❃ ❬ ❩ ❄ ❅ ❂ ❀ ❁ ❘ ❀ ❱ ❈ ❵ ❊ P ❚ ❘ ❀ ❅ ❄ ❃ ❂ ❁ ❘ ❀ ■ ●■ ❉ ❳ ❏ ● ❘ ❪ ❂ ❯ ❀ ✿ ❫ ✿❀ ❊ ❀ ✿ ❄ ❅ ❛ ◆ ❁ ✿ ❀ ■ ❯ ❂ ● ❄ ❳ ❘ ❋ ❭ ❊ ❀ ✿ ❊ ❈ ❆❇ ❃ ❂ ❲ ❑ ❘ ❂ ❅ ❄ ❃ ❂ ❁ ■ ❈ ❂ ✿ ■ ❄ ❀ ❃ ❃ ❀ ❃ ❂ ❁ ❘ ◗ ❯ ❘ ◆ ❙ ❀ ❉ ❖ ❳ ◆ ■ ❯ ◗ ❁ ◗ ❨ ❱ ❏ ❘ P ■ 182 3 ✴❦✯ b = 2. ✰✲✱ ❏✥❯ ✩✫✪ 01011. 11101. ✄✆☎ q✷✿ q✷✿ - buddy directory, 181 d directory 3 ❀▲❑ ❀▲❑ b ✰✲✱ directory? ❋❍●■ ✩✫✪ d ✼✲✽ 1. d < D 2. d = D ✣✥✤ page). ✄✆☎ - ✾✷✿❀ ❜❞❝❡ ✵✷✶

  91. r ✘ ✑ ✥ ✔ ☞ ✠ ✞ ✚ ✣ ✏ ✤ ✞ ✎ ✏ ✙ ☞ ✔ ✠ ✙ ✓ ✚ ✒ ✔❨ ✠ ✎ ✢ ☎ ✏ ✒ ✔ ✓ ✌ ❲ ✞ ✠ ❚ ❯ ❱ ✡ ✓ ✚ ✥ ✏ ☎ ✏ ✡ ☞ ✠ ✌ ☞ ✤ ✆ ☛ ✓ ✠ ✌ ✡ ✒ ✓ ✏ ✘ ✎ ✏ ✤ ✒ ✢ ✏ ▲ ✠ ✞ ✠ ✡ ✏ ✓ ✏ ✡ ✢ ✓ ✏ ☛ ✗ ✠ ☞ ✌ ✆ ✠ ✞ ✆ ✓ ✔ ✓ ✚ ✞ ✆ ✛ ❬ ✒ ✌ ✠ ✟ ✞ ✠ ✠ ✞ ✏ ✡ ✌ ✡ ✓ ☞ ✠ ✌ ✠ ☞ ✆ ☛ ✓ ✌ ❩ ✡ ✒ ✓ ✏ ✘ ✌ ✦ ✞ ☞ ✙ ✆ ✡ ☎ ✡ ❙ ❖ ☛ ✌ ✎ ✍ ✞ ✦ ✠ ✎ ✢ ✚ ✞ ✠ ☎ ✆ ✟ ☞ ✆ ☛ ✛ ✎ ✏ ✤ ☎ ✆ ✟ ✣ ✞ ✏ ✤ ✞ ✒ ✌ ✠ ✠ ✞ s ✢ ✞ ✍ ☎ ✆ ✡ ✆ ✛ ☞ ✠ ✡ ✠ ✗ ✏ ✌ ✎ ✏ ✒ ✤ ✞ ✑ ✛ ✎ ✆ ☛ ✡ ✚ ✣ ✆ ✡ ✔ ✟ ❏ ❀ ❀ ❁ ✲ ❂ ❃ ✲❄ ❅ ❀ ✸ ❀ ❆ ❇ ❈ ❁ ●❍ ✲ ■ ❏ ◆ ❖ P ◗ ❍ ❏ ❍ ❘ ◆ ■ ❙ ❍ ✿ ✸ ☎ ✳ ✡ ✠ ✒ ✆ ✙ ✟ ☎ ✩ ✪ ✫ ✬ ✭ ✱ ✲ ✴ ✾ ✵ ✱ ✶ ✷ ✲ ✳ ✴ ✸✹ ✸ ✺ ✻ ✼ ✽ ✳ ✢ ✡ ✆ ☞ ✦ ✞ ☞ ✙ ✆ ✡ ☎ ✡ ❩ ✞ ❴ ✆ ✞ ✡ ✍ ✑ ✕ ✑ r ✔ ✒ ✤ ✞ ✍ ☛ ✌ ✔ ✒ ✔ ☞ ✌ ✓ ✌ ✤ ☛ ✓ ✠ ✌ ✡ ✒ ✓ ✏ ✘ ✞ ✆ ✎ ✡ ✌ ❴ ✏ ✣ ✍ ✞ ✆ ✡ ☞ ✠ ✙ ✑ ☞ ✠ ✌ ✠ ✞ ✠ ✠ ☞ ✠ ❴ ✡ ✠ ✒ ✤ ❴ ☞ ✆ ☞ ☛ ✡ ✓ ✚ ✞ ✒ ✓ ✘ ✞ ✏ ✙ ✠ ☞ ✙ ✆ ✡ ☎ ✡ ❩ ✞ ✧ ✑ ✏ ☞ ✌ ✆ ☛ ✓ ✠ ✌ ✡ ✒ ✓ ✏ ✘ ✞ ✆ ✎ ✡ ✤ ✞ ❴ ✣ ✍ ✞ ✆ ✡ ☞ ✠ ✙ ✑ ☞ ✠ ✌ ✠ ✆ ✠ ✠ ☛ ✌ ✆ ☛ ✔ ✎ ✆ ☛ ✌ ✦ ✒ ✔ ✌ ✠ ✎ ✓ ☛ ✠ ✌ ✠ ✓ ✎ ✏ ☛ ✏ ✘ ✞ ✞ ✠ ✆ ✟ ✢ ✡ ✠✡ ✆ ✌ ✏ ✤ ✎ ✟ ✞ ✠ ☞ ✠ ☞ ✠ ✌ ✠ ☞ ☛ ✆ ✓ ✠ ✌ ✡ ✒ ✓ ✏ ✘ ❭ ✧ ❪ ✌ ✔ ✣ ✞ ✎ ✌ ♣ ❜❤ ✐ ❞ ❜❥ ❞ ❥ ❤ ♥ ❞ ❥ ❡ ♥ ❧ ♦ ❤ ❞ q r ✔ ✒ ✤ ✞ ✍ ☛ ✌ ✔ ✒ ✔ ☞ ✠ ❝ ❛❜❝ ✏ ✡ ✙ ✘ ✍ ✒ ✣ ✔ ✓ ✠ ❬ ❴ ☛ ✠ ✓ ✓ ☞ ❵ ✢ ✧ ❭ ✠ ✌ ✔ ✌ ✏☞ ✑ ✙ ✎ ✤ ✙ ✏ ✎ ☎ ✘ ✣ ➩ ➵ ➴ Ð ➩ ➶ ➸ ➺ ➭ ➺ Ñ ➹ ➽ ➥ ➫ ➧ ➘ ➥ ➽ ➨ ➹ ➱ ➥ ➷ ➥ ➻ ➹ ➦ ➮ ➻ ➨ ➨ ❒ ➥ ➩ ✃ ➘ ➶ ➽ ➮ ➦ ➥ ❐ ❒ ❮ ❰ Ï ➩ ➽ ➷ ➬ ➽ ➨ ➹ ➱ ➥ ➺ ➼ ➨ ➳ ➵ ➳ ➼ ➥ ➹ ➲ ➨ ➨ Õ ➲ ➳ ➩ ➨ ✒ ➽ ➥ ➽ ➘ ➥ ➩ ➹ ➳ ➱ ➥ ➽ ➮ ➪ ➥ ➩ ➫ ➘ ➥ ➽ ➮ ➩ ➨ ➭ ➩ ➽ ➱ ➼ ➘ ➧ ➻ ➥ ➹ ➽ Ô ➽ ➥ ➨ ➳ ➵ ➩ ➮ ➨ ➲ ➦ ➨ ➮ ➺ ➱ ➭ ➫ ➘ ➩ ➽ ➘ ➥ ➩ ➶ ➮ ➱ ➈ ⑨ ❼ ➂ ➃ ➄ ❽ ❼ ➇➈ ➉ ➍ ➎➏ ➉ ➑ ➇➒ ➁ ➍ ➑ ➉ ➌ ➋ ➍ ➓ ➏ ➔ ➌ ➑ → ➒ ➑ ❼ ③➀ ↔ ⑤ t ✉ ✈ ② ③ ④ ⑤ ⑥ ② ⑦ ⑧ ③ ④ ⑨⑩ ❿ ⑨ ❶ ❷ ❸ ❹ ④ ❺ ⑨ ③ ❻ ❼ ❽ ③ ❾ ➣ ↕ ➱ ➺ ➼ ➘ ➧ ➩ ➴ ➫ ➨ ➩ ➳ ➵ ➸ ➺ ➨ ➴ ➹ ➽ ➨ ➹ ➩ ➥ ➩ ➸ ➺ ➮ ➷ ➹ ➶ ➩ ➩ ➽ ➩ ➆ ➦ ➎ ➙ ➎ ➉ ➌ ➈ ➓ ➉ ➛ ➉ ➑ ➞➟➠ ➞➡ ➢ ➫ ➨ ➭ ➩ ➯ ➺ ➨➻ ➥ ➳ ➨ ➧ ➨ ➚ ➥ ➩ ➸ ➹ ➧ ➬ ï â û ✙ î ý â ➥ ✙ ã ó ç è æ í ç ✏ ñ ✎ ò å í ó ê è û ê ✠ ó û ☛ é â û ê â ï â ø è ✆ ✒ ✎ í å ü ✛ æ å ✡ äå â ✌ ç ✢ þ ✘ í ñ ✏ å ☎ ✏ ✎ æ é â é è ù ä ✡ è û ✠ ✍ ✌ ✆ ✄ ☞ ✆ ✟ ✞ ✠✡ ✌ ☞ ✏ ò ✠✗ ê è ø ù â ✓ ä è û ë â ✑ æ ✑ é ✞ æ ✓ ✆ ✒ è ø ÿ ë ê ó ✙ ð å ç è é è ê æë ✠ â î ï ä ✆ è ✞ ç ï è í ç ñ ò ✟ ✆ å æ â ✏ Ü ➵ Ò ✡ ✠ Ö × Ø Ù Ø Ú ➝ Û Ú á Ý Ú ➞ ✞ Þ ➶ ➷ Ð ➺ ➧ ➺ßà ✟ ☛ í ✎ ï ð í ú å í ó í ê ó ä ë ø æ ç å ù ä ì ù ó ë å æ ✟ ù æ å ✓ ï ✠ ã ø ÷ å å ç å æ ✆ å ê â ê å æ ù ð ☎ ä è ù æ ✛ è 184 ✚✜✛ 3 ☎✝✆ ☎✝✆ ✍✜❴ ✟❫✎ ❦♠❧ ❑▼▲ ✑✜❳ ❉❋❊ ❡❣❢ ✮✰✯ - 183 ❰ÓÒ 3 ✔✖✕ óôöõ ✥✝✦ ➪✥➶ b ➼➾➽ directory ➼➾➽ �✂✁ ➐✲➑ – ì✥í ➵✥➸ ➼➾➽ ➷❍➬ ➊✫➋➌ ➲❍➳ i ✠★✧ ➅✥➆ i ✚✜✛ ➧❦➨➩ ✇✆① - ã❍äå ☎✝✆✞ 2i ➤❍➥ ➜❍➝

  92. s ß ✡ å ✄ ✆ ç é ☎ á Þ ç é ☞ è å ✄ ☎ á å ☞ ✞ ç ☛ Û Û é ● ú ü ❍ û ø ■ þ ✞ Û ÿ å ☞ ✞ ç ß Ö ❏ é Û ç ß ÜÝ ó ✌ ç è ç Ü é Ö è Ý ✞ ❑ ✞ û ▲ ï ▼ þ ÿ à é ç ✂ è æ Ý Þ Ý ß Ü é Û ✁ å ✞ Ö é ✞ ☞ Þ ✠ Û é å ✡ å ✁ é ç ò ø ✆ ✢ ✢ ✣ ✢ ✤ ✥ ✦ ✧ ✘ ★ ✗ ✘ ✩ ✪ ✫ ✗ ✬ ✭ ✗✮ ✯ ✪ ✢ ✙ ✗ ✰ ✍ ç ☞ Û é è à ÿ Û ✌ ✎ ✜ t ✑ ✒ ✖ ✗ ✘ ✙ ✚ ✖ ✛ ✪ ✱ ❋ ❇ ❁ ✽ ✸ ❂ ✷ ✹ ✾ ✻ ❂ ✿ ✹ ❂ ✽ ❈ ✾ ✷ ❉ ✸ ✽ ✻ ❊ ✷ ✷ ✲ ✹ ✫ ✪ ✶✷ ✸ ✹ ✽ ✾ ✿ ❀ ✷ ✷ ❀ ❁ ✽ ✸ ❂ ✷ ✹ ✾ ✻ ❂ ❃ ã ✆ ✁ Û ✞ ã ✡ Û ✂ ç ✆ å é ç Ö ✞ Û é å ✡ å ✁ é ç ☞ ç é á ç ß ✆ Ö Ù ä ç å ☞ ✞ ß é ã è é ß ç Ö ÿ å è ß Û æ ☎ à æ å ß ÿ å è ß é é Ö à ã è é ß Û Ü Ý Ü à ✌ ☛ Ü ç é è ç Ü é Ö è ç á ✄ Û ß ç ✡ � ç è ß ä ✆ ☎ é é ❏ ç ✄ ✂ ç ç ✁ é ã ÿ Û ✁ ✁ ✡ ✞ Ö ä Ý á ✄ ✁ ✞ ç ✞ ☞ ✠ Û ◆ é ç é Û ✁ ç ✡ � ã ☎ ✡ é Û ✁ å ☞ ✞ ç ß ÿ Û é ✠ á é ☛ ❖ å Ý ✞ ✌ ■ þ ✞ Û á å ☞ ✞ ç ß ã è é ß Û ✡ å ✞ é é ç ß Ý á é å ✡ å ✁ ç è ☞ Û ß æ ç è ç Ü é Ö æ ✏ é ➸ ➨ ➽ ➺ ➾ ➯ ➲ ➷ ➬ ➮ ➺ ➺ ➺ ➽ ➾ ➭ ➧ ✃ ➼ ➾ ➻ ➾ ❰ ➾ ➾ ➚ ➧ ➯ ➪ ➦ ➻ ➶ ➫ ➹ ➺ ➵ ➘ ➭ ➫ ➴ ➽ ➫ ➵ ➼ ➻ ➳ ➺ ➼ ➺ ☎ ➱ ➺ ➽ ➾ ➭ ➧ ✃ ➾ ➯ ➻ ➯ ➪ ➦ ➯ ➺ ➽ ➾ ➭ ❰ ➼ ➯ ➼ ➶ ➯ ➩ ➵ Ï ➯ ❰ ➧ ➫ ➯ ➲ ❐ ➳ ➯ ❰ ➦ ➧ Ð ➻ ➯ ➼ ➧ ❐ ⑩ ❼ ❽ ❾ ④ ❿ ➀ ④➁ ➂ ❽ ❽ ⑩ ➃ ➄ ➅ ❾ ❽ ➈➉ ➊ ➋ ➎ ④ ❻ ➐ ⑧ ✉ ✈ ✇ ③ ④ ⑤ ⑥ ⑦ ③ ⑨ ⑤ ④ ⑤ ⑥ ⑩❶ ⑩ ❷ ❸ ❹ ❺ ➏ ➑ ➽ ➦ ➛ ➢ ➞ ↔ ➤ ➜ ➛ ➟ ➥ ➧ ➝ ➳ ➻ ➺ ➻ ➺ ➼ ➺ ➻ ➫ ➝ ➟ ➉ ➍ ➋ ➉ ➒ ➎ ➊ ➓ ➉ ➋ ➏ ➓ ↕➞ ➔ → ➣ ↔ ↕ ↔➙➛ ➜ ➝ ↔ ➧ ➶ ➲ û û ø ô ã ß ü ö ì ëú ý Þ ß Ñ Û �✁ ✂ ó ø ✄ í ✂ ☛ ✡ ß å ê ëì å ô ê ðñ ô õ ö õ ï Ö ✁ è ß ✠ ✄ å Ý ✞ ✄ á å Û ÿ å è ß ÿ é ✝ ÿ è ✄ ç é å ☎ ✡ Ü ✆ ç é Ü ✁ é ß Ü Û ✄ ✁ ✂ ç ÿ Ö ➴ ➺ ➘ ➚ ❰ ➸ Ð ➭ ➲ ➾ Ü ß ÓÔ ÕÖ Ô × ç ➻ ➹ ☎ ã ➺ ➽ ➺ ➻ ➼ Ð ✆ ➽ Ò ➴ ➸ ➚ ➼ ➯ ➸ ➻ ✆ é ✄ Ü é ã ✞ é ç â æç ß Ü ✆ ÜÝ ÚÛ å ☛ è ✄ ✁ ✄ ç Û é Ü ç ✞ Ø Ù Ù 186 3 ❄❆❅ à✜á h(K à✜á r 2 r ✺✼✻ m Þ✝ß m ✳✵✴ ✓✕✔ - i , 185 i r c 1 0 − = p i ∑ 3 ä✝å ✝✟✞ ÷ùø ASCII ❐❮❒ ➠❋➡ ➌▼➍ h(K) = K mod m à✜á ò❋ó ➵✜➸➺ Þ✝ß ➶❣➱ ➆❋➇ ➭✝➯➲ î❋ï ①✰② ➨✜➩➫ - c j p r o o o m K þ❫ÿ

  93. P ✥ ➮ ➵ ➘ ➚ ➻ ➵ ➮ ❒ ➚ ➮ ➶ ➶ ➱ ➘ ➚ ➘ ➹ ➵ ➮ ➱ ➻ Õ ➳ ➳ ➶ ➵ Ñ ➶ ➷ Ô ❮ ➚ ➱ ➴ Õ ➶ ➻ ➸ ✃ ➮ Ï ➪ ➻ ➵ ➘ Ñ ✥ ➴ ➻ ➴ ➚ ➹ ➚ ➻ ➴ ➚ ➾ ✧ ➪ ➹ Õ ➪ ➮ ➱ ✢ ❮ ➳ ➶ Õ ➮ ➚ ✢ ➚ Ô ➴ ➮ Ô ❮ ➵ ➴ ➵❮ ➚ ➳ ➘ ➵ ➹ ➶ ✦ ➮ ➘ ➶ Õ Ñ ➶➱ ➻ ➵ ➵ Ñ ➵ ➚ ➪ ✂ � ø ú ø ✁ ýù ✂ ø úþ ü ✄ þ ☎ ✆ ✡ ☛☞ ✏ ✍ ✑ ✟ ✒ ✓ ÿ ý ✏ ï ß ê ä Þ ë ì í Þ ◗ Þð ú ñ ì ä ì ò ó ô í ì øù ✍ ✡✔ ➘ ➵ ➚ ➚ ✤ ➵ ➳ ❒ ❰ ➪ ➴ ❮ ➵ ➹ ❐ ➹ ✥ Ñ ➶➱ ➷ ➬ ➾ ➶➱ ➘ ✣ ✓ ✚ ✕ ☞ ✖ ✏ ✍ ✏ ✡ ✑ ✙ ✟ ☛ ✃ ✓ ✝ ✘ ✛✘ ✜ ➻ ✢ ➻ Ï ➳ ➹ Ò è ➮ ➘ ➶ ✢ ❰ ➸ ➵ ✦ ➶ Õ ➱ ➵ ➻ ❰ ➷ Ï ➪ ➴ ➶ ➮ Õ ➻ ➳ ✦ ➾ Ñ ➷ ➶ ❐ ➶ ➪ ➬ ❒ ➚ ➹ ➚ ➚ ➳ ➹ ➶ ➳ ➹ ➮ ➘ ➬ ➻ Ô ➶ ➶ ➘ ➾ ➚ ❐ ➶ ➘ ➬ ➻ ➵ ➵ ➾ ✥ ➵ ✣ ➶➱ ➻ ➻ ➵ ➘ ➹ ❮ ✥ Ò ❐ ➴ ➴ ✃ Ï ➻ ❮ ➹ ✧ ➻ ❐ ➚ ➻ ➚ ➻ ➹ ➵ ➹ ✢ ➻ Ï ➮ ➴ ➽ ❮ ➵ ➬ ✩ ★ ➻ ➹ ❮ ➬ ➪ ✢ ➹ ➵ ➴ Ô ➚ ❒ Õ ➳ ➚ ➮ Ô ➴ ➬ ➾ ➚ ➹ ➻ ➚ ➻ ❒ ❒ ➶➱ ➻ ➵ ➻ ➵ Ñ ➴ ➱ ➷ ➴ ➚ ✃ ❐ ➹ ✥ ✦ ➶ ➳ ➚ ➪ Ï ➮ Ò ➮ ➵ ❐ ➚ ❐ ➶ ➘ ➬ ➻ ➵ Õ ✢ ➻ ➶ ➾ ➵ ❮ ➾ ➚ ➹ ✧ ➻ ➹ Õ ➱ ➵ ➻ ➵ ➳ ➴ ❰ ➮ ➚ Ï ➻ ❮ ➴ ➶ ➵ ❮ ➚ ➳ ➘ ➵ ➴ ➹ ➮ ➘ é î ç ➓ r ➦ ➐ ➅ ➧ ③ ↔ ➈ ➥ ➀ ↕ ➐ ➅ ➧ ➨ ➅ ➀ ➜ ➤ ↔ ✇ ➅ ➈ ➙ ➉ ➋ ↕ ➍ ➓ ➢ ➈ ➀ ④ r ➛ ➜➝ ➞ t æ ➉ ➍ ❣ ❫ ❢ ❦ ❳❥ ✐ ❤ ❳ ❢ ❧ ❡ ❳ ❫ ❞ ❨ ❝ ❜ s ❢ ♠ ➊ ➍ ➇➈ ➀ ❿ ➋ ➌ ➅ ➂ ➩ ♥ ② ① ✇ t s qr ❢ ❣ ➐ ↕ ❫ ➇➈ r ③ ➀ ⑥ ⑤ ➂➆ ✈ ➉ t ➊ ➇➈ ➀ ❿ ➋ ➌ ➅ ① ❽ ❹ r ⑤ r t ① ✈ ⑤ ⑨ ② ✇ r ✇ r ❷ s ✇ ✈ ❸ ④ t ➑ ➋ ➂➆ ➀ ➈ ➀ ➋ ➉ ➊ ➅ ➍ ➇ s ↔ ➃ ➀ ➈ → ➀ ➓ ➣ ➑ ➍ ➁ ➒ ➉ ➊ ➇➈ ➀ ➓ ➈ ➔ → ➅ ➉ ➋ ➂ ➅ ➂ ⑤ ➇➈ ❵ ❛ ➫ ➶➱ ➾ Ñ Ò Ó ➬ ❒ ➻ ❙ ➴ Ô ➻ ✃ ❐ ➹ ➵ ❮ ➸ ➹ ➳ ➽ ➾ ➶ ➱ ❒ ➵ ➵❮ Ò ❿ ❰ ➳ Ï ➻ ❚ ❒ ➴ ➘ ➱ á Ù Ú Ý Þ ß à Ý × â ã Þ ß à äå ä Ø Ö ❰ ➘ ❮ ➵ ➴ ➵ ❮ ➚ ➳ ➵ ❘ ➴ ➹ ➮ ➘ ➶ Õ Ò ➻ ➴ ➮ ➒ ➀ ➓ ➯ ❿ ➧ ➍ ➃ ➧ ➀ ➶➱ ➈ ➓ ➯ ➙ ➉ ➃ ➃ ❩ ➂ ➂ ➋ ➉ ➊ ➇➈ ➈ ❿ ➣ ➈ ➉ ➀➭ ➙ ➉ ➊ ➉ ❿ ❫❴ ⑤ ❨ ➪ ➵ ❳ ➚ ➘ ➷ ❲ ➶➹ ❳ ➽ ➚ ➸ ➹ ➼ ➴ ➴ ❐ ❪ ❭ ➹ ❲ ❬ ❩ ➵➻ ➲ ➳ ❨ ✃ ➶➱ 188 3 ✗✠✘ ❐❆➹ ➾❶➚ ❐❆➱ ➾❶➚ ❐❆➹ û✼ü ✌✎✍ õ✵ö÷ Û✕Ü - ❐❆➹ ✝✞✠✟ ABC, BCA, CBA, 187 I mod (r-1)) mod (r-1) i, r k +a, r k - j r j ) mod (r-1) = 1. 3 ⑦❆⑧ ➊❼➐ ❾✵❿ c )mod (r-1) ➬❼➮ ⑩❶① ➂➍➏➎ r- i r c i 1 0 = 0 ( − 1 0 i 1 − r i mod (r-1) = (1 + (r-1) ∑ = = − = p Ð✼Ñ➺Ò i a. p i = ( ∑ c i mod (r-1) = ∑ ➾❶➚➪ k m = r- ✉✼✈ m m m = r- ♦✵♣ ➄❼➅ i r ➟➡➠ 1 ❯✕❱ 0 − - = ➁❶➂➃ p i ∑ ❺❼❻ ➵➺➸ ❾✵❿

  94. ✪ ⑦ ❽ ➁ Þ ß ➓ ➏ ➏ ➏ ➓ à á âã Û ❶ ➇ ➀ ➋ ➉ ❾ ➉ ➊ ❻ ❺ ➌ ➉ ➁ ➃ ➉ ➉ ➅ ➁ ➍ Ü ➉ ➍ ❻ ➁ ➀ ➁ ➊ ❽ ❻ ❺ ➌ ❽ ➃ ➉ ❿Ý ❻ ❺ ❷ ❼ ❽ ➉ ❽ ❻ ➍ ➂ ➁ ➄ ➇ ➀ ❿ ❸ ↔ ❸ ➙ ➓ æ ❼ ❷ ➁ ➊ ➓ ❸ ➉ → ➍ ❿ ❸ ❿ ❿ ❿ ❺ Ý ❻ ❶ ➀ ➒ ➂ ❿ ❽ ❺ ❸ ➇ ❹ ❹ ➁ ➀ ❸ ➇ ➃ ❺ ❻ ➂ ❸ ❺ ➇ ➋ ❸ ❺ Ü ➭ ➭ ➥ ➻ ➼ ➽ ➥ ➾ ➚ ➥➪ ➶ ➼ ➼ ➦ ➹ ➘ ➴ ➽ ➼ ❒ ➷ ✃ Ñ ❐ ❰ ➺ ➸ ➬ ➦ ❸ ➇ ➋ ➏ ➜ ➝ ➞ ➟ ➠ ➤ ➥ ➧ ➵ ✫ ➤ ➩ ➫ ➥ ➦ ➧ ➭➯ ➭ ➲ ➳ Ô Õ ➅ ❷ ❸ ➙ ❻ Ü ➅ ➒ ➂ ❿ ❹ ➁ ➄ ➎ ❸ ❻ ❺ ➇ ❺ ➇ ❺ ❽ ❺ ➇ ❿ ➌ ↔ ❿ Ö Ö × Ð ÑØ Ô Ö ➬ ❒ Ñ Õ Ù ➬ ❐ ➀ Ð Ù ➱ Ú Û Ü → ➙ ➌ ➁ ➄ ➇ ❸ ç ➁ ❼ ❸ ❺ ➉ ➊ ❻ ❽ ❾ ➉ ❾ ❺ ❷ ❽ ➄ ❾ ❻ ➇ ➀ ➋ ➉ ❾ ❽ ➊ → ❺ ➅ ❺ ❺ ➂ ➇ ❿ ➊ ➉ ❾ ❽ ➁ ➊ æ ❿ ➍ ➁ ❽ ➊ ➄ ➏ ë ❸ ❺ ➀ ➒ ➂ ➁ ❻ ➎ ➓ ❹ ➇ ❾ ➂ ➃ ➉ ❿ ❸ ➉ ❿ ➉ ➍ ➂ ➌ ➁ ➁ ➊ ➉ ❾ ❿ ➎ ❻ ❺ ❸ é ê ↔ ❺ ❾ æ ❷ ❸ ❻ ❼ ❽ ❾ ❽ ❺ ➒ ❻ ❺ ❿ ❻ ➍ ➂ ➁ ➄ ➇ ➀ ❿ ❸ ↔ ❸ ➙ ❻ ➃ ➁ ❸ ➁ ➙ ❻ ➓ ➅ ❼ ❻ ➁ ➒ ❻ ❺ ❷ ➉ ↔ ➁ ➉ ❽ ❼é ê ↕ ❽ ➁ ❷ ➁ ➀ ❸ ❸ ➛ ↔ ❺ ➇ ➋ ❹ ❿ æ ❿ ➍ ➂ ➁ ➄ ❸ ❿ ❺ è ➁ ➌ ❿ ❽ ❸ ➇ ➙ ❻ ➇ ➀ ➍ ❽ ❷ ➌ ↔ ❾ ➂ ❿ ➎ ➉ ❿ ➉ ➍ ➂ ➇ ➁ ❺ ➊ ➉ ❾ ➇ ❺ ❽ ➋ ❽ ❾ ❻ ❺ ➁ ❻ Ý ❻ ❻ ➉ ➊ ❻ ❺ ➌ ❽ ➃ ➉ ❿ Ý ❽ ➎ ❾ ➄ ➇ ➀ ➋ ➉ ❾➄ ➅ ❷ ➁ ➌ ❿ ❽ ➨ ↔ ❷ ⑥⑦ ⑧ ⑨ ⑩ ❶ ❷ ❸ ❾ ❽ ❺ ❿ ❸ ④ ➅ ❸ ❺ ➄ ➆➇ ❺ ➇ ➉ ➊ ❻ ❽ ⑤ ♣ ➉ ❥ ♦ ❝ ❞ ❡ ❦ ❣ ❤ ② ❝ ❧③ ❡ ❦ ♦ ❤ ❦ ❥ ❧ ❣ ♠ ❦ ❤ ♥ ❝ ♠ ❾ ❾➄ ♦ ➉ ❼ ➅ ❿ ➁ ❾ ❿ ➇ ❽ ➒ ➀ ❿ ❾ ❿ ❽ ➁ ➊ ❷ ➌ ➁ ➂ ➌ ➋ ➅ ➅ ❷ ❿ ➇ ➅ ❺ ❽ ❺ ➇ ❿ ➌ ➅ ❺ ❽ ❸ ➉ ➁ ➑ ➍ ❿ ➎ ❻ ❺ ❸ ➅ ❸ ➇ ➌ ⑩ r ❝ ❽ ❁ ❃ ❄ ✳❅ ❆ ❁ ✹ ❁ ❇ ❈ ❉ ❂ ▼ ❂ ❊ ❏ ❚ ❑ ❖ ❲ ❋ ❳ ❨ ❩ ❘ ✳ ❁ ❲ ✳ ✬ ✭ ✮ ✲ ✳ ✴ ✵ ✶ ✲ ✷ ✸ ✴ ❀ ✵ ✹✺ ✹ ✻ ✼ ✽ ✾ ✴ ✿ ✹ ✳ ❚❬ ❊ ❦ ❤ ❤ ♥ ❝ ♠ ♦ ♣ ❧ q ♥ r ❧ ❦ ♠ ♥ ❤ ❞✈ ❦ ✇ ❦ ❦ q ① s ❡ ❦ ❣ ❋ ❍ ❨ ❋ ▼ ❚ ❳ ❭ ❋ ❨ ❑ ❘ ❭ ❪ ❧ ❫ ❴ ❞❡ ❤ ✐ ❝ ✐❥ ❦ ❤ ❦ ❥ ❺ ❻ ➁ ➒ ❸ ➇ ➋ ❿ ❷ ❸ ➀ ❿ ➂ ➅ ❻ ❿➇ ❺ ❺ ❷ ❼ ➒ ❻ ❺ ❻ ❺ ❻ ❽ ❽ ➉ ❷ ❾ ❽ ❺ ❸ ➉ ❽ ➁ ➉ ➍ ➉ ❽ ➅ ➁ ❺ ❿ ➎ ❻ ❺ ❸ ❷ ➁ ➀ ➍ ➌ ➎ ❺ ➁ ➊ ➓ ❾ ➅ ➒ ❹ ➁ ↔ ➁ ➄ ❺ ❽ ➁ ➃ ↔ ➁ ➊ ➀ ❿ ➍ ❿ ❸ ❺ ❷ ➛ ↔ ↕ ➀ ➙ ➉ ❽ ❿ ❻ ❺ ❺ ❷ ➁ ❽ ❿ ➁ ➁ ➍ ❻ ➋ ➉ → ❾ ➅ ❾ ❿ ➎ ➉ ❻ ❷ → ❸ ❺ ➅ ❺ ❿ ➎ ❻ ❺ ➅ ❺ ❸ ➇ ➃ ➏ ➂ ➋ ⑩ ➌ ❽ ❻ ➇ ❸ ❸ ❺ ➇ ❿ ❷ ➋ ➀ ➋ ❺ ❻ ➋ ❷ ➒ ➉ → ❾ ➅ ➣ ➌ ❼ ❼ ❾ ❷ ➀ ➊ ❿ ➁ ❷ ❸ ❷ ➋ ❸ ❻ ❷ ❺ ❹ ❽ ➇ ↔ ❺ ❺ ➌ ❽ ➅ ❾ ➇ ❸ ❺ ❺ ❺ ❻ ❺ ❽ ❽ ❾ ❸ 190 m-1} ❼■❽ 3 x ≠ y m. |{h ∈ H: h(x) = h(y)}| / |H| ≤ 1/m. Ï❙Ð ❮P❰ ➷✠➬ ➋✠➌ x,y ✃▲❐ ➮■➱ ➷✠➬ ❹P❺ m Ò✠Ó ä✠å ➡✱➢ - m H 189 ➃✉➄➈ ➄➔➓ 3 ➃➐➏ s✉t ◗❙❘ ➃✉➄ ➀▲➁➂ ◆P❖ ❏▲❑ ●■❍ ❊✠❋ ➀▲➁ ❼■❽ (at run time). ♣✉❡ ❹P❺❻ ❯✠❱ ❢✠❣ ➋✠➌ ✯✱✰ - ❵✠❛ ❜✠❝

  95. ì ❆ ✽ ❄ ❇ ❃ ❊ ❇ ❩ ❣ ❯ ❈ ❋ ❆ ❂ ✿❁ ❄ ✽ ❃ ✿ ❁ ■ ✽ ❆ ✐ ❇ ❇ ❆ ✾ ✽ ❄ ❬ ❊ ❇❈ P ❇ ❉ ❏ ❊ ❦ ❣ ❥ ❇❙ ❆ ❊ ✾ ✿❁ ✐ P ❂ ❁ ■ ❬ ❊ ❋ ■ ❁ ❂ ✿❁ ❅ ❃ ❇ ❃ ❂ ❇ ❃ ❇ ❂ ❁ ✽ ✿❁ ✾ ✽ ❄ ❇ ❊ ❑ í ■ ❁ ❃ ❇❙ ❆ ❈ ❇ ❉ ✽ ❊ ❇ ❇ P ❁ ❂ ❆ ❬ ❆ ✾ ❁ ❄ ❁ ❃ ❁ ❂ ▼ ❋ ❁ ❭ ■ ❈ ❇ ❂ ❁ ❄ ❏ ■ ❁ ✽ ❆ ❇ ❃ ❃ ❄ ❣ ✈✇ ④ ③ ✇ s ③ ⑥ ④ ⑤ ④ ③ ②③ ① ✉ ✈ st qr ♦ ♣ ♦ ♥ ❧♠ ❯ ✐ ❇ ❊ ❑ ⑦ t ❆ ① s ③ ② ✈ ⑤ s ① ⑦ ❶ ⑦ ✈ ④ ⑧ ⑧⑨ ③ ⑦ ✈ ⑧ ⑩ s ③ ⑧ t ✈ s ③ ④ ❂ ❁ ✽ ❉ ❊ ❑ ❁ ❙ P ✿ ❙ ❑ ✽ ❑ ✾ ✽ ❬ ❊ ❂ ❇ ❃ ❇ ❊ ❂ ❑ ❬ ❄ ▼ ✿❁ ✾ ❋ ✽ ❊ ◗ ❄ ✽ ■ ❁ ❆ ❁ ■ ❙ ❇ ❊ ❂ ❑ ❆ ❆ ■ ✽ ❃ ✽ ❇❙ ❆ ❇ P ❇ ❉ ❏ ❊ ❬ ❩ ❁ ú ❁ ▲ ❆ ❂ ❁ ■ ❢ þ ❂ ❄ ❇❈ ❂ ❄ ❃ ❂ ✹ ✸ ✷ ✴ ✶ ✵ ÿ ý ❆ ✱ ✂ ■ ❁ ❈ ❇❙ ❆ ❘ ◗ ❂ P ❂ ❖ ❇ ■ ▲❈ ◆ ❂ ✽ ❂ ❄ ✽ ❃ ü õ ✰ ❄ ô ✁ ✖ ✒ ✛ ✂ ✑ ☛ ✄ ✔ ☎ ☞ ✆ ô ✝ ✞ ✂ ✄ ✡ ✠ ✟ ✂ ✣ ✤ ✯ ★ � ú ô ✩ ✎ ★ ✙ ✒ ✥ ☞ ✤ ✥ ✛ ✔ ☞ ✥ ☞ ☛ ✣ ✛✧ ✙ ✦ ✽ ✽ ❁ ❁ ■ ❭ ■ ❂ ❖ ❇ ❉ ❁ ■ ❂ ó ◗ ✽ ✾ ❂ ❁ ■ ô õ ❬ ● ✽ ❖ ❑ ❩ ð ❡ ❞ ❴ ❛ ❵ ❫ î ❫❴ ❪ ï ❯ ❬ ✿ ▲ ✽ ❑ ❭ ❆ ❄ ✿❁ ❙ ❑ ö ❚ ❯ ❯ ❄ ❁ ✽ ❉ ◗ ■ ❁ ● ❂ ÷ ú ❯ ❲ ❯ ❱ õ ö úû ✼ ● ❂ ô ❋ ❈ ▲❨ ❆ ó ❳ ❱ ø ❚ ❇❙ ❂ ❁ ■ ❳ ù ❱ (SETS WITH SPECIAL UNION-FIND OPERATIONS) b ∈ 191 . ✾❀✿ ❏✏❑ 3 ≤ b < N} h a,b (x) = ((ax+b) mod N) mod m a ∈ {1,…,N- ❋❀❈❍● ❉✓❊ ✘✚✙ ❅✏❆ H = {h a,b : 1 ≤ a < N ✾❀✿ ✕✗✖ ❉✓❊ ✑✓✒ ✾❀✿❁ ✍✏✎ m ✼✬✽ ☛✌☞ ❏❀▼ ✸✻✺ ❏✏❑ ❜✏❝ K ❏❀❤ {0,…,N- ❈❍● ✜✌✢ m - ñ✱ò - ✭✌✮ ❋✏❆ ✪✬✫ ✲✏✳✴ ✾❀✿ ❋✏❆

  96. ❷ ❭ ▼ ❪ ❫ ▼❴ ❵ ❬ ❙ ❬ ❛ ❜ ❝ ❬ ❬ ❢ ✐ ❢❥ ❦ ❧ ❢♠ ♥ ♦ ❦ ♣ ❦ ❭ ❩ ❤ ❘ ❉ ❊ ❋ ● ▲ ▼ ◆ ❖ P ▲ ◗ ▼ ▼ ◆ ❖ ❙❚ ❙ ❯ ❱ ❲ ❳ ◆ ❨ ❙ ♠ q ✳ ❺ ✉ ⑤ t s ❷ ❸ ② ❸ t ⑨ ❹ ❻ r ❾ ➁ ➂ ➁ ➃ ✉ ③ r ⑦ ➄ ✇ ❸ ⑤ rst r⑩ ✉ t s ② ③ st ④ r⑤ s t ⑨ ⑤ ❶ ⑥ ⑨ t ❶ ❷ ⑦ s t r⑩ ⑤ ⑥ ❂ ✻ ❷ ❁ ✾ ★ ✽ ✿ ✫ ✻ ❀ ✸ ❁ ✧ ★ ✭ ✧ ✹ ★ ✿ ❄ ❅ ✻ ✺ ✮ ✾ ✫ ✻ ✫ ✵ ★ ✰ ☛ ✄ ✂ ✕ ✥ ☛ ✦ ✧ ★✩ ★✩ ✯ ✳ ✾ ✴ ✲ ★ ✩ ✱ ✫ ✺✻ ✧ ★ ❸ ✻✽ ✧ ✭ ✫ ★ ✻ ❀ ✸ ❁ ✧ ★ ❁ ✭ ✹ ★ ✿ ✲ ✿ ✽ ✺ ✻ ✫ ✶ ✻ ❁ ✧ ✾ ❈ ❇ ✫ ✽ ✮ ✸ ★ ✯ ✰ ✳ ✴ ✲ ✵ ✶ ★✩ ✱ ✫ ✹ ★ ✺ ✻ ✧ ★ ✼ ✻✽ ✾ ✵ ✧ ✫ ✾ r⑩ ❸ ☎ ➤ ✩ ✶ ✿ ★ ✽ ✵ ✧ ➟ ✧ ✺ ✫ ✺ ★ ✾ ✧ ❆ ✧ ✺ ✻ ✺ ✶ ✵ ✧ ➟ ✿ ✶ ✶ ★ ➞ ❁ ✧ ➟ ✹ ★✩ ✾ ❆ ✧ ★ ✩ ❁ ➢ ✧ ★ ✭ ❁ ✧ ★ ✻➠ ✫ ✸ ★✩ ➡ ✹ ★ ➝ ★ ➧ ✧ ➟ ✯ ★ ✩ ✾ ❆ ✯ ✶ ✾ ✧ ✻ ✫ ✾ ✺ ✻ ✵ ✧ ➟ ✧ ✺ ✯ ❂ ➦ ✺ ✿ ➝ ✩ ✶ ✿ ★ ✽ ✵ ✧ ➟ ✧ ✺ ✧ ✹ ✽ ✿ ✫ ✻ ✧ ★✩ ✾ ✮ ✻ ➥ ✹ ➦➧ ✮ ✮ s s t ❶ ✉ ⑤ s➌ ⑤ ❸ ✉ t ➇ ➌ ➎ ② ⑦ r ❶ r ⑩ ✇ t ❸ t ➏ s ❸ ❸ r st r⑩ ⑦ ✉ ⑩ ➇ r ❶ rs ✉ s ⑦ r ❷ ⑩ ⑤ ⑥ ❶ ⑤ r ❸ ✉ ⑤ t s ✇ ⑦ ✺ r⑩ t ❸ r ⑩ t ➇ ➌ r⑤ ✈ ➎ ⑦ t ② ⑩ ❸ r ➒ ❹ ➓ ❽ ↔ ➀ ↕➙ ❁ ❸ ➑ ✉ s ✈ ➉ ❸ rs ➅ ✉ ✈ q rst ✉ t ② r ➅ ⑤ r ✇ s➑ ➈ ❶ r ➇ ⑩ ❶ ➇ ✝ ✼ ✂ Ý à ß á ×âã Ý ä Ù å Ï Ô Ù ÝÞ â á × Ü ç Ý â è ä Ð á ß Ú Õ ➺ ➴ ➳ ➹ ➪ ➳ ➾ ➪ ➻ ➫ ➼ ➳ ➽ × ➾ ❮ ❰ Ò Ó Ô Õ Ö ×Ø Ù Õ é × ➩➫ è Ý Ù Þ í á × Ü ç Ý â ä â Ð á é Õ × Þ ðñò õ â Ù ö÷ ì Ù Þ Ù å Ò Ô Õ ç à Ö ×ê × Ö ä Õ á × Þ ß Þ ß ã Ý á ä ë Ý ➺ ➲ ï ➎ ➑ ➀➒ ➓ ➎ ➆ ➎ ➔ → ➣ ➏ ➝ ➀ ➙ ➞ ➟ ➠ ➙ ➡ ➢ ➤ ➟ ➥ ➟ ➐ ➏ ➜ ➁ ❹ ❺ ❿ ➀ ➁ ➂ ➃ ❿ ➄ ➅ ➀ ➂ ➎ ➆➇ ➆ ➈ ➉ ➊ ➋ ➁ ➌ ➆ ➀ ➍ ➡ ➦ ➼ ➬ ➺ ➘ ➺ ➫ ➾ ➚ ➯ ➭ ➲ ➚ ➷ ➺ ➨➩ ➲ ➾ ➳ ➭ ➴ ➪➮ ➩ ➳ ➺ ➫ ✃ ➵ ➫ ➭ ➺ ➯ ➫ ➲➳ ➩ ➫ ➺ ➻ ➫ ➺ ➲ ➳ ➚ ➭ ➯ ➭➪ ➺ ➚ ➭ ➶ ➩➴ ➲ ➾ ➲ ➳ ï á ö Ô Ý â á × Þ Ö � ä Ý Ø ✁ Ø ✂ ✄ ☎ ✆ ✝ ✞ ✂ ☎ ✝ ☛ ç Ý ✂✌ ä Õ × Þ Þ Õ × ß á à Ö ë ú Ý â á â Ø Ù Ü í ç × Þ ☞ ✍ Ó ✚ ✆ ✂ ☎ ✝ ☛ ✘ ✂ ✎ ✗ ☛✙ ✣ ✒ ✤ ✆ ✝ ☛ ✑ ✂ ✄ ☎ ✆ ✝ ✞ ☎ ✑ ✆ ✂ ✎ ✌ ☛ ✏ ☛ ✑ ☎ ✆ ✝ ✞ ☎ ✖ ✔✌ ✕ ☎ ✔ ✠ ☛ ✞ ✂ ✝ ☎ ☞ Ð Ú Ö þ â þ æ Ù ß æ é æ ì þ æ æ Ý × ç Ù × ì Ý é Õ Ø × Ø è ô ä ü ì þ æÿ æ � â Ù × ú á � ú ó ü é à ï à ë Ø Ù ❰ ê û Ò à Ý ð ã â ý Ù á Ý ä × Ý ß Ù 194 Insert 3 ❷⑧❸ ❷①⑤ ➈⑧➉ j- Print(DeleteMin(S)); ⑥①➍ ⑥⑧⑦ for (j = 0; j < n; j++) for (j = 0; j < n; j++) ➄➆➅ MakeEmptySet(); ❣✬❤ t➋➐ ⑨➋➊ ❞✷❡ n DeleteMin ✈①✇ ✈①✇ ❿✷➀ ➛➜✻ ❍■❑❏ ❿✓➣ - ❼✬❽ ➔✓→ S = ∅ , false 193 ∅ . S. 3 ➹❀➘ ✱✢✲ ➵❀➱ ✛✢✜ info true ✪✬✫ key ✸✡✹ ✟✡✠ ➽✗➾ ✟✡✠ ❆✡❇ ✟✡✠ ➼✏➳ info. ✖✡✗ ➪❒❐ Û❀Ü MakeEmptySet(): ✒✓✄ ✵✷✶ ➛✗➜ IsEmptySet(S): DeleteMin(S): Insert(K,I,S): ➵✏➸ FindMin(S): ↔✏↕➙ ✱✢✲ ★❃❂ ø✏ù è✗× ó✌ô ä❀æ ❻❼❾❽ ÏÑÐ ✭✡✮ - ➧❀➨➩➫ ÏÑÐ ✪✬✫ î✬ï

  97. ➨ ❋ ❉ ✶ ✵ ❸ ❉ ✶ ✶ ❋ ✵ ❄ ✾ ❸ ✽ ❉ ❋ ✹ ✷ ✼ ✷ ✸ ✷ ✻ ✷ ❂ ✷ ❂ ✷ ✼ ✶ ❶ ❉ ✶ ❊ ✽ ✹ ✵ ❄ ❷ ❀ ❋ ✵ ✼ ✹ ✽ ✼ ✶ ❆ ❊ ❸ ❉ ✷ ❋ ❸ ❉ ❋ ✸ ✶ ⑨ ❋ ✼ ✿ ✼ ✶ ❄ ✺ ❽ ✽ ❄ ❾ ✷✸ ❇ ✿ ✵ ❋ ● ✼ ✷ ❄ ❅ ✽ ✵ ❋ ❂ ✼ ✷ ✾ ✶ ❁ ✽ ✼ ✷ ✵❹ ✽ ❸ ❹ ✷ ✸ ❉ ✼ ✿ ❉ ❺ ✹ ✵ ❋ ✵ ✿ ✼ ✶❻ ❼ ✻ ✹ ✷ ✼ ✽ ⑩ ✘ ❈ ✼ ✸ ✹ ✺ ✻ ✹ ✷ ✼ ✽ ✹ ✶ ✵ ✾ ✿ ● ✼ ✶❀ ❊ ✽ ❍ ✺ ❄ ✹ ✶ ❑ ▲ ▼ ✷ ❇ ❙ ✹ ❃ ✽ ✼ ✶ ✵ ❋ ✵ ✻ ❅ ● ✷ ✸ ✺ ✶❀ ✻ ✹ ✷ ➩ ✽ ✹ ✶ ✵ ✾ ✼ ✿ ✼ ◆ ❚ ☛ ④ ✐ ❥ ❝ ❜ ❦ ♥ ♠ ♦ s ♦ t ✇ ✇ ❜ ♠ ⑤ s ✇⑥ ♦ ♠ ⑤ ♣ ⑤ ⑦② ⑧ ✧ ❤ ❩ ❯ ❫ ❱ ❲ ❙ ❳ ❨ ❚ ❯ ❱ ❩❬ ❩ ❭ ❪ ❴ ❜ ❯ ❵ ❩ ❚ ❛ ❜ ❝ ❚ ❞ ❡ ❚❢ ❣ ✸ ✽ ❊ ❹ ❾ ✷ ✵ ✻ ✷ ✸ ❋ ✵✶ ✼ ✹ ❸ ✷ ✾ ✸ ❊ ✽ ✽ ❂ ❉ ✶ ✵ ✶ ➅ ❺ ✷ ❊ ❄ ❁ ✷ ✼ ✿ ✹ ❂ ❃ ✶ ✻ ✹ ✷ ❀ ✷ ✻ ✵ ✵ ✷ ❋ ● ✻ ✷ ✼ ✽ ❄ ✾ ❊ ❾ ✷ ✸ ✶❀ ✻ ✁ ✱✲ í ý ó ýù ò ✰ ➂ ✆ ø ò ö ý ☎ ✁ ö ñ ó î ø ô õ ø ù î ✝ õ ✁ ✻ ❀ ✹ ✷ ✼ ✽ ✹ ✶ ✵ ✾ ✼ ✿ ✼ ✶ ■ ý ✭ ✫ ✆ ð þ î û î ò î ü ó ✾ ✶ ❂ î ✵ ✼ ✷ ✸ ❄ ✾ ❊ ❾ ✷ ✸ ■ ✭ ò ✵ î ✆ ý ✮ õ ö ò î ✰ ✂ ó ➁ ❋ ✶ ✆ ● ❉ ✶ ✵ ❊ ✵ ❄ ✹ ✾ ✼ ✽ ✹ ✿ ❂ ✻ ❁ ✿ ✽ ❄ ✽ ❂ ❉ ✿ ❀ ✼ ➀ ❉ î ó ✵ ✝ ý ✁ ö ñ ó î ø ô õ ø ù î ➃ ö ✽ ❄ ✺ ❽ ✽ ❊ ✷ ❉ ✷ ✻ ✺ ✼ ✁ ò ý ò ó ò ✰ ù ýù ☎ û ù î ✁ ýù ö ☎ ø ✁ ö í ý ô õö ó ýù ò ✰ ➂ ✆ ✺ ✼ ✷ ✠ ✍ ✞ ✞ ☛ ✌✒ ✓✔ ✌ ✕ ✌ ☛ ✘✙ ✙ ☛ ✔ ✘ ✕ ✘ ✚ ✣ ✚ ✤ ✘✙ ☛ ✥ ✌ ✞ ✕ ø ò í î ï ðñ ò í ó ô õö ø ù ✠ ö ✝ ✞ ✏ ✑ ✞ ✞ ☛ ✌ ✍ ✞ ✟ ✠ ☛ ó û ó ô õ î ð ø ô õ ø ù î ✫ ò ✆ ✬ ö ó û î ù î ü í ö ✷❉ ñ ✘ ý ✦ ✣ ☛ ✌ ✚ ✘ ✦ ✛ ✦ ✧ ÷ ø ð î ó î ð ý õ ✂ ✆ ò í î ï ý ✆ ý Ô ➸❒ ❮ ➮ ➚ ➮ ❰ Ï Ð ➱ ➮ Ñ Ó ✃ Õ Ø Õ Ù Ü ß Ü Ó à Ø Üá ❐ ➸ Ó ➺ ➫ ➭ ➵ ➸ ➺ ➻ ➼ ➵ ➽ ➾ ➸ ➻ ➱ ➚➪ ➚ ➶ ➹ ➘ ➴ ➺ ➷ ➚ ➸ ➬ ➮ Õ à ö ö û î ò ÿ í î ð ó ý ó ò î þ ð ù � û ù î ✁ ýù ö ò ☎ ✁ î ð Ö íó à â Þ ã æ å é è ê ë î ÷ ö ô õ ö ú û î ù î ü ó ý õ ó í õö þ ù ✯ ✷ î ð õ � þ ý ✸ ✆ þ ý ✻ ò ✁ î ð ð � ✸ ù ò ✮ ý ✆ ✴ ✱✲ ✰ ✶ ✵❄ ö ✽ ✻ ❃ ❂ ✳ ✹ û ✿ ✼ ❁ ✷ ✶❀ ñ ✵ ò î ø ✼ î ✾ ✵ ✁ ✶ ✰ ö ✹ � ✽ ✝ ✼ ✷ ✹ ✹ ✷✸ û ò ✼ û ✮ ✿ ý ✆ û î ð õö ð î ö í ✁ ☎ ù ✮ ï î õ ✁ ñ ✵✶ ù ✝ ✁ ýù ö ò ❀ ☎ ✁ ö ✆ ✽ ù ❂ ✭ î ❆ ò ö ý ✮ ✼ î û ✸ ð þ î û î ò ÿ ✵ í î ✶ � ó ý ❄ ó ò ö ❇❈ î ð ù ✹ õ ✷ ✰ ✵ ✆ û þ î ö ò ✽ þ ❅ ☎ ✵ ✁ ö ✁ ö ñ ð î ❀ ý ð î ò ☎ ✯ ✁ ò þ û ù î ✼ õ ò ✰ ÿ ò 196 ✂✄✁ ✾✢❊ 3 ❂✗✶ FindMin ✂✄✁ ①③② ô①õ ✉✢✈ ❷✗❉ Heaps) ♣rq ô①õ ✾✢✼ ❧✢♠ ✷✪➄ ❖P❘◗ - ❅❿✶ ●✗❀ ❋③✸ – DeleteMax() 195 B- DeleteMin()? FindMax 3 ð ✪✩ Red-black ✛✢✜ ✂✄✁ (3) ô①õ ø⑧ù Delete? Insert(), FindMin ❈❏■ Ý✬Þ FindMax ô①õö ✟✗✖ ✟✡✠ Ú✷Û (2) ï➆ðñò - ✠✎★ LookUp? ì✬í Ö➆× ✌✎✍ DeleteMax? ✞☞☛ ✂✄✁ ✂✄✁ ✺✢✻ ✟✡✠ Ò✷Ó ç①è ô①õ ➯➲❑➳ AVL - (1) ä⑧å ✟✡✠

  98. ➆ ➚ ➶ ❒ ➻ ❒ ➸ ➽ ➾ ➶ ü ➹ ➱ ➘ ➵ ➬ Ö ➶➘ ➹ ➻ ➻ û ø é ç ù ë ú í ➚ ➽ ➹ ➶ ➴ ➴ Ð ➸ ➪ ➽ ➹ ì ➻ ❮ ➹ ➱ ➘ ý ➻ ➺ ➬ ❒ ➚ ➸➪ ➽ ➻ ➽ ➼ ➺ ➼ ➼ ➶ ➚ ❒ ➻ ➽ ➾ ➚ ➶ ➻ Ñ ➶ ➘ ➶ ➮ Ñ ➹ ➶ ➮ ë è Ñ çè ➫ ➩ ➭ ➯ ➲ ➯ æ ç ➝ é ê èë ì ë í î ➧ ➞ ➻ ➟ å ➔ ➎ ➜ ➝ ➞ ➇ ➠ ➦ ➎➡ ➢ ➝ ➔ ➝ ➤ ➥ ➽ ➵ ó ➱ ➼ ➽ Ñ ➚ ➸ Ð ➹ ➵ ➹ ➬ Ö ➶ Ø ð ñ çò ➶ ➻ ➼ ï ➺ Ò ❒ ➼ ❮ ➹ ➶ ➱ Õ ➺ ➶ ➹ ➶ Ð ➹ ➵ ➾ ➵ ➹ ➙ Ð ➵ Ñ ÿ ➴ ➱ ➽ ➶ ➹ ➻➼ ➱ ➽ ➻ ➵ ➼ ➺ Ò ➮ ➵ ➼ ➷ ➶ ➚ ➹ ➶ ➴ ➴ ➶ Ò ➾ ❒ ➻ ➚ ➻ ➮ ➼ Õ ❒ ❮ ➪ ➺ Õ ✃ ➸➪ ➽ ➻ Ð ➻ Ñ ➚ ➽ Õ ❐ ➴ ➴ ➸ Ø ➶ ➼ ➹ ➼ ➶ ï ➱ ➹ ➸➪ ➮ ➶ ❒ ➵ ➼ ➸ ❐ Ò ➽ ➾ � ➱ ➸ ➻ ➚ ➚ ➶ ➴ ➶ ➽ ÿ ❮ ➸➪ ➮ ➽ ➻ ➹ ➱ ➚ ➬ ❒ ➶ þ ➶ ➵ ➵ ➱ ➮ ➵ ➸ ➹ ➻ ➶ Ø ➼ Ñ ➹ ➱ ➹ ➶ ü ➼ ❒ ➮ ➽ ➹ ➻ ➚ ➶ Õ ✃ ➸➪ ➻ ➻ Ð ➻ ➺ Ñ ➽ ÿ ➸ ➬ ➼ ➼ ➶ ➶ ❒ ➼ ➺ ➶ Ð ➬ ➷ ➵ ➼ ➶ ➹ ➸ ➮ ➶➼ ❒ ➽ ➏ ➎ ↕ ➵ ➹ ➻ Ð ➬ ➷ ➸➪ ➸ ➶ ➼ Ñ ➹ ➼ ➺ ❮ ✃ Ð ➽ Ð ➶ ➸ ❒ ➸ ➹ ➼ ➺ ✃ ➮ ➬ ➵ ➾ ➮ ➻➼ ➚ ➶ ➻ ➻ ➹ ➶ ➷ Ô ➻ ➚ ❒ ➼ ↔ ➵ ➶ ➸ ➪ ➽ ➻ ➹ ➱ ➵ ➼ ➚ ➼ ➹ ➷ ➽ Ò ➹ ➸ ➸ ➘ ➸➪ ➬ ❒ ➾ ➚ ❒ ➵ ➸➪ ➻ ➮ ➸ Ö ➎ ➛ ➔ ➎ ➜ ➝ ➞ ➟ ➙ ➠ ➎➡ ➢ ➝ ➔ ➝ ➤ ➏ ↕ ➦ ➍ ➈ ➉ ➍ ➎ ➏ ➐ ➑ ➒ ↔ ➓ ➎ ➏ ➐ ➔→ ➔ ➣ ➥ ➞ ✃ ➘ ➸ ➪ ➵ ➷ ➬ ➻ ➚ ➼ ➶ ➪ ➴ ➸ ➮ ➸ ➼➱ ➹ ➻ ➝ ➸➺ ➧ ➫ ➩ ➲ ➯ ➳ ➵ ➻➼ ➴ ➽ ➸➪ ➶ ➪ ➹ ➸ ➼ ➬ ➶ ➶ ➽ ➚ ➻ ➬ ➚ ➶ ➼ ➻ ➽ ➵ ➼ ➺ × ➘ ❒ ➱ ➶ ➶ ❒ ❐ ➮ ➹ ➸ ➚ ❒ ➵ ➚ ➸ ➮ ➸ ➵ ➻ ➬ ➼ ➹ ➸ ➒ ➍ ➎ ➏ ➐ ä ➍ ➓ ➈ ➎ ➏ ➐ ➔→ ➔ ➣ ➉ ➇ ➻ ➸ ➹ ➶ ➷ ➽ ➾ ➚ ❒ ➆ ➾ ➚ ❒ ➵ ➸ Ø ➺ ➽ ❐ ➷ ➮ ➻ ❒ ➶ ➶ ➹ ➻ Õ ➽ ❐ ➸➪ Ó ❮ ➵ ➴ ➽ ❰ ➸ Õ ➵ × ➹ ➶ ➶ ➴ ➸ ➶ ➚ ➵ ➹ ➼ ➸ ❒ ➚ ➬ Ü áâ ã ➴ ➴ ❮ ➾ Ö ➵ à ➱ ➸ ➹ Ð ➪ ➸ ➹ ➮ ➶ 198 ❒③➸ 3 ❮✢Ð ❮✢Ð DeleteMin() ➾ ✁� ➬Ï❰ ö✗÷ ÿ❿➸ ➨✢➩ (swapping ô✡õ ➊➋❘➌ - ❮✢Ð 197 ❐✗➚ ➶ÙØ Insert() & DeleteMin() 3 O(log n). ➾✗➚ ➾✗➚ ➶✪❰ ➾✗➘ ➭③➯ DeleteMin() ❮✗Õ ❐✗Ó ❮Ï❰ ➾✗➚ ➨✢➩ ➊➋❘➌ ÝÞ✄ß - Ú✗Û

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