d diochnos di uoa gr

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Apr 30, 2023 •174 likes •1.34k views

  1. � ✸ ✻ ✺ ✆ ✌ ✍ ✠ ✓ ✹ ✍ ☛ ✎ ✠ ✿ ✠ ✆ ✠ ✓ ✟ ✠ ✁ ✁ ✞ ✎ ☛ ✞ ✓ ✄ ✆ ☎ ✞ ✩ ✆ ☛ ✎ ✔ ✆ ✠ ✏ ✍ ✄ ✟ ☛ ✎ ✟ ☛ ☛ ✟ ✏ ✁ ✠ ✝ ✠ ✟ ❀ ✻ ☛ ✞ ✄ ✻ ✑ P ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ ❉ ❊ ❊ P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❋ ❈❉ ☛ ✝ ✞ ✟ ✆ ✠ ☎ ✻ ✠ ✸ ✞ ✁ ✁ ✠ ✺ ❇ ✆ ✌ ✍ ✠ ✓ ✹ ✍ ☛ ✎ ✠ ✸ ✟ ❃ ✿ ☛ ✸ ✂ ✎ ✔ ✻ ✿ ✄ ✆ ✡ ✿ ✎ ✌ ✓ ✔ ✠ ✁ ✻ ✞ ❂ ✞ ✌ ✍ ✞ ✎ ✌ ✻ ✍ ✄ ✿ ✁ ✧ ✔ ★ ✭ ✤ ✳ ✤ ✬ ✮ ✧ ✰ ✧ ✮✯ � ✩ ★ ✰ ✧ � ★ ✆ ✫ ✣ ✵ ✮ ✩ ✥ ✬ ✮ ✧ ✠ ✁ ✁ ✏ ✍ ✹ ✻ ✎ ✔ ✸ ✿ ✆ ✠ ✔ ✂ ✠ ✍ ☛ ✄ ✻ ✔ ✸ ✿ ✄ ✆ ✡ ✿ ✎ ✌ ✓ ✓ ✞ ✻ ✑ ✿ ✁ ✞ ✺ ✆ ✌ ✍ ✠ ✓ ✹ ✍ ☛ ✎ ☛ ✿ ✺ ☎ ✠ ✻ ✎ ☛ ✌ ✁ ✻ ✍ ✔ – p. 5/43 ❚❱❯ ▲❆▼ ❄❆❅ ✏✑✸ Translocation: ✒✝☛

  2. � ✞ ✞ ✓ ✄ ☎ ✻ ☛ ✞ ✟ ✆ ✠ ☎ ✻ ✠ ✸ ✁ ✎ ✓ ✸ ✠ ✎ ☛ ✍ ✹ ✠ ✁ ✍ ✌ ✆ ✺ ✝ ✠ ☛ ✞ ✟ ✌ ✩ ✠ ✟ ❀ ✻ ☛ ✞ ✟ ✆ ✠ ✄ ✻ ✺ ✆ ✍ ✁ ✿ ✁ ✠ ✟ ✓ ✠ ✆ ✸ ✠ ✠ ✎ ☛ ✍ ✹ ✓ ✟ ❀ ✁ ◆❖ ❃ ❇ ❈❉ ❊ ❋ ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ● ☛ ❘ ▼ ❙ ❉ ▼ ■ ❅ ◗ ▼ ■ ❈ P ❊ ❉ P ✟ ✎ ✻ ✌ ✓ ☛ ✞ ❂ ☛ ✍ ✓ ✞ ✏ ✍ ✌ ✆ ✺ ☛ ✿ ✎ ☛ ✆ ✍ ✹ ✓ ✠ ✍ ✌ ✺ ☛ ✠ ✏ ❂ ✄ ✓ ✞ ✠ ✝ ✏ ✂ ✌ ✞ ❂ ✞ ✻ ✠ ✔ ✞ ✓ ✌ ✎ ✿ ✡ ✆ ✍ ✎ ✿ ☛ ✌ ✆ ✺ ✞ ✁ ✿ ✁ ✌ ✁ ✠ ✿ ✄ ✍ ✻ ✄ ✻ ✟ ✰ ★ ★ ✩ � ✮✯ ✧ ✧ � ✮ ✬ ✤ ✳ ✤ ✭ ✧ ★ ✔ ✮ ✎ ✁ ✁ ✧ ✰ ✧ ✬ ✆ ✥ ✩ ✮ ✵ ✣ ✫ ✍ ✠ ✓ ✿ ✄ ✔ ✸ ✿ ✄ ✆ ✡ ✿ ✎ ✌ ✓ ✔ ✸ ✑ ☛ ✏ ✄ ☛ ☛ ✎ ✞ ☛ ✟ ✍ ✆ ✏ ✠ ✆ ✔ ✎ ☛ ✍ ✻ ✠ ✁ ✻ ✌ ✎ ✻ ✠ ☎ ✿ ✔ ✑ ☛ ✎ ☛ ✍ ✹ ✂ ✍ ✺ ✎ ✔ ✠ ✆ ✿ ✸ ☛ ✔ ✻ ✹ ✍ ✓ ✞ ✻ – p. 5/43 ❚❱❯ ▲❆▼ ❄❆❅ ✏✑✸ Translocation: Fission: ✒✝☛

  3. � ✓ ✓ ☛ ✞ ❂ ☛ ✍ ✌ ✏ ✞ ✍ ✌ ✆ ✺ ☛ ✻ ❀ ✟ ✿ ✎ ✸ ✍ ☛ ✎ ☛ ✍ ✹ ✓ ✠ ✌ ☛ ✆ ✺ ✠ ✏ ❂ ✄ ✓ ✞ ✟ ✠ ✶ ☛ ✞ ☛ ✻ ☎ ✄ ✓ ✞ ✎ ✆ ✞ ✁ ✁ ✠ ✟ ✓ ✠ ✆ ✟ ✠ ✎ ✆ ☛ ✍ ✹ ✓ ✠ ✍ ✌ ✺ ☎ ✝ ✠ ✁ ✁ ✞ ✸ ✠ ✻ ✟ ✏ ✸ ❈❉ ❍ ❋ ❍ ❍ ● ❋ ❊ ❇ ❏ ❃ ✟ ☛ ✍ ✌ ✓ ✏ ✍ ■ ❑ ✆ ◗ ▼ ❙ ❉ ▼ ■ ❅ ❘ ■ ❉ ❈ P ❊ ❉ P ▼ ● ◆❖ ✌ ✺ ✠ ✎ ✞ ✝ ❂ ✻ ✝ ✓ ☛ ☛ ✠ ✍ ✹ ✓ ✠ ✍ ✌ ✆ ✺ ✡ ✻ ✠ ✂ ☎ ✻ ☛ ✻ ☎ ✻ ✝ ✠ ✞ ✎ ✆ ☛ ✿ ✞ ☛ ✟ ✞ ☛ ✩ ✿ ✠ ✎ ✁ ✠ ✿ ✄ ✍ ✻ ✌ ✞ ☛ ✍ ✌ ✞ ❂ ✞ ✻ ✠ ✂ ✁ ✿ ✓ ☛ ✻ ✠ ☎ ✁ ✿ ✑ ☛ ✎ ✍ ✁ ✹ ✓ ✠ ✍ ✌ ✆ ✺ ✞ ✔ ✌ ✌ ✮✯ ★ � ✧ ★ ★ ✩ � ✧ ✫ ✰ ✧ ✮ ✬ ✤ ✳ ✤ ✭ ✆ ✣ ✎ ✁ ✿ ✡ ✆ ✄ ✿ ✻ ✔ ✎ ✁ ✵ ✧ ✰ ✧ ✮ ✬ ✥ ✩ ✮ ✎ ✎ ✔ ✟ ✟ ☛ ☛ ✍ ✎ ✞ ☛ ✄ ✁ ✍ ✏ ✠ ✆ ✻ ✎ ☛ ✏ ✠ ☛ ✞ ✺ ✍ ✻ ✄ ✠ ✆ ✟ ☛ ✝ ✠ ✻ ❀ ✟ ✓ ✠ ✹ ✆ ☛ ✌ ✹ ✠ ✆ ✿ ✸ ✔ ✎ ✻ ✍ ✂ ✓ ✞ ✻ ☛ ✺ ✔ ✍ ✔ ✠ ✿ ✡ ✑ ✸ ✔ ✓ ✌ ✎ ✿ ✆ ✏ ✄ ✿ ✸ ✔ ✻ ✄ ✍ ✆ – p. 5/43 ❚❱❯ ▲❆▼ ❄❆❅ ✏✑✸ Translocation: Fission: Fusion: ✒✝☛

  4. � ✌ ✂ ✠ ✻ ✞ ❂ ✡ ✍ ✎ ☛ ✄ ✻ ✞ ✁ ✁ ✠ ✓ ✝ ✻ ✄ ✡ ❂ ✠ ✝ ✸ ☛ ✄ ✠ ❂ ✔ ✂ ✔ ✏ ✍ ✠ ✄ ✄ ✻ ☛ ✁ ☛ ✍ ☎ ✞ ✄ ✠ ✞ ✝ ✿ ✏ ✂ ✞ ✻ ✡ ✿ ✡ ✞ ❂ ✠ ✝ ✸ ✟ ✕ ☛ ✠ ✍ ✄ ✎ ✆✞ ✟ ✗ ✔ ✻ ☛ ✔ ✟ ✟ ✹ ✻ ☛ ✻ ☛ ☛ ✞ ✎ ☛ ✎ ✞ ✆ ✄✌ ☛ ❂ ✿ ✁ ✺ ✣ ✓ ✎ ✠ ✿ ✗ ✞ ✸ ✎ ✌ ✻ ✂ ✠ ✻ ✻ ✠ ✍ ✎ ✻ ✠ ✝ ☛ ✿ ✏ ✠ ✍ ✂ ✠ ✻ ✞ ❂ ✡ ✌ ☎ ✞ ✝ ✏ ☛ ✿ ☛ ✞ ✎ ✠ ✻ ✄ ✎ ☛ ✞ ✿ ✆ ✠ ✟ ✎ ✸ ✁ ✝ � ✁ ✁ ✔ ✁ ✠ ✺ ✔ ✞ ✹ ✻ ☞ ✠ ✻ ✸ ✓ ❂ ✁ ✟ ✏ ✽ ✆ ✏ ☎ ✔ ☛ ✍ ☛ � ✿ ✏ ✓ ✎ ✞ ✞ ✻ ■ ▼ P ❉ ❊ P ❈ ◗ ◆❖ ❘ ❅ ■ ▼ ❉ ❙ ▼ ● ❉ ✹ ❊ ✻ ✔ ✟ ❃ ❇ ❈❉ ❋ ❑ ● ❍ ❍ ❋ ❍ ■ ❏ ✞ ✄ ✏ ✟ ✠ ✻ ✞ ❂ ✞ ☛ ✹ ✸ ✻ ☛ ✻ ☛ ✟ ☛ ✂ ✄ ☛ ✝ ✁ ✠ ✝ ✸ ✎ ✠ ✸ ✎ ✿ ☛ ✆ ✞ ✂ ✠ ✻ ✎ ✎ ☞ ✌ ✻ ✎ ✞ ✍ ✄ ✎ ✿ ☛ ✡ ✓ ✄ ✞ ✎ ✠ ✓ ☛ ✻ ✞ ✄ ✆ ✄✌ ✻ ✁ ☎ ✍ ✿ ✡ ✌ ✸ ✍ ✿ ✠ ✆ ✄ ✞ ✞ ✄ ✄ ✏ ✿ ✌ ✸ ✿ ✆ ✞ ✄ ✻ ☛ ✎ ✠ ✝ ✸ ✆ ✻ ✝ ✑ ✞ ❂ ✡ ✌ ✍ ☛ ✂ ☛ ✡ ✻ ✄ ✎ ☛ ✞ ✾ ✓ ✍ ✠ ☎ ✆ ✠ ✟ ✄ ✞ ✍ ✻ ✠ ✠ ✝ ✻ ☛ ☛ ✿ ✠ ✿ ✎ ✿ ✝ ✄ ✆✞ ✁ ✞ ☎ ✠ ✍ ✝ ✄ ✏ ✁ ✠ ✝ ✸ ☛ ✻ ✂ ✎ ✟ ✬ ✰ ✦ ✮ ✧ ✁ ✠ ✯ ✻ ✎ ☛ ✸ ❂ ✞ ☛ ✣ ✠ ✞ ✧ ✩ ✯ ✰ ✩ ✤ ✮ ✥ ✁ ✴ ✧ ✛ ✤ ✧ ✰ ✩ ✿ ✓ ☛ ☎ ✠ ✝ ✍ ✄ ✎ ☛ ✞ ✸ ✁ ✁ ✠ ✝ ✻ ☎ ✻ ✿ ✄ ✎ ✝ ✹ ✓ ✄ ✞ ✎ ✠ ✸ ✎ ✿ ☛ ✆ ✞ ✂ ✠ ✻ ✻ ☛ ✗ ✍ ✠ ✍ ✞ ✡ ✠ ✝ ✄ ✠ � ✿ ✏ ✓ ✎ ☛ ✻ ✻ ✎ ✎ ✎ ✆✞ ✓ ✗ ☛ ✝ ✗ ✔ ✓ ✎ ✠ ✄ ✆ ✎ ✓ ✔ ✍ ✟ ✟ ✎ ✞ ✟ ✞ ✄ ✾ ✆ ✆ ✞ ✡ ✄ ✞ ✎ ☛ ✿ ✓ ✡ ✝ ✍ ☎ ✠ ✁ ✡ ✠ ✞ ✝ ✍ ✄ ✍ ✄ ✟ ☛ ✄ ✟ ✔ ✔ ✒ ✻ ✂ ✌ ✂ ☛ ✓ ✞ ✄ ✸ ✔ ✎ ☛ ✍ ✎ ✁ ✠ ✠ ✝ ✍ ✄ ✓ ✎ ❂ ✹ ✄ ✄ ✆ ✠ ✄ ✆ ☎ ☛ ✠ ✍ ☛ ✟ ☛ ✁ ✗ ✔ ✄ ✍ ✎ ✆✞ ✟ ✗ ✟ ✼ ✞ ✓ ✄ ✠ ✝ ✍ ✄ ✻ ☛ ✺ ☎ ✍ ✝ – p. 6/43 ❚❱❯ ✒✝✠ ✒✝☛ ✔ ✒✑ ✏✑✸ ▲❆▼ ✏ ✷✎ ☎ ✷✿ ❄❆❅ ✒✝☛ ’ ✒✝✠ ✒✝☛ ✗✑✸ d ✏✑✸ ✗✑✸ ✏ ✑✸ ❀✖✺

  5. � ☛ ✌ ✻ ✄✌ ✆ ✞ ✎ ✎ ✿ ☛ ✟ ☛ ✻ ☛ ✻ ✹ � ✏ ☛ ✾ ✌ ✿ ✄ ✟ ✔ ✻ ✞ ✸ ✿ ✡ ✆ ✌ ✻ ✏ ✁ ✠ ✟ ✞ ✎ ✏ ✠ ✁ ✏ ✻ ✞ ✂ ✿ ✸ ✝ ✸ ☛ ✎ ✻ ✠ ✻ ✝ ✎ ❂ ✻ ✞ ✻ ✠ ✂ ✸ ✄ ✎ ✠ ✠ ✂ ✞ ✆ ☛ ✿ ✸ ✝ ✸ ✠ ☎ ❋ ■ ❍ ❋ ❍ ❍ ● ❊ ❑ ❈❉ ❇ ❃ ✟ ✗ ✾ ✆ ❏ ❉ ✍ ◗ ▼ ❙ ❉ ▼ ■ ❅ ❘ ■ ◆❖ ❈ P ❊ ❉ P ▼ ● ✠ ✗ ✿ ✡ ✏ ✁ ✠ ✿ ✞ ☛ ✻ ✄ ✏ ☛ ✍ ✔ ✁ ☎ ✏ ✆ ❂ ✓ ✟ ✎ ✞ ✻ ✄ ✂ ✠ ✓ ✏ ☛ ✟ ✞ ✍ ✄ ✓ ✠ ✁ ✠ ✯ ✞ ✍ ✁ ✠ ✎ ✓ ✞ ✺ ✞ ✄ ✁ ✠ ✎ ✞ ☛ ✻ ✡ ✄ ✿ ✗ ✞ ✹ ☛ ✟ ☛ ✻ ☛ ✻ ✟ ✸ ☛ ✞ ❂ ✞ ✻ ✠ ✂ ✠ ✁ ☛ ★ � � ☎ ✧ ✮ ✦ � ✰ � ✣ � � ✢ ✤ ✭ ✥ ✧ ✭ ✤ ✵ ✴ ✳ ✤ ✭ ✩ � ✫ ✥ ✪ ✦ ★ ★ ☎ ✂ ✦ ✎ ☛ ✎ ✁ ✺ ✝ ✠ ✁ ✔ ✁ ☛ ☛ ☛ ✞ ✍ ✄ ✍ ✝ ✞ ✡ ✂ ☎ ✍ ✁ ✁ ✁ ✞ ☛ ✞ ✄ ✠ ✓ ✻ ✌ ✡ ❂ ✞ ✻ ✂ ✠ ✆ ✎ ✆ ✿ ✎ ✞ ☛ ✠ ✻ ✟ ✏ ✠ ✎ ✞ ☛ ✝ ✠ ✄ ☛ ✻ ✆ ☛ ✎ ✄ ✍ ☛ ✝ ✿ ✠ ✻ ✄✌ ✻ ☎ ✍ ✞ ✿ – p. 7/43 ❚❱❯ ✬✜✭ ★✜✩ ▲❆▼ ❄❆❅ ✒✝✠ ’ ✔ ✒✑ ✽✖✠

  6. � ✎ ✡ ✂ ✻ ☛ ✺ ✡ ✄ ✁✂ ✠ � ✠ ✏ ✄ ✿ ☛ ✔ ❂ ✄ ✟ ✟ ✗ ✾ ✆ ✠ ✻ ✞ ✗ ✏ ✿ ✠ ✸ ✞ ✎ ✄ ✍ ☛ ✻ ✾✌ ✍ ✓ ✿ ✸ ✞ ✄ ✓ ✗ ✆ ✔ ✎ ☛ ✆ ☛ ✍ ✟ ✡ ✸ ☛ ✍ ✔ ✁ ☎ ✏ ✆ ✿ ✠ ✎ ✌ ✡ ✿ ✄ ✟ ✔ ✻ ✞ ✾ ✸ ✡ ✯ ✌ ✄ ✻ ✞ ✓ ✻ ✄ ✂ ✠ ✓ ✏ ✎ ☛ ✞ ✍ ✄ ✠ ☛ ✁ ✠ ✟ ✓ ✏ ❂ ✏ ✁ ✠ ✿ ✞ ✠ ✄ ✏ ✎ ❈❉ ❇ ❃ ✟ ✾ ✠ ✆ ✎ ✓ ✞ ✻ ❋ ☛ ✟ ✞ ✎ ☛ ✍ ✌ ✓ ✠ ✍ ✌ ❊ ● ✺ ❊ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❉ ❍ P ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ✆ ✏ ✸ ✄ ❂ ✞ ✻ ✠ ✂ ✻ ✌ ✎ ✞ ✆ ✞ ✓ ✌ ✔ ✎ ✓ ✆ ✠ ✾ ☛ ✞ ❂ ✞ ✠ ✡ ✻ ✿ ✔ ☛ ☛ ✺ ✞ ✻ ✠ ✍ ✻ ✏ ✔ ✂ ✆ ✍ ✄ ☛ ✻ ✻ ☛ ✓ ✏ ✠ ✆ ✠ ✿ ✻ ✆ ✁ ✹ ✞ ✎ ☛ ✎ ☛ ✟ ☛ ✻ ☛ ✻ ✟ ✄✌ ☛ ✞ ❂ ✞ ✻ ✠ ✂ ✸ ✗ ✁ ✿ ✆ ✻ ✎ ✞ ✠ ✻ ✠ ✍ ✞ ✄ ✟ ✠ ✆ ✿ ☛ ✿ ✎ ✻ ✏ ✠ ✎ ✞ ☛ ✿ ☛ ✝ ✠ ✠ ✓ ✻ ☎ ★ ☎ ✂ ✦ ✥ ✧ ✰ ✥ � � ✧ ✦ ✮ ✦ ★ � � ✣ � � ✢ ✤ ✭ ★ ✪ ✞ ✄ ✺ ✞ ✁ ✄ ✠ ✎ ✞ ☛ ✻ ✡ ✠ ✫ ✞ ✁ ✭ ✵ ✴ ✳ ✤ ✭ ✩ � ✤ ✝ ☎ ☛ ✠ ☛ ✿ ✻ ✸ ✌ ✝ ✠ ✎ ✸ ✝ ✁ ✞ ✏ ✻ ✞ ✂ ✏ ✍ ✿ ✝ ✸ ☛ ✆ ✂ ✻ ❂ ✟ ☛ ✻ ☛ ☛ ✻ ✹ ✟ ☛ ✞ ✞ ✠ ✻ ✎ ✠ ✂ ✸ ✞ ✆ ✄ ✎ ✄✌ ✻ ✎ � ✎ ✄ ✝ ✠ ✁ ✔ ✁ ✁ ☛ ☛ ✞ ✍ ✿ ✡ ✍ ✝ ✠ ✂ ☎ ✆ ✎ ☛ ✎ ✄ ✺ ☛ ✠ ✞ ✻ ✞ ✿ ☎ ✍ ☛ ✁ ✏ ✁ ✁ ☛ ✂ ✞ ✍ ✄ ✓ ✻ ✌ ✡ ❂ ✞ ✻ ✠ ☛ – p. 7/43 ❚❱❯ ✬✜✭ ✒✝✠ ★✜✩ ▲❆▼ ❄❆❅ ☎ ✑✸ ✒✝✠ ’ ☎ ✑✸ ☎ ✑✸ ✔ ✒✑ ✽✖✠

  7. � ✟ ✞ ✓ ✎ ✆ ✠ ✾ ✁ ✻ ✎ ✔ ❂ ✄ ✟ ☛ ✄ ✎ ☛ ✡ ✌ ☛ ☛ ✿ ✏ ✺ ✆ ✍ ✟ ✠ ✓ ✌ ✍ ☛ ✎ ✞ ✎ ☛ ✞ ✺ ✸ ✔ ✄ ✻ ☛ ✓ ✏ ✠ ✁ ✔ ✓ ✔ ✍ ✄ ✎ ✝ ✎ ✎ ✠ ✠ ✝ ✠ � � ✂ ✿ ✆ ✎ ✏ ✞ ✒ ✔ ✟ ✄ ☛ ✿ ✺ ✻ ✿ ✸ ✸ ✠ ✿ ✠ ✡ ✄ ✠ ✎ ✞ ❂ ✞ ☛ ✾ ✠ ✆ ✞ ✄ ✎ ✄ ☛ ✎ ✔ ✯ ✗ ✓ ✞ ✍ ✸ ✓ ✏ ✍ ✾✌ ✻ ☛ ✓ ✔ ✠ ☛ ✠ ✏ ✓ ☛ ✻ ✻ ✄ ✠ ✆ ✔ ✂ ✔ ✏ ✻ ✍ ✆ ✿ ✓ ✻ ✄ ✞ ✆ ✞ ✎ ✌ ✂ ✍ ✠ ✻ ✞ ❂ ✡ ✌ ✻ ✠ ✆ ☛ ✄ ✝ ✁ ✠ ✂ ✄ ✻ ✎ ✄ ✞ ✟ ☛ ✻ ☛ ✟ ☛ ✾ ✓ ☛ ✻ ✓ ✄ ✻ ☛ ✾ ☛ ✄ ✔ ✡ ✺ ✆ ✗ ✓ ✞ ✍ ✎ ✓ ✠ ■ ▼ P ❉ ❊ P ❈ ◗ ◆❖ ❘ ❅ ■ ▼ ❉ ❙ ▼ ● ❉ ✟ ❊ ✄ ✝ ✟ ❃ ❇ ❈❉ ❋ ❑ ● ❍ ❍ ❋ ❍ ■ ❏ ✏ ✆ ✏ ✠ ✄ ✞ ❂ ✹ ✻ ✍ ✻ ✔ ✞ ✺ ☛ ✿ ☛ ✆ ✠ ✸ ✓ ✓ ✄ ☎ ✁ ✔ ✍ ☛ ✄ ✓ ☛ ✔ ✸ ☛ ✿ ✏ ✓ ✎ ✝ ✡ ✠ ✏ ☛ ✾ ✠ ✏ ✔ ✁ ✓ ✌ ✔ ☛ ✝ ✎ ✗ ✍ ✿ ✻ ✡ ☛ ✆ ☛ ✻ ✎ ✞ ✓ ✎ ✠ ❂ ✾ ✹ ✻ ✂ ✠ ✻ ✞ ✆ ✆ ✿ ☎ ✆ ✠ ✟ ✄ ✞ ✍ ✻ ✞ ✠ ✝ ✻ ☛ ✍ ✄ ✎ ✿ ☛ ✎ ✝ ✞ ✆ ✄✌ ✻ ✿ ✠ ☛ ✎ ✿ ☛ ✞ ✎ ✠ ✏ ✻ ☛ ✆ ☛ ✻ ✠ ✻ ✞ ❂ ✡ ✌ ✓ ☛ ✄ ✍ ✞ ☛ ✞ ✁ ✁ ✂ ✡ ☎ ✞ ✂ ✠ ✝ ✍ ✄ ✍ ☛ ✺ ☛ ✁ ✁ ✔ ✁ ✠ ✝ ✎ ✎ ☛ ✪ ✦ ✂ ☎ ★ ★ ✦ ✫ ✧ ✤ � ✩ ✭ ✤ ✳ ✴ ✥ ✰ ✭ � ✭ ✤ ✢ � � ✣ � ✥ ★ ✦ ✮ ✧ ☎ � � ✵ ✁ ☛ ✞ ✸ ✂ ✠ ✻ ✞ ❂ ☛ ✁ ✟ ✹ ✻ ☛ ✻ ☛ ✟ ✗ ✿ ✞ ✎ ✠ ✄ ✡ ✻ ☛ ✞ ✠ ✠ ✄ ✁ ✞ ✺ ✞ ✓ ✎ ✁ ✞ ✍ ✏ ✞ ✿ ✠ ✁ ✏ ❂ ✓ ✻ ✟ ✠ ✁ ✠ ✓ ✄ ✍ ☛ ✡ ☛ ✠ ✟ ✄ ☎ ✌ ✸ ✎ ✿ ✄ ✆ ✏ ☎ ✁ ✔ ✍ ☛ ✞ ✎ ✻ ✄ ✿ ✏ ✎ ✠ ✠ � ✁✂ ✡ ✔ ✺ ☛ ✻ ✂ ✡ ✻ ✄ ☛ ❂ ✏ ✟ ✓ ✠ ✂ ✄ ✻ ✞ ✗ ✄ ✍ ✠ ✆ ✾ ✗ ✟ ✟ ✔ ✿ ✞ ✝ ✞ ✎ ✻ ✠ ✂ ☛ ✸ ✾ ✄ ✎ ✻ ✠ ✂ ✸ ✿ ✞ ✝ ✁ ✠ ✝ ✸ ✎ ✠ ✞ ✞ ✸ ✿ ☛ ✂ ✏ ✆ ❂ ☛ ✻ ✻ ✸ ✡ ✆ ✞ ✌ ✻ ✏ ✁ ✠ ✿ ✏ ✿ � ✌ ✄✌ ✟ ✟ ✹ ✻ ✻ ☛ ✻ ☛ ☛ ✆ ✎ ☛ ✎ ✠ ✻ ✞ ✏ – p. 7/43 ❚❱❯ ✬✜✭ Watterson, et al. (phylogeny ✒✝✠ ★✜✩ ▲❆▼ ☎ ✑✸ ❄❆❅ ☎ ✑✸ ✏ ✑✆ ☎ ✑✸ ✒✝✠ ’ ☎ ✑✸ reconstruction) ☎ ✑✸ ✔ ✒✑ ✽✖✠

  8. � ✔ ✄ ✓ ✔ ✂ ✠ ✞ ✍ ❂ ✔ ✏ ✠ ✎ ✝ ☛ ✸ ✠ ✎ ✎ ✍ ✍ ✿ ✔ ✎ ☛ ✆ ☛ ✿ ✝ ✠ ✔ ✄ ✓ ✔ ✒ ✹ ✔ ✁ ✞ ✂ ☛ ✗ ✄ ☎ ✌ ❂ ☎ ✞ ☛ ✟ ✠ ✁ ✎ ☛ ✻ ✠ ✍ ✠ ✎ ✏ ✎ ✝ ✓ ✻ ✁ ✓ ☎ ✠ ✆ ✿ ✝ ✠ ✎ ✔ ✏ ✎ ✁ ✄ ✟ ✔ ✓ ✏ ✁ ✔ ✆ ✡ ✠ ❇ ❋ ❍ ❍ ● ❋ ❊ ❈❉ ❃ ■ ✟ ✸ ☛ ✡ ✂ ✠ ✁ ✠ ❍ ❏ ✘ ■ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ❈ ❑ P ❊ ❉ P ▼ ● ◆❖ ❉ ✞ ✟ ✎ ✎ ✞ ✻ ✏ ✆ ✺ ☛ ✡ ☛ ✝ ✓ ✄ ✁ ✄ ✎ ☛ ✎ ✞ ☛ ☛ ✎ ✞ � ✎ ✓ ✞ ✂ ✠ ✁ ✠ ✿ ✸ ✠ ✔ ✎ ☛ ☎ ✍ ✠ ✎ ✻ ✣ ☛ ✎ ✞ ✁ ✿ ✠ ✎ ✓ ✞ ✺ ✁ ✸ ✄ ✠ ✎ ✞ ☛ ✻ ✡ ✄ ✗ ☛ ✞ ✂ ✻ ✄ ✡ ❂ ✞ ✻ ✠ ✻ ✻ ✹ ✾ ✠ ✆ ✎ ✓ ✞ ✎ ✠ ✁ ✆ ✥ ★ ☎ ✂ ✦ ✥ ✧ ✰ � ✦ � ☎ ✧ ✮ ✦ ★ � � ★ ✪ ✭ ✩ ✵ ✴ ✳ ✤ ✭ ✩ � ✱ ✣ ✫ ✰ ✥ ✤ ✰ ✭ ☎ ✮✯ ✤ ✿ ✌ ✠ ✁ ✠ ✎ ✺ ✆ � ✌ ✔ ✁ ✠ � ✞ ☛ ✞ ✍ ✄ ✟ ✻ ✌ ✝ � ❂ ✁ ✍ ✻ ☛ ✟ ✔ ✒ ✗ ✠ ✂ ✺ ✓ ☛ ✞ ☛ ✟ ✞ ✠ ✡ ✓ ✞ ✄ ✝ ✠ ✂ ☎ ✆ ✎ ☛ ✎ ✍ ✄ ☛ ✻ ✝ ✠ ✻ ☎ ✍ ✞ ✄ ✻ ✍ ✝ ✡ ✠ ✁ ✞ ✠ ✂ ☛ ☛ ✔ ✺ ☛ ✁ ✁ ✍ – p. 8/43 ❚❱❯ ✗✑✸ ✬✜✭ ▲❆▼ Hannenhalli ★✜✩ ✏ ✑✆ ❄❆❅ Pevzner, Bafna ✒✝✠ ✁✄✂ ☎ ✑✸ ✽✖✠

  9. � ✻ ✞ ✎ ☛ ✆ ✔ ✄ ✡ ☛ ☛ ✞ ✔ ✿ ✆ ✞ ✆ ✔ ✎ ✎ ✟ ✸ ✠ � ✻ ☛ ❂ ✞ ✟ ✗ ✂ ✻ ☛ ✞ ❂ ✞ ☛ ✟ ✗ ☛ ✻ ✔ ☛ ✔ ❈ ◆❖ ● ▼ P ❉ ❊ P ■ ❑ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❉ ❏ ✻ ❃ ✎ ✞ ✓ ✎ ✆ ✠ ✾ ✟ ❇ ■ ❈❉ ❊ ❋ ● ❍ ❍ ❋ ❍ ✍ ✠ ✝ ✟ ✿ ✏ ✁ ✠ ✞ ✿ ✠ ✠ ✠ ✍ ✍ ✞ ✎ ✞ ✎ ✔ ✸ ✝ ✎ ☛ ✦ ✧ ✣ ✩ ✪ ✥ ✴ ✧ ✥ ✝ ✁ ✯ ✢ ✤ ✭ ✆ ✯ ✤ ✁ ✠ ✿ ✆ ✁ ✄ ✎ ✔ ✍ ✄ ✁ ✔ ☛ ✁ ✄ ✔ ✁ ✠ ✝ ✺ ✞ ✹ ✻ ✠ ✍ ✍ ✸ ✝ ✔ ✁ ✓ ✡ ✠ ✺ ☛ ✓ ✓ ☛ ✏ ✔ – p. 9/43 ❚❱❯ ☎ ✷✎ ✒✝☛ ▲❆▼ ✒✝✠ ✒✝☛ ❄❆❅ (inversion) ✗✑✸ ✏✷✿

  10. � ✄ ✡ ✺ ✝ ✠ ✁ ✔ ✁ ✁ ☛ ✄ ✒ ✞ ✟ ✓ ✟ ☛ ✓ ☎ ✍ ✞ ✆ ✠ ✾ ☛ ✞ ✍ ✏ ✁ ☛ ❃ ✿ P ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❊ ❉ ▼ ❇ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ● ❋ ❊ ❈❉ ✩ ✠ ✠ ✆ ✎ ✸ ✠ ✍ ✓ ✏ ✟ ✁ ✁ ✭ ✵ ✰ ✮ ✌ ✯ ✵ ✤ ✳ ✤ ✭ ✩ � ✦ ✥ ✦ ✴ ✎ ✻ � ✞ ✞ ☛ ✎ ✄ ✡ ✡ ✻ ☛ ✾ ✍ ✄ ✠ ❂ ✰ ✡ ✻ ✠ ✂ ☛ ✻ ❀ ✟ ✠ ✭ ✞ ☛ ✟ ✩ – p. 10/43 ❚❱❯ ▲❆▼ Hannenhalli ❄❆❅ Pevzner, Bafna ★✜✩

  11. � ☛ ✎ ✑ ✗ ✟ ✞ ✍ ✍ ☛ ✂✆ ✞ ✻ ✎ ✡ ✄ ☛ ✡ ✺ ✝ ✠ ✁ ✔ ✁ ✁ ✏ ✄ ✔ ✂ ✞ ❂ ✠ ✎ ✓ ✞ ☛ ✎ ✄ ✻ ✡ ✔ ✔ ✓ ✞ ✺ ✡ ✠ ✎ ✓ ✞ ✎ ✻ ☛ ☛ ✓ ✓ ✞ ✄ ✍ ✝ ✠ ✡ ✡ ✺ ✞ ✠ ✎ ✓ ✎ ☛ ✻ ☛ ✄ ✎ ✏ ✎ ✑ ✗ ✩ ✞ ✁ ✎ ✂ ☛ ✎ ✆ ✄ ✏ ✿ ✝ ✔ ✻ � ☛ ✍ ✔ ✓ ✠ ✞ ✞ ❂ ✠ ✎ ✓ ☛ ✞ ❂ ✡ ✻ ✞ ✎ ✝ ✎ ❈❉ ❇ ❃ ✟ ☛ ✺ ✞ ✠ ✎ ✓ ✡ ✻ ❋ ☛ ✞ ☛ ✟ ✠ ✍ ✒ ✆✞ ☛ ✸ ✝ ❊ ● ✎ ❊ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❉ ❍ P ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ✠ ✄ ✔ ✻ ✒ ✞ ✎ ✄ ✍ ✔ ✎ ✄ ✍ ✝ ✠ ✹ ✓ ✞ ✓ ✞ ☛ ✁ ✿ ✞ ☛ ✟ ☛ ✍ ✄ ✔ ✍ ✎ ✞ ✞ ✆ ✄ ❂ ✞ ☛ ✟ ✞ ✆ ✄ ✓ ✿ ✆✞ ☛ ✞ ✄ ✎ ✟ ✠ ✆ ✿ ✝ ✠ ✟ ✟ ✟ ✠ ✞ ✟ ✄ ✓ ☛ ✓ ☎ ✍ ✞ ✆ ✾ ✄ ☛ ✞ ✍ ✏ ✁ ✠ ✿ ✠ ✎ ✞ ☛ ✒ ☛ ✄ ✟ ✻ ☛ ☛ ✞ ✝ ✠ ✻ ☎ ✍ ✞ ✄ ✠ ✁ ✆ ✽ ✟ ☛ ✡ ✺ ✝ ✠ ✁ ✔ ✁ ✎ ✡ ✞ ✤ ✟ ✁ ✁ ✭ ✵ ✰ ✆ ✮ ✵ ✤ ✳ ✭ ✓ ✩ � ✦ ✥ ✦ ✴ ✩ � ✯ ✰ ✭ ✏ ✍ ✡ ✂ ✻ ☛ ✾ ✍ ✄ ✠ ✞ ❂ ✡ ✻ ✠ ☛ ✠ ✻ ❀ ✟ ✠ ✞ ☛ ✟ ✻ ✌ ✎ ✸ ✎ ✻ ✍ ✡ ✒ ✆✞ ☛ ✻ ☛ ✻ ☎ ✠ ✞ ❂ ✻ ✏ ✠ ✂ ✠ ✻ ✄ ✍ ✏ ✡ ✄ ✻ ☛ ✍ ✠ ✍ ✁ ☛ ✻ ✡ ✄ ☛ ✡ ✺ ✝ ✠ ✁ ✔ ✁ � ☛ ✔ ✓ ☛ ✆ ✄ ✏ ✿ ✝ ✔ ✻ ✾ ✞ ✄ ☎ ✓ ✞ ✂ ✠ ✁ ✠ ✿ ✝ ☛ ✍ ✔ ✁ ✏ ✞ ✆ ✿ ✠ ✎ ✡ ✄ ✎ ✓ ✞ ✿ ✌ ✎ ✄ ✎ ✟ ✄ ✏ ✞ ✞ ✍ ✠ ✄ ✎ ✓ ✓ ✠ ✞ ✎ ✟ ✻ ☛ ✑ ✞ ✒ ✺ – p. 10/43 0 ✏ ✑✸ ❚❱❯ [0 , N ] . ▲❆▼ [1 , N ] Hannenhalli ❄❆❅ ✏ ✷✿ Pevzner, Bafna ☎ ✷✎ ☎ ✷✎ ★✜✩ N + 1

  12. � ✸ ✞ ✍ ✄ ✎ ✠ ✝ ☛ ✆ ✆✞ ✒ ✍ ✠ ✟ ☛ ✞ ✄ ☛ ✓ ✟ ✎ ✄ ✞ ☛ ✆✞ ✎ ❂ ✄ ✆ ✞ ✟ ☛ ✞ ✞ ✻ ✆ ✞ ✗ ✓ ✟ ✠ ✿ ✞ ✑ ✞ ✠ ✞ ✝ ✿ ✏ ✄ ✟ ✂ ✎ ☛ ✡ ✓ ✎ ✠ ✞ ✺ ✟ ✠ � ✿ ✏ ☎ ✞ ✠ ✁ ✠ ✿ ☛ ✂ ✠ ✡ ✺ ✞ ✓ ✔ ✡ ✓ ✻ ✄ ✎ ☛ ✞ ✓ ✎ ✞ ✠ ✗ ✂✆ ☛ ✍ ✍ ✩ ✟ ✑ ✎ ✎ ✏ ✎ ✄ ✔ ☛ ✻ ✎ ❂ ✝ ✎ ✍ ✄ ✎ ✔ ✍ ✄ ✞ ✠ ✒ ✄ ✓ ✔ ✿ ✠ ✝ ✻ ✞ ✟ ✞ ✓ ✎ ✔ ✍ ☛ ☛ ✹ ✞ ✿ ✁ ☛ ✞ ✓ ✞ ✆ ✓ ☛ ✄ ✄ ✟ ✾ ✆ ✞ ✡ ✎ ✎ ☛ ✞ ✍ ✄ ✿ ✆ ✏ ✝ ✓ ✔ ✞ ✓ ✍ ☎ ✻ ☛ ✟ ☛ ✼ ✠ ✂ ✄ ✂ ✠ ✻ ✏ ✔ ✠ ◗ P ❉ ❊ P ❈ ■ ❘ ● ❅ ■ ▼ ❉ ❙ ▼ ▼ ◆❖ ✍ ❋ ☛ ✟ ❃ ❇ ❈❉ ❊ ● ❉ ❍ ❍ ❋ ❍ ■ ❏ ❑ ✁ ✎ ✔ ✗ ✞ ☛ ✿ ✏ ✎ ✍ ✍ ✎ ☛ ✎ ✠ ✝ ✄ ✻ ☛ ✻ ✆ ✝ ☛ ✁ ✁ ✔ ✁ ✠ ✺ ✠ ✡ ✄ ✸ ✿ ✆ ✠ ✺ ✠ ✂ ☛ ✞ ✞ ☛ ✄ ✡ ✻ ☛ ✓ ✡ ✿ ✆ ✠ ✓ ☛ ✻ ❂ ✻ ☛ ✑ ✻ ✞ ✓ ✍ ✠ ✏ ✠ ✠ ✿ ✏ ✎ ✄ ✎ ☛ ✂ ✞ ✞ ✻ ☎ ✆ ✠ ✟ ✄ ✞ ✍ ✻ ✟ ✠ ✝ ✻ ☛ ☛ ✻ ✽ ☛ ✞ ✄ ☛ ✓ ✄ ✟ ✞ ✒ ☛ ✡ ✁ ✁ ✔ ✁ ✠ ✝ ✺ ✎ ✍ ☎ ✓ ✿ ✠ ✁ ✠ ✂ ✞ ✎ ☛ ✞ ✟ ✞ ✑ ✡ ✻ ✝ ✍ ✄ ✄ ✎ ✌ ✿ ✞ ✓ ✎ ✡ ✔ ✎ ✠ ✿ ✆ ✏ ☎ ✁ ✓ ✍ ✞ ✓ ✵ ✭ ✁ ✁ ✟ ✏ ✍ ✆ ✠ ✸ ✎ ✌ ✻ ✟ ✰ ✮ ✞ ✦ ✭ ✰ ✯ � ✩ ✴ ✥ ✵ ✦ � ✩ ✭ ✤ ✳ ✤ ☛ ✠ ✞ ✁ ☛ ✞ ✎ ✠ ✿ ✠ ✏ ✄ ✍ ✞ ☛ ✾ ✠ ✆ ✎ ✡ ✟ ✡ ❀ ✻ ☛ ✂ ✠ ✻ ❂ ✡ ✞ ✠ ✄ ✍ ✾ ☛ ✻ ✎ ☛ ✓ ✠ ✎ ✠ ✞ ✺ ✡ ✡ ✝ ✞ ✍ ✄ ✎ ☛ ✂ ✠ ✓ ✎ ✡ ✞ ☛ ✞ ✎ ✝ ✟ ✁ ✟ ✻ ✗ ✑ ✎ ✏ ✎ ✄ ☛ ✻ ❂ ✡ ✁ ✆ ☛ ✓ ✔ ☛ ✁ ✔ ✏ ✁ ✠ ✝ ✺ ✡ ☛ ✄ ✄ ✿ ✞ ✞ ☛ ✓ ✎ ✠ ❂ ✞ ✓ ✝ ✎ ✔ ✍ ☛ � ✻ ✔ ✻ ✟ ✄ ✠ ✺ ✻ ✠ ☎ ✠ ✄ ✞ ❂ ✡ ✏ ✻ ✠ ✂ ✍ ✻ ✻ ✍ ✟ ✞ ✒ ✄ ✄ ✄ ✾ ✄ ☛ ✻ ✞ ✡ ✏ ✍ ✓ ☛ ✞ ✄ ☛ ✡ ✺ ✝ ✠ ✁ ✔ ✁ ✁ ☛ ✔ ✓ ☛ ✠ ✆ ✏ � ☛ ✆✞ ✒ ✍ ✏ ✠ ✻ ✔ ✝ ✿ – p. 10/43 0 ✏ ✑✸ ❚❱❯ ☎ ✑✆ [0 , N ] . ▲❆▼ [1 , N ] Hannenhalli ❄❆❅ ☎ ✷✎ ✏✑✸ ✏ ✷✿ Pevzner, Bafna ✏✑✸ ☎ ✷✎ ☎ ✷✎ ★✜✩ DNA N + 1

  13. � ✸ ✞ ✸ ☛ ✁ ✁ ✔ ✁ ✠ ✝ ✺ ✡ ✄ ✠ ✓ ✄ ☛ ✻ ✎ ✞ ✓ ✎ ✆ ✠ ✾ ✗ ✍ ✄ ✎ ✡ ☛ ✠ ✁ ✣✤ ✁ ✠ ✝ ✺ ✡ ☛ ✠ ✕ ✿ ✆ ✓ ✠ ✔ ✍ ☛ ✓ ✍ ☎ ✻ ✠ ✞ ☛ ✆✞ ✒ ✍ ✎ ✎ ☛ ▼ ❋ ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● P ❈❉ ❉ ❊ P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❊ ❇ ✆ ☛ ✄ ✞ ✎ ✔ ✻ ✿ ☛ ✆ ☛ ✿ ✞ ✻ ✌ ✁ ❃ ✁ ✔ ✁ ✠ ✝ ✺ ✡ ☛ ✓ ✎ ✔ ✻ ✠ ✁ ✔ ✻ ✄ ✑ ☎ ✺ ✠ ✝ ✻ ✿ ✆ ✠ ✟ ✏ ✂ ✞ ✞ ❂ ✏ ✠ ✿ ✆ ✠ ☎ ✁ ✗ ✍ ☛ ✔ ☛ ✞ ✿ ✎ � ✥ ✮ ☎ ✭ ✰ ✤ ✥ ✰ ✣ ✩ ✱ ✁ ✿ ✠ ✏ ✍ ☛ ✒ ✔ ✍ ☛ ✎ ✞ ✟ ✗ ✓ ✟ ✓ ✎ ☛ ❀ ✸ ✠ ✻ ✎ ✞ ✓ ✎ ✆ ✠ ✾ ✟ ✟ ✓ ✎ ✎ ✌ ✿ ✌ ✸ ☎ ✺ ✠ ✝ ✍ ✄ ✎ ✔ ☛ ✔ ✓ ✿ ☛ ✞ ✻ ✆ ☛ ✠ ✆ ✡ ✡ ✠ ✄ ✝ ✍ ✄ ✎ ✔ ✻ ✠ ✿ ✿ ✆ ✞ ✆ – p. 11/43 ❚❱❯ π ′ = ( π 1 , . . . , π j , . . . , π i +1 , π i , . . . , π n ) π = ( π 1 , . . . , π i , π i +1 , . . . , π j , . . . , π n ) (Inversion) ▲❆▼ ✗✑✸ ❄❆❅ ✲ ✄✂ ☎ ✷✿ ρ ( i, j ) � ✄✂ ✔ ✂✁ ☎ ✷✿

  14. � ✠ ☛ ✍ ✔ ✓ ✠ ✆ ✽ ✻ ✍ ✔ ✎ ✓ ☛ ✡ ✺ ✝ ✓ ☎ ✁ ✎ ✁ ✔ ✁ ✁ ☛ ✸ ✞ ✓ ✻ ✡ ✠ ✍ ✒ ✆✞ ☛ ✞ ✠ ✠ ✔ ✝ ✎ ✗ ✾ ✠ ✆ ✎ ✓ ✞ ✻ ✄ ☛ ✄ ✠ ✸ ✄ ✣✤ ✺ ✝ ✍ ✎ ✁ ✆ ✁ ☛ ✌ ✻ ✞ ✿ ☛ ☛ ☛ ✿ ✻ ✔ ✎ ✞ ✄ ✆ ✎ ✠ ✺ ✁ ❈❉ ❍ ❋ ❍ ❍ ● ❋ ❊ ❇ ❏ ❃ ✠ ✻ ✔ ✎ ✓ ☛ ■ ❑ ✺ ◗ ▼ ❙ ❉ ▼ ■ ❅ ❘ ■ ❉ ❈ P ❊ ❉ P ▼ ● ◆❖ ✡ ✝ ✡ ✎ ✎ ✄ ✍ ✗ ✾ ✠ ✆ ✓ ✎ ✞ ✎ ✻ ☛ ✄ ✠ ✸ ✄ ☛ ✆ ✠ ✞ ✁ ✔ ✁ ✁ ☛ ✌ ✻ ✿ ✄ ☛ ✆ ☛ ✿ ✻ ✔ ✎ ✞ ✠ ✡ ✔ ✆ ✎ ☛ ✍ ✗ ✁ ☎ ✠ ✿ ✓ ✠ ✏ ❂ ✞ ✄ ✂ ✏ ☛ ✎ ✠ ✻ ✝ ✠ ✁ ✡ ✆ ✠ ☛ ✞ ☛ ✄ ✆ ✿ ☛ ✡ ✠ ✿ ✠ ✟ ✆ ✄ ✣ ✍ ✏ ✿ � ✁ ✱ ✩ ✰ ✒ ✥ ✤ ✰ ✭ ☎ ✮ ✥ ☛ ✔ ✿ ✞ ✻ ✝ ✠ ✺ ☎ ✑ ✞ ✿ ✍ ✠ ✟ ✓ ✗ ✟ ✞ ✎ ☛ ✍ ✓ ✎ ✡ ✓ ✠ ✆ ✿ ✕ ✠ ☛ ✺ ✍ ✝ ✠ ✁ ✔ ✁ ✁ ☛ ✔ ☛ ✔ ✠ ✁ ☛ ✸ ✞ ✎ ✓ ✡ ✍ ✓ ✒ ✆✞ ☛ ✞ ✠ ✻ ☎ ✍ ✻ ✔ ✎ ✓ ✞ ☛ ✓ ✎ ✆ ✸ ✠ ✄ ✾ ✔ ✟ ✟ ✎ ❀ ✔ ✻ ✌ ✍ ✝ ✠ ✺ ☎ ✸ ✿ ✆ ✻ ✌ ✎ ✿ ✆ ✞ ✎ – p. 11/43 ❚❱❯ π ′ = ( π 1 , . . . , − π j , . . . , − π i +1 , − π i , . . . , π n ) π ′ = ( π 1 , . . . , π j , . . . , π i +1 , π i , . . . , π n ) π = ( π 1 , . . . , π i , π i +1 , . . . , π j , . . . , π n ) (Inversion) ▲❆▼ ✗✑✸ ❄❆❅ ✲ ✄✂ ☎ ✷✿ ☎ ✷✿ ρ ( i, j ) ρ ( i, j ) � ✄✂ ✔ ✂✁ ☎ ✷✿

  15. � ☎ ✽ ✆ ✠ ✓ ✔ ✍ ☛ ✓ ✍ ✻ ✡ ✠ ✞ ☛ ✆✞ ✒ ✍ ✠ ✡ ✁ ✏ ✟ ✠ ✔ ✔ ✾ ✚ ✻ ✟ ✕ ✔ ✿ ✆ ✠ ✓ ✍ ✍ ☛ ✓ ✍ ☎ ✻ ✠ ✞ ☛ ✆✞ ✒ ✓ ✿ ✆ ❊ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ P ❉ P ❍ ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❋ ❍ ✠ ✺ ✁ ✝ ✌ ✻ ✝ ✍ ✞ ✟ ✠ ✏ ✆ ● ✏ ✻ ✠ ✝ ✂ ❃ ❇ ❈❉ ❊ ❋ ✠ ✹ ✎ ✳ ✢ ✧ ✭ ✥ ✰ ✩ ✭ ✢ ★ ✆ ✩ ✰ ✁ ✟ ✁ ✁ ☛ ☎ ✺ ✠ ✝ ✻ ✻ ✦ ✥ ✓ ✰ ✣✤ ✧ � ✣ ✩ � ★ ✲ ✮ ✧ ✧ ✴ � ✩ ✪ ✧ ✭ ✰ ✤ ✮✯ ✰ ✵ � ☛ ✟ ✞ ✓ ✻ ✞ ❂ ✡ ✌ ✻ ✠ � ✿ ✏ ✎ ✻ ☛ ✓ ✔ ✍ ☎ ✓ ✌ ☛ ✻ ✎ ✞ ✠ ✂ ✻ ✁ ✠ ✝ ✻ ✍ ✄ ✎ ☛ ✆ ✏ ☛ ✎ ✁ ✝ ✹ ✔ ✺ ✞ ✠ ✁ – p. 12/43 ❚❱❯ (Inversion Distance) ▲❆▼ ⇒ NP-Hard = ❄❆❅ ⇒ =

  16. � ✍ ✓ ✞ ✎ ✻ ☛ ✄ ✠ ✆ ✻ ✏ ✍ ✔ ✓ ✔ ✎ ✠ ✏ ◗ ✍ ✞ ✓ ✔ ✆ ✺ ✸ ✾ ✠ ❘ ✁ ✟ ❅ ■ ✍ ✻ ✿ ✁ ✁ ✠ ✿ ✔ ✓ ✏ ▼ ✌ ✚ ✡ ✠ ✍ ✒ ✆✞ ✝ ✻ ✞ ✻ ✆ ☛ ✼ ✂ ✝ ✠ ✏ ✝ ✆ ✺ ✏ ✠ ✟ ✞ ✍ ✠ ✠ ✞ ✍ ❊ ✠ ✟ ✞ ✎ ☛ ✔ ❃ ✒ ☛ ✍ ✞ ☛ ✟ ✟ ❉ ✠ ❍ ◆❖ P ❉ ❑ ❏ ■ ❋ ❇ ❍ ❍ ● ❋ ❊ ❈❉ P ✟ ✞ ✏ ✂ ✏ ✁ ✠ ✞ ☎ ✿ ✝ ☛ ✞ ☛ ✎ ✡ ✄ ✠ ✸ ✆✞ ✿ ✠ ✾ ✠ ✆ ✔ ✁ ✏ ✟ ✿ ☛ ❈ ✸ ✠ ■ ✁ ☛ ✠ ▼ ☎ ☛ ✻ ✻ ✝ ✠ ✺ ☛ ✞ ✁ ✁ ✟ ✁ ✩ ✳ ✟ ✻ ★ ✏ ✁ ✔ ✁ ✁ ☛ ✎ ✆ ✠ ☛ ✎ ✄ ✍ ✻ ✝ ✆ ✢ ✝ ✲ ✩ � ✧ ✰ ✧ ✮ ★ ✧ � ✩ ✣ � ✧ ✣✤ ✪ ✭ ✭ ✦ ✩ ✰ ✥ ✭ ✧ ✢ ✰ ✰ ✥ ✴ � ✵ ✰ ✮✯ ✤ ✠ ✺ ✻ ☛ ✞ ✠ ✻ ☎ ✍ ✓ ✍ ✆✞ ✔ ✓ ✠ ✆ ✿ ✔ ☛ ✒ ✟ ✓ ☎ ✍ ✓ ☛ ✍ ✔ ✠ ✍ ✆ ✽ ✟ ❉ ❙ ✡ ✠ ✕ ▼ ✞ ✡ ✏ ✿ � ✠ ✻ ✌ ❂ ✎ ✞ ✻ ✠ ✂ ✻ ✹ ✓ ☛ ✻ ✞ ✹ ✾ ✠ ✆ ✎ ✓ ✎ ✓ ✻ ☛ ✌ ✓ ☎ ✍ ✔ ● – p. 12/43 ❚❱❯ ✏ ✑✸ (Sorting by reversals) (Inversion Distance) ▲❆▼ ⇒ NP-Hard ✏ ✑✸ = ❄❆❅ ☎ ✑✸ ⇒ = inversion reversal ✏✑✆ ✏✑✆

  17. � ✟ ✔ ✒ ☛ ✍ ✞ ☛ ✟ P ✠ ✆✞ ☛ ✠ ✾ ✠ ✆ ✔ ✁ ✿ ✏ ✿ ✍ ✎ ❈ ✁ ✞ ✎ ✻ ☛ ✻ ✌ ✎ ✻ ✌ ✏ ✞ ✔ ✓ ✄ ❉ ❀ ✟ ❊ ✠ ✟ ☛ ✸ ✎ ✁ ✞ ✓ ✔ ✆ ✺ ◗ ✸ ✠ ❘ ✟ ✠ ❅ ✚ ✾ ✠ ✆ ✎ ✓ ✞ ✎ ✍ ✿ ✠ ✠ ■ ✁ ✟ ✸ ✝ ✠ ✂ ✏ ✁ ✞ ✠ ☎ ✏ ✿ ☛ ✞ ☛ ✎ ✡ ✄ ✞ ✓ ✆ ☛ ✓ ✄ ✻ ✄ ✂ ✠ ✁ ✝ ✾ ✄ ✻ ✞ ✌ ✎ ✞ ✍ ✔ ✁ ☎ ✠ ✆ ✎ ✟ ✔ ❋ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ● ❊ ✞ ❈❉ ❇ ▼ ❃ ✟ ☛ P ✻ ☎ ❂ ✿ ✓ ✠ ✸ ✄ ✓ ✻ ✏ ✠ ✂ ✔ ❂ ✠ ✄ ☛ ✡ ✠ ✿ ✠ ✞ ✠ ✻ ✹ ✾ ✎ ✆ ✏ ✟ ✁ ✔ ✎ ✞ ✄ ✻ ✏ ✁ ✠ ✝ ✞ ✄ ✞ ✶ ✟ ✔ ✓ ✔ ✍ ✏ ✻ ✻ ✄ ◆❖ ✎ ✁ ✔ ✁ ✁ ☛ ✎ ✏ ✆ ☛ ✄ ✝ ✍ ✻ ✝ ✠ ✻ ✞ ✟ ☛ ✻ ✠ ✺ ✝ ✠ ✍ ✔ ✓ ☛ ✎ ✓ ✏ ✿ � ✻ ✞ ✌ ✡ ❂ ✞ ✻ ✠ ✂ ✻ ✹ ✻ ✠ ✓ ✰ ✮✯ ✤ ✰ ✭ ✧ ✪ ✩ � ✧ ✧ ✵ ✮ ✲ ★ � ✩ ✣ � ✧ ✣✤ ✰ � ✺ ★ ☎ ☛ ✁ ✁ ✟ ✁ ✩ ✳ ✆ ✢ ✴ ✭ ✩ ✰ ✥ ✭ ✧ ✢ ✦ ✰ ✥ ☎ ✌ ✍ ✓ ✍ ✝ ✻ ✌ ✝ ✁ ✠ ✿ ✔ ✏ ✟ ✁ ■ ▼ ✡ ✠ ✍ ✒ ✆✞ ☛ ✞ ✠ ✠ ✻ ✠ ✻ ✏ ✍ ✔ ✓ ✔ ✍ ✏ ✞ ✏ ✆ ☛ ✼ ✂ ✝ ✠ ✻ ✏ ✆ ✺ ✞ ✻ ☛ ▼ ✍ ✔ ✓ ✠ ✆ ✿ ✔ ✕ ✟ ✻ ✓ ✹ ✾ ✠ ✆ ✎ ✓ ✞ ✎ ✻ ☛ ✍ ☎ ✟ ✍ ✓ ☛ ✍ ✔ ✓ ✠ ✆ ✽ ❉ ☎ ❙ ✡ ✠ ✍ ✒ ✆✞ ☛ ✞ ✠ ✻ ● – p. 12/43 ❚❱❯ ❂✑✆ ✏ ✑✸ (Sorting by reversals) (Inversion Distance) ▲❆▼ ⇒ NP-Hard ✏ ✑✸ = ❄❆❅ ☎ ✑✸ ⇒ = inversion reversal ✏✑✆ ✏✑✆ ✏ ✑✆

  18. � ✍ ✔ ✍ ✏ ✻ ✞ ✆ ☛ ✎ ✄ ✝ ✔ ✠ ✺ ✏ ✎ ✞ ✿ ✄ ☛ ✻ ✓ ✌ ✎ ✄ ✻ ✞ ☛ ✻ ☎ ✞ ✄ ✆ ✡ ❂ ✁ ✻ ✝ ✠ ✺ ☎ ✞ ☛ ✟ ✞ ✄ ✓ ✾ ✠ ✓ ✞ ✎ ✻ ☛ ✸ ✗ ✁ ✿ ✎ ✆ ✓ ✞ ✂ ☎ ✍ ✠ � ✞ ☛ ✎ ✠ ✹ ☛ ✠ ✎ ✔ ✎ ✡ ☛ ✆ ☛ ✿ ✞ ✾ ☛ ✻ ✡ ✄ ✝ ✠ ✿ ✻ ✹ ✌ ✆ ✡ ✻ ❍ ● ❋ ❊ ❈❉ ❇ ❃ ✞ ☛ ✡ ❋ ✄ ☛ ✎ ☛ ✍ ✗ ✁ ☎ ✠ ❍ ❍ ✿ ❈ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ P ■ ❊ ❉ P ▼ ● ◆❖ ❉ ❑ ❏ ✆ ☛ ✞ ✄ ☛ ✞ ✍ ✄ ✍ ✏ ✠ ☛ ✻ ✍ ✠ ✏ ✠ ✆ ✠ ✿ ✕ ☛ ✍ ✂ ✎ ✞ ✼ ✎ ✌ ✻ ✹ ✾ ✠ ✆ ✎ ✓ ✞ ✻ ☛ ☛ ☛ ✡ ✒ ✝ ✠ ✁ ✠ ✟ ☛ ✻ ✎ ✄ ✠ ✠ ✍ ✔ ❂ ✠ ✡ ✠ ✿ ✠ ✎ ✞ ☛ ✍ ✔ ✁ ☎ ✏ ✆ ✿ ✠ ✓ ✄ ✞ ✄ ✠ ✁ ✠ ✡ ✎ ✄ ✍ ✑ ☎ ✠ � ✏ ✠ ✎ ✠ ✎ ✓ ✄ ✟ ✔ ✒ ✎ ☛ ☛ ✩ ★ � ✩ ✣ � ✭ ✵ ✰ ✂ ✮ ✮ ✩ ✣ ✳ � ✣ � ✂ ✧ ✭ ✦ ✆ ✻ ✦ ✡ ❀ ✁ ✮✯ ✪ ✩ ✥ ✲ ✰ ✂ ✰ ✪ ✥ ✧ ✁ ✪ ✩ � ✭ ✵ ✟ ✎ ✞ ☛ ✎ ☛ ✍ ✗ ✁ ☎ ✠ ✆ ✿ ✼ ✿ ✠ ✡ ✡ ☛ ✂ ☎ ✍ ☛ ✻ ☛ ✠ ✄ ✄ ✞ ✠ ✞ ☛ ✻ ✡ ✄ ✻ ✏ ✎ ✝ ☛ ✻ ✻ ✏ ✠ ✆ ✠ ✿ ✍ ☎ ☛ ✓ ✄ ✺ ✞ ✎ ☛ ✡ ✟ ✻ ✔ ✞ ✓ ✎ ✻ ✄ ✍ ✞ ✄ ✻ ✞ ☛ ✟ ✟ ✆ ✞ ✿ ✎ ✄ ☛ ✎ ✄ ✝ ✠ ✿ ✔ ✓ ☛ Pancake American Mathematical Monthly Pancake Problem Waiter: “The chef in our place is sloppy, and when he prepares pancakes they come out all in different sizes. Therefore, when I deliver them to a customer, on the way to the table I rearrange them (so the smallest winds up on top, and so on, down to the largest at the bottom) by grabbing several from the top and flipping them over, repeating this (varying the number I flip) as many times as necessary.” ✒✝✠ ✒✝✠ ✽✖✠ Gates Papadimitriou (5 N + 5) / 3 . ☎✝✆ ☎ ✷✎ NP-Complete. ❄❆❅ ▲❆▼ ❚❱❯ – p. 13/43

  19. � ✸ ✍ ✄ ✻ ☛ ✎ ✞ ✎ ✠ ☛ ✆ ✞ ✻ ✠ ✍ ✝ ✁ ✓ ✆✞ ✝ ✒ ✡ ✄ ✸ ☛ ✒ ☎ ✍ ✹ ✻ ✟ ☛ ✞ ✒ ✗ ✠ ✮ ■ ▼ P ❉ ❊ P ❈ ◗ ◆❖ ❘ ❅ ■ ▼ ❉ ❙ ▼ ● ❉ ✝ ❊ ✍ ✄ ✟ ❃ ❇ ❈❉ ❋ ❑ ● ❍ ❍ ❋ ❍ ■ ❏ ✠ ✁ ✠ ✰ ✩ � ★ ✲ ✮ ✧ ✩ � ✂ ✮✯ ✧ ✪ ✰ ✟ ✣ ✣ ✩ ✯ ✩ ✤ � ✰ ✦ ✰ ✂ ✩ ✢ ✧ ✤ ✞ ✤ ✧ ✱ � ✆ ✪ ✟ ✠ ✻ ✞ ✟ ☛ ✻ ✄ ✍ ✝ ✠ ✄ ✍ ✁ ✄ ✁ ✍ ✮✯ ✧ ✧ ✭ ✰ ✤ ☛ ✰ ✮✯ ✵ � ✴ ✥ ✩ ✪ ✝ – p. 14/43 ❚❱❯ Pancake ▲❆▼ ✁✄✂ ❄❆❅ ❀✖✺

  20. � ✞ ✺ ✄ ✞ ✻ ☛ ✟ ✞ ✻ ✄ ✍ ☛ ✄ ✒ ✄ ✎ ✞ ✟ ✠ ☛ ✆✞ ✒ ☎ ✍ ✠ ✠ ✎ ☛ ✆ ✞ ✻ ✠ ✍ ✗ ✓ ✝ ✔ ✍ ✄ ✟ ✼ ✠ ✿ ✆ ✏ ☎ ✁ ✍ ☛ ✞ ❊ ■ ❏ ❑ ❉ ◆❖ ● ▼ P ❉ P ❋ ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❍ ❍ ☛ ✗ ✿ ✠ ✝ ✓ ✡ ☛ ✿ ✆ ✠ ✓ ✍ ❍ ✌ ✻ ☞ ✟ ❃ ❇ ❈❉ ❊ ❋ ● ✮ ✸ ✎ ✩ ✰ ✩ ✂ ✮✯ ✧ ✪ ✰ ✆ � ✪ ✮ ✁ ✧ ✧ ✭ ✰ ✤ ✮✯ ✰ ✵ � ✧ ✲ ✥ ✧ ✩ ✤ � ✰ ✦ ✰ ✯ ✂ ✢ ✤ ★ ✞ ✤ ✧ ✱ ✩ ✣ � ✣ ✩ � ☛ ✴ ✩ ✻ ✝ ✒ ✡ ✄ ✸ ☛ ✆✞ ✒ ✍ ✹ ✟ ✁ ☛ ✞ ✒ ☎ ✁ ✠ ✝ ✍ ✄ ✻ ✪ ✠ ✠ ✞ ✮✯ ✁ ✟ ✠ ✝ ✍ ✄ ✻ ☛ ✟ ✟ ✻ ✄ ☛ ✄ ✠ ✍ ✍ ✝ – p. 14/43 ❚❱❯ Pancake ✏ ✑✸ ▲❆▼ ✏ ✑✸ ✁✄✂ ❄❆❅ ❀✖✺

  21. � ✿ ☛ ✍ ✔ ✁ ☎ ✏ ✆ ✠ ✿ ✎ ✏ ✎ ✝ ☛ ✄ ✓ ☛ ☛ ✔ ✎ ✓ ✞ ✎ ✻ ☛ ✔ ✄ ✠ ✞ ✎ ✻ ✞ ✿ ✡ ✄ ✎ ✸ ✓ ✆ ✄ ✞ ✄ ✺ ✞ ✠ ✎ ✓ ✟ ❂ ☞ ✻ ✌ ✍ ✗ ✓ ✠ ✹ ✔ ✔ ✔ ✍ ✏ ✻ ✞ ✆ ☛ ✎ ✆ ✸ ✞ ✆ ✿ ✸ ✌ ✍ ✏ ✎ ✠ ✮ ❊ ■ ❍ ❋ ❍ ❍ ● ❋ ❈❉ ❑ ❇ ❃ ✟ ✸ ✔ ✓ ✄ ❏ ❉ ✞ ◗ ▼ ❙ ❉ ▼ ■ ❅ ❘ ■ ◆❖ ❈ P ❊ ❉ P ▼ ● ✒ ✎ ✾ ✄ ✎ ✹ ✆ ✿ ✠ ✎ ✞ ✺ ✓ ☎ ✆✞ ✄ ✿ ☛ ✻ ✗ ✠ ✎ ✄ ✠ ✍ ✸ ✔ ✻ ☎ ✍ ✓ ❂ ✠ ✸ ✔ ✎ ✠ ✡ ✄ ✺ ✞ ✆ ✿ ☛ ✩ ✝ ✠ ✟ ✁ ✮✯ ✪ ✥ ✄ ✴ � ✵ ✰ ✮✯ ✤ ✰ ✍ ✻ ✧ ✄ ✡ ✝ ✠ ✁ ✠ ✟ ☛ ✍ ☛ ✄ ✍ ✝ ✠ ✻ ✞ ✟ ✭ ✧ ✄ ✢ ✩ ✱ ✧ ✤ ✞ ✤ ✧ ✂ � ✯ ✰ ✦ ✰ � ✤ ✩ ✣ ✣ ✁ ✮✯ ✪ ✩ � ✆ ✰ ✪ ✧ ✂ ✩ ✩ ✰ ✧ ✮ ✲ ★ � ✡ ✒ ✸ ✞ ✄ ✻ ✞ ✟ ☛ ✻ ✄ ✍ ✺ ☎ ☛ ✍ ✔ ✁ ☛ ✞ ✄ ✆ ✍ ✓ ✝ ✠ ✿ ☛ ☛ ✠ ✒ ✒ ✆✞ ☛ ✠ ✟ ✞ ✎ ✄ ✏ ☎ ✿ ✒ ✠ ✄ ✍ ✝ ✠ ✁ ☎ ✞ ✎ ☛ ✟ ✻ ✹ ✍ ✒ ✆✞ ☛ ✻ ✞ ✸ ✼ ✟ ✄ ✍ ✝ ✠ ✓ ✗ ✍ ✠ ✻ ✞ ✆ ☛ ✎ – p. 14/43 ❚❱❯ Pancake ✏ ✑✸ ▲❆▼ ✏ ✑✸ ✁✄✂ ❄❆❅ ❀✖✺

  22. ☎ ☛ ✡ ☛ ✸ ☛ ✟ ✠ ✁ ✠ ✝ ✒ ✡ ✸ ✺ ✟ ☛ ✞ ✎ ✔ ✸ ☛ ✻ ✎ ✡ ✓ ☛ ✝ ✠ ☛ � ✄ ✻ ☛ ✿ ✆ ✠ ☎ ✁ ✗ ✍ ✎ ✎ ☛ ✍ ✄ ✎ ☛ ✆ ✏ ✍ ✞ ☛ ✸ ✎ ✞ ✠ ❉ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ P ❊ ● P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❍ ❋ ✺ ✟ ✔ ✸ ✎ ☛ ✝ ✎ ✠ ✎ ✞ ✟ ☛ ✠ ❊ ✁ ✠ ✝ ✒ ✡ ☛ ✸ ✟ ❃ ❇ ❈❉ ✏ ✍ ✂ ✿ ❀ ✓ ✎ ✌ ✿ ✌ ✸ ✓ ✎ ☛ ✆ ✁ ✠ ☎ ✁ ✗ ✍ ☛ ✎ ☛ ✍ ☛ ✔ ✟ ✮✯ ✿ ✦ � ✩ ✴ ✦ ✥ ✦ � ✣ ✩ � ★ ✮ ✚ ✆ ✰ ✵ ✭ � ✩ ✪ ✁ ✧ ✳ ✩ ☛ ✸ ☛ ✻ ✟ ✄ ✡ ✻ ☛ ✄ ✒ ✄ ✡ ✂ ✄ ✞ � ✟ ✗ ✁ ✏ ✓ ✔ ✓ ✎ ☛ ✿ ✆ ✠ ✓ ✆ ✔ ✠ ☛ ✁ ✁ ✻ ✏ ✁ ✠ ✝ ✺ ✠ ✡ ✄ ✸ ✁ ✠ ✟ ✺ ☛ ✞ ✞ – p. 15/43 π 2 . ❚❱❯ π 1 ✗✑✸ ▲❆▼ ❄❆❅ I = 1 2 . . . N ☎✝✆ ★✜✩

  23. ☎ ✠ ✎ ✞ ✓ ✎ ❀ ✟ ☛ ✡ ✒ ✝ ✁ ✿ ✠ ✟ ☛ ✗ ✟ ✞ ✎ ✠ ✎ ✝ ☛ ✠ ✆ ✻ ✿ ✄ ✺ ☎ ✡ ✄ ✁ ✠ ✺ ✓ ☛ ☛ ✏ ✸ ☛ ✍ ✝ ✠ ✿ ☛ ✍ ✔ ✁ ☎ ✎ ✔ ✻ ✻ ☛ ✍ ✔ ✺ ✓ ☛ ✎ ✄ ✍ ☛ ✄ ✡ ✍ ✏ ✠ ✆ ✠ ✿ ✍ ✸ ☛ � ✺ ✎ ✓ ✎ ✒ ✓ ✔ ✠ ✺ ✏ ✎ ✓ ✓ ☛ ✡ ✝ ✠ ✠ ✁ ✠ ✟ ☛ ✻ ✔ ✎ ✄ ✍ ✝ ✞ ☛ ✠ ✁ ❊ ❈❉ ❇ ❃ ✂ ☛ ✡ ✒ ✝ ✠ ✠ ● ✟ ☛ ✗ ✟ ✞ ✎ ✠ ✎ ✝ ☛ ✎ ❋ ❍ ✞ P ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ ❊ ❍ ❉ P ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ✔ ☛ ✟ ☛ ✍ ✞ ☛ ✟ ✸ ✞ ✄ ✓ ☎ ✒ ✎ ✻ ✄ ✍ ✸ ✞ ✎ ✄ ✍ ✞ ✄ ✻ ✞ ✏ ✠ ✻ ☛ ✡ ✄ ✠ ✏ ❂ ✸ ✞ ✎ ✏ ✿ ☛ ✡ ✿ ✍ ✔ ☛ ✞ ✿ ☛ ✆ ✹ ✎ ✝ ✠ ✞ ✡ ✎ ✔ ✞ ☛ ✟ ✸ ✄ ✡ ✺ ✝ ✠ ✁ ✁ ✆ ✁ ☛ ✞ ✠ ✻ ✏ ✠ ✁ ✠ ✺ ✓ � ✟ ✿ ✗ ✠ ✓ ✿ ☛ ✎ ✓ ✔ ✓ ✏ ✁ ✟ ✄ ✞ ✻ ✄ ✂ ✡ ✄ ✒ ✄ ☛ ✻ ✡ ☛ ☛ ✂ ✮ ✳ ✧ ✁ ✪ ✩ � ✭ ✵ ✰ ✆ ✦ ✚ ★ � ✩ ✣ � ✦ ✥ ✦ ✴ ✩ � ✩ ✮✯ ✸ ✆ ☛ ✍ ☛ ✎ ☛ ✍ ✗ ✁ ☎ ✠ ✿ ✁ ☛ ✎ ✓ ✸ ✌ ✿ ✌ ✎ ✓ ❀ ✟ ✔ ✆ ✠ ✎ ✝ ✠ ✁ ✠ ✟ ☛ ✟ ✞ ✎ ✠ ✝ ✡ ☛ ✎ ✸ ✔ ✺ ✞ ✠ ✎ ✓ ✏ ✎ ✒ ☛ ☛ ✍ ✞ ✎ ✻ ☛ ✸ ☛ ✞ ✍ ✌ ✓ ☎ ✡ ✸ ✎ ☛ ✞ ✂ ✏ ✎ ✝ ☛ ✞ ❁ ✟ ✻ ✡ ✸ ✎ ✔ ✞ ✍ ✏ ✆ ☛ ✎ ✄ ✍ ☛ ☛ ✎ ✍ ✗ ✁ ☎ ✠ ✆ ✿ ☛ ✻ ✄ ✍ ✸ ☛ ✟ ✸ ✠ ✝ ☛ ✒ ✸ ✡ ☛ ☛ ✠ ✡ ✟ ☛ ✞ ✎ ☛ ✺ ✝ ✁ – p. 15/43 π 2 . ❚❱❯ π 1 ✗✑✸ ▲❆▼ π 2 ψ ( π 2 ) , ψ ❄❆❅ ψ ( π 1 ) ψ ( π 2 ) = I. I = 1 2 . . . N ☎✝✆ ★✜✩

  24. ☎ ✞ ✠ ✏ ❂ ✸ ✞ ✎ ✏ ✿ ☛ ☛ ✡ ✍ ✔ ☛ ✿ ✡ ☛ ✆ ✹ ✎ ✝ ✠ ✿ ✠ ✻ ✏ ✍ ✞ ☛ ✟ ✄ ✻ ✞ ✝ ✠ ✓ ✗ ✆ ✔ ✎ ☛ ✆ ☛ ✿ ✂ ☛ ✡ ✒ ✠ ☛ ✠ ✞ ✔ ✎ ☛ ✝ ✎ ✎ ✁ ✞ ✟ ✗ ☛ ✟ ✠ ✸ ✄ ✍ ✎ ✠ ✿ ☛ ✍ ✔ ✁ ☎ ✏ ✆ ✿ ✠ ✎ ✞ ✓ ❀ ✍ ✟ ☛ ✡ ✒ ✝ ✠ ✁ ✠ ✟ � ✗ ✟ ✞ ✎ ✝ ☛ ✓ ✞ ☎ ✒ ☛ ✎ ✄ ✍ ✸ ✞ ✎ ✄ ✍ ✞ ✄ ✻ ✟ ✸ ✁ ☛ ✿ ☛ ✓ ✺ ✠ ✄ ☛ ✡ ☎ ✺ ✄ ✞ ✻ ✝ ✄ ✎ ✎ ✄ ✸ ✠ ✻ ✏ ✆ ✺ ✸ ✠ ✻ ✄ ✍ ✏ ✠ ✞ ✻ ☛ ✿ ☛ ✠ ✸ ✌ ✻ ☎ ✍ ✠ ✿ ❀ ✝ ✠ ✡ ☛ ✏ ▼ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❊ ❉ P ● ✞ ❍ ❃ ❇ ❈❉ ❊ ❋ ● ❍ ◆❖ ❋ ❍ ■ ❏ ❑ ❉ ✺ ✎ ✍ ✎ ✓ ✡ ☞ ✠ ✞ ☛ ✟ ☛ ✎ ✝ ☛ ✍ ✓ ✞ ☛ ✄ ✠ ✏ ✻ ✠ ✿ ✌ ✸ ✍ ✍ ✄ ✎ ☛ ✓ ✺ ✔ ✟ ✎ ✓ ✓ ✸ ☛ ✡ ✒ ✝ ✠ ✁ ✠ ✟ ☛ ✸ ✔ ✎ ✔ ✌ ☛ ☛ ✞ ✍ ✄ ✍ ✞ ☛ ✿ ✻ ✁ ✗ ☛ ✻ ✞ ✂ ✠ ☛ ✝ ✒ ✎ ✓ ✔ ✓ ✏ ✁ ✗ ✟ ✞ ✻ ✄ ✂ ✡ ✄ ✄ ✿ ☛ ✻ ✡ ✄ ✟ ✆ � ✞ ☛ ✟ ✸ ✄ ✡ ✺ ☛ ✆ ✠ ✎ ✺ ✝ ✎ ✸ ☛ ✞ ✍ ✏ ✆ ☛ ✎ ✄ ✍ ☛ ☛ ✠ ✻ ✔ ✂ ✠ ✏ ✍ ✄ ☛ ✍ ✿ ✆ ✠ ☎ ✁ ✗ ✝ ✁ ✡ ✭ ✎ ✓ ❀ ✟ ✁ ✮✯ ✚ ✩ ✳ ✧ ✁ ✪ ✩ � ✵ ✿ � � ✩ ✴ ✦ ✥ ✦ ✣ ✰ ✩ � ★ ✦ ✮ ✆ ✌ ✌ ✔ ☛ ✁ ✁ ☛ ✞ ✠ ✻ ✏ ✠ ✁ ✠ ✺ ✓ ☛ ✿ ✸ ✸ ☎ ✓ ✎ ☛ ✿ ✆ ✠ ✁ ☛ ✗ ✍ ☛ ✎ ☛ ✍ ☛ ☛ ☛ ✸ ☛ ✎ ✄ ✍ ☛ ✻ ✄ ✍ ✏ ✠ ✆ ✠ ✿ ✍ ☛ ✺ ✻ ✌ ✸ ✞ ☛ ✸ ☛ ✎ ✡ ✞ ✓ ✎ ✠ ✞ ✺ ✓ ✔ ☎ ✺ ✡ ☛ ✓ ✓ ✎ ✏ ✠ ✝ ✔ ✓ ✎ ✔ ✻ ✎ ✒ ✠ ✍ ✍ ☛ ✎ ✡ ✓ ✠ ✝ ✄ ✁ ✎ ✔ ✻ ☛ ✟ ✠ ✓ ✍ ✍ ✸ ✎ ✝ ✎ ✸ ✔ ✺ ✞ ✠ ✎ ✓ ✡ ✎ ✻ ☛ ✔ ✎ ✎ ✞ ☛ ✟ ✸ ☛ ✡ ✒ ✝ ✠ ✁ ✠ ✟ ☛ ✠ ☛ ✞ ✒ ✡ ✎ ☛ ✞ ✂ ✏ ✎ ✝ ☛ ✞ ❁ ✸ ☛ ✡ ✟ ✠ ✁ ✟ ✠ ☛ ✟ ✝ – p. 15/43 π 2 . π 2 . ❚❱❯ π 1 ψ ✗✑✸ ✏✑✸ ✏✑✸ ▲❆▼ O ( N ) . π 2 ψ ( π 2 ) , ψ ❄❆❅ ψ ( π 1 ) ψ ( π 2 ) = I. I = 1 2 . . . N ☎✝✆ ★✜✩ ☎✝✆ ψ − 1 � ✖☛

  25. � ✆ ☛ ✸ ☛ ✻ ✎ ✞ ✓ ✎ ✠ ✻ ✾ ✿ ☛ ✡ ✆ ✻ ✠ ✝ ✍ ✎ ✠ ✠ � � � � ✁ � ☎ ✎ � ✂ ✄ ✡ ✼ ✏ ✎ ✄ ✾ ☛ ✆ ✍ ✏ ✄ ✂ � P ❑ ❉ ◆❖ ● ▼ P ❉ ❊ ❈ ■ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❏ ❍ � ✂ ✁ � ✠ � ✁ � ☎ ✎ � � ❋ ✄ ❃ ❇ ❈❉ ❊ ❋ ● ❍ ❍ ✁ ✠ ✂ ✄ ✁ � ✸ ❂ ✠ ✏ ✚ ✍ ☎ ✂ ✻ ☛ ✿ ☛ ✆ ✞ ✠ ❂ ✄ ✭ ✦ ✂ ✰ ✩ � ✧ ✣ ✆ ✳ ✯ ✤ ✛ ✮ ✥ ✧ ✰ ✧ ✝ ✤ ✭ � ✮ ✦ ✞ ✍ ✩ ✔ ❂ ✡ ✧ ✻ ✄ ✎ ☛ ✞ ☛ ☛ ✟ ✭ ✠ ✁ ✠ ✝ ✒ ✡ ☛ ✸ ✍ ☛ ✓ ✎ ☛ ✆ ✞ ✥ ✻ ✏ ✍ ✔ ✔ ✞ ✸ ✟ ✟ ❀ ✓ ✎ ✌ ✏ ✎ ✮ – p. 16/43 ❚❱❯ ▲❆▼ − ❄❆❅ ☎ ✑✸ − ✄ ✒✑ ⊲

  26. � ✻ ✓ ✎ ✆ ✠ ✾ ✿ ☛ ✡ ✆ ✠ ✎ ✝ ✍ ✄ ✠ ✂ � ✁ � ✠ ✞ ✻ ✁ ✄ ☎ ✎ � � � ✄ ✼ ✏ ✎ ✾ ☛ ☛ ✆ ✍ ✏ ✡ ✠ ✻ ✎ ☛ ✸ � � ✁ ❊ ■ ❏ ❑ ❉ ◆❖ ● ▼ P ❉ P ❋ ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❍ ❍ ☎ ✁ ✎ � ✂ � ✄ ✂ ☎ ✁ ✠ ✎ ❍ � ✂ � ✄ ❃ ❇ ❈❉ ❊ ❋ ● � ✂ � ✭ ✍ ✩ ✄ ✮ ☎ ✩ ✻ ✰ ☛ ✿ ✏ ☛ ✆ ✞ ❂ ✠ ✄ ✞ ✂ ✍ � ✠ ✎ ✤ ✯ ✤ ✛ ✣ ✮ ✧ ✰ ✧ ✝ ✭ ❂ � ✮ ✦ ✥ ✦ ✂ ✧ ✁ � ✸ ☛ ☛ ✆ ✧ ✄ ✎ ☛ ✞ ✔ ☛ ✟ ✠ ✁ ✠ ✡ ✝ ✒ ✡ ☛ ✠ ✂ � ✭ ✁ � ✻ ❂ ✆ ✟ ✞ ✚ ✻ ✏ ✍ ✔ ✓ ✔ ✸ ✟ ✸ ❀ ✥ ✓ ✎ ✌ ✏ ✎ ✞ ✍ ☛ ✳ – p. 16/43 ❚❱❯ ▲❆▼ − ❄❆❅ ☎ ✑✸ − − − − − − ✄ ✒✑ ⊲ ⊲

  27. � ✁ ✡ ✆ ✻ ✠ ✝ ✍ ✄ ✠ ✂ � � ✿ ✠ � ✁ � ☎ ✎ � ✂ � ✄ ☛ ✾ ☎ ✡ ✄ ✼ ✏ ✎ ✄ � ☛ ✆ ✍ ✏ ✠ ✠ ✻ ✎ ☛ ✸ ☛ ✻ ✎ ✞ ✓ ✎ ✆ ✂ ✁ ✂ ❊ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ P ❉ P ❍ ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❋ ❍ ✠ � ✁ ✎ � ✂ � ✄ ✂ ☎ ✁ ✠ ✎ ● � ✁ � ✂ � ✄ ❃ ❇ ❈❉ ❊ ❋ � ✾ � � ✆ ✿ ✣ ☛ ✧ ✆ ✞ ❂ ✄ ✞ ✂ ✳ ✩ ✍ ✎ ☛ ✮ ✎ ✩ ☛ ✰ ✆ ☛ ✻ ✻ ✥ ✮ ✧ ✰ ✧ ✝ ✤ ✭ � ✮ ✦ ✦ ☎ ✂ ✯ ✁ � ✸ ❂ ✠ ✏ ✍ ✄ ✞ ✏ ✛ ✠ ✔ ☛ ✟ ✠ ✁ ✠ ✝ ✒ ✡ ☛ ✂ ☛ � ✁ � ✠ � ✧ ✁ � ☎ ✭ ✞ ✎ ✍ ✚ ✭ ✔ ✓ ✔ ✸ ✟ ✟ ❀ ✓ ✎ ✌ ✄ ✏ ✎ ✞ ✍ ☛ ✸ ❂ ✡ ✻ ✥ ✤ – p. 16/43 ❚❱❯ ▲❆▼ − ❄❆❅ ☎ ✑✸ − − − − − − − − − ✄ ✒✑ ⊲ ⊲ ⊲

  28. � � ✠ ✂ � ✁ � ✠ � ✁ � ☎ ✎ ✂ ✍ � ✄ ✂ ☎ ✁ ✠ ✁ ✎ � ✂ � ✄ ✝ ✂ ✻ ☛ ✆ ✍ ✏ ✡ ✠ ✻ � ☛ ✸ ☛ ✎ ✠ ✞ ✓ ✎ ✆ ✠ ✾ ✿ ☛ ✡ ✆ ✻ ✄ ☎ ✄ ❉ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ P ❊ ● P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❍ ❋ ✁ ✁ ✠ � ✎ � ✁ � ✂ � ✄ ✂ ☎ � ❊ ✠ � ✎ � ✁ � ✂ � ✄ ❃ ❇ ❈❉ ✾ ✎ ✻ ✆ ✞ ❂ ✄ ✞ ✂ ✍ ☛ ✎ ☛ ✳ ✆ ✆ ✞ ✣ ✎ ✧ ✏ ✍ ✔ ✓ ✔ � ✸ ✯ ✤ ✟ ✁ ✧ ✝ ✤ ✭ � ✮ ✦ ✥ ✦ ✂ ✮ � ☛ ✸ ❂ ✠ ✏ ✍ ✄ ☎ ✻ ☛ ✿ ✛ ✩ ✮ ✧ ✠ ✝ ✒ ✡ ✧ ☛ ✠ ✂ � ✁ � � ✁ ✁ � ☎ ✭ ✎ � ✂ � ✄ ✼ ✏ ✠ ✠ ✟ ✍ ✩ ❀ ✰ ✓ ✭ ✎ ✌ ✏ ✎ ✞ ✚ ☛ ✟ ✸ ❂ ✡ ✻ ✄ ✎ ☛ ✥ ✞ ✔ ☛ ✰ – p. 16/43 ❚❱❯ ▲❆▼ − ❄❆❅ ☎ ✑✸ − − − − − − − − − − − ✄ ✒✑ ⊲ ⊲ ⊲ ⊲

  29. � ✠ ✁ � ☎ ✎ � ✂ � ✄ ✂ ☎ ✁ ✁ ✠ ✎ � ✂ � ✄ ✂ ☎ ✁ ✠ � ✎ � � � � ✠ ✠ ✻ ✎ ☛ ✸ ☛ ✻ ✎ ✞ ✓ � ✆ ✾ ✁ ✿ ☛ ✡ ✆ ✻ ✠ ✝ ✍ ✄ ✠ ✂ � ✁ ✂ ✏ P ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ ❉ ❋ ❊ P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ● ❊ � � ✄ ✂ ☎ ✁ � ✠ � ✎ � ✁ � ✂ ✄ ❈❉ ✂ ☎ ✂ ✁ ✎ ✠ � ✁ � ✄ ❃ ❇ ✡ ✎ ✍ ✄ ❂ ✠ ✏ ✍ ✄ ☎ ✻ ☛ ✿ ☛ ✆ ✞ ❂ ✞ � ✂ ✍ ☛ ✎ ☛ ✆ ✞ ✻ ✏ ✆ ✔ ✓ ✔ ✸ ✁ ✟ ✳ ✭ ✧ ✥ ✚ ✭ ✰ ✩ ✮ ✩ � ✧ ✣ ✆ ✯ ✂ ✤ ✛ ✮ ✧ ✰ ✧ ✝ ✤ ✭ � ✮ ✦ ✥ ✦ ✸ ✍ ✔ ✁ ✻ ✁ ✄ � ✎ ✠ ☛ � ✞ � ☎ ✟ ☛ ✟ ✠ ✁ ✂ ✠ ✝ ✒ ✡ � ✡ ☛ ✎ ☛ ❀ ✾ ✓ ✄ ✎ ✌ ✏ ✎ ✏ ✎ ✞ ✼ ✄ ✍ � ☛ ✂ ✸ � ❂ ✠ – p. 16/43 ❚❱❯ ▲❆▼ − ❄❆❅ ☎ ✑✸ − − − − − − − − − − − − − − − − ✄ ✒✑ ⊲ ⊲ ⊲ ⊲ ⊲

  30. � ✠ ☎ ✁ ✠ ✁ ✎ � ✂ � ✄ ✂ ☎ ✁ � ✄ ✎ � ✁ � ✂ � ✄ ✂ ☎ ✁ � ✠ � ✂ � � ✝ ✞ ✓ ✎ ✆ ✠ ✾ ✿ ☛ ✡ � ✻ ✠ ✍ ✂ ✄ ✠ ✂ � ✁ � ✠ � ✁ � ☎ ✎ � ✎ ✁ ✻ ▼ ❋ ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● P ❈❉ ❉ ❊ P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❊ ❇ � ✂ ✂ � ✄ ✂ ☎ ✂ ✁ ✎ ✠ � ✁ � ✄ � ❃ ✠ � ✎ � ✁ � ✂ � ☎ � ✁ � ✄ ✎ ✆ ☛ ✍ ✏ ✍ ✄ ☎ ✻ ☛ ✿ ☛ ✆ ✞ ❂ ✄ ✞ ✂ ☛ ❂ ✓ ✓ ✸ ✟ ✟ ✸ ✔ ✔ ✎ ✍ ✏ ✻ ✞ ✆ ☛ ✠ ✸ ✌ ✯ ✭ ✧ ✥ ✚ ✭ ✰ ✩ ✮ ✩ � ✧ ✣ ✆ ✳ ✤ � � ✁ ✂ ✦ ✥ ✦ ✮ ✭ ✛ ✤ ✝ ✧ ✰ ✧ ✮ ✎ ❀ ☛ � ✆ ✒ ✡ ✏ ☛ ✏ ✠ ✂ � ✁ � ✠ ✁ ✁ � ☎ ✎ � ✂ ✍ � ✄ ✼ ✏ ✎ ✄ ✾ ✠ ✝ ✡ ✞ ✄ ✻ ✡ ❂ ☛ ✸ ✻ ☛ ✍ ✎ ✠ ✔ ✞ ☛ ✎ ✟ ☛ ✠ ✎ – p. 16/43 ❚❱❯ ▲❆▼ − ❄❆❅ ☎ ✑✸ − − − − − − − − − − − − − − − − ✄ ✒✑ ⊲ ⊲ ⊲ ⊲ ⊲ ⊲

  31. � ✞ ☛ ✞ ❂ ✻ ✏ ✠ ✂ ✆ ✝ ✠ ✍ ✠ ✔ ❂ ✝ ✠ ✿ ✾ ✠ ✆ ✎ ✓ ❂ ✺ ✎ ✓ ✆✞ ✞ ✂ ✝ ✄ ✡ ✠ ✎ ✑ ✞ ✎ ✞ ❀ ✟ ✟ ✠ ✍ ✒ ✆✞ ☛ ✠ ✟ ✞ ✻ ☛ ✞ ✞ ✂ ✝ ✄ ✡ ☛ ✻ ☎ ✍ ✓ ✁ ☛ ✠ ✎ ☛ ✻ ☛ ✓ ✣ ✆ ✿ ☛ ✆✞ ✄ ☛ ✾ ✻ ✝ ✠ ✡ ✡ ✆ ☛ ✟ ✏ ✠ ☛ ✡ ✑ ☛ ✍ ✞ ✓ ✗ ✆ ✺ ✞ ☛ ✻ ✟ ✆ ✟ ☎ ❋ ❊ ❈❉ ❇ ❃ ✄ � ✂ � ✁ ✎ ❍ � ✠ � ✁ � ✂ ✠ ☛ ✡ ✺ ● ❍ ✠ P ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ ❊ ❋ ❉ P ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ✝ ✁ ✡ ✎ ✠ � ✁ � ✂ ✠ ✗ ✾ ✠ ✆ ✓ ✁ ✞ ✎ ✻ ☛ ✻ ✔ ✎ ✞ ✄ ✡ � � ✔ ✁ ✁ ✁ ☛ ✻ ✔ ✎ ☛ ✍ ✓ ✄ ☎ ☎ ✎ ✠ ✿ ☛ ✄ ✍ ✄ � ✂ � ✎ ✼ ✠ ☛ ☎ ❂ ✆✞ ✞ ✂ ✝ ✄ ✡ ☛ ✻ ✞ ☛ ✮ ☛ ✻ ✡ ✄ ✁ ✸ ☛ ✡ ✺ ✞ ❂ ✠ ✌ ✎ ✻ ☛ ✄ ✥ ✍ ✑ ✻ ✻ ✡ ✠ ☛ ✆ ✄ ✟ ☛ ✻ ✹ ✟ ✞ ✺ ✝ ✁ ✒ ✯ ✧ ☛ � ✓ ✠ ✆ ✿ ☛ ✻ ❀ ☛ ✟ ✠ ✪ ✍ ✓ ✆✞ ✁ ✁ ✧ ✣✤ ✻ ✎ ✔ ✞ ✁ ✁ ☛ ✸ ☛ ✞ ✍ ✭ ✆✞ ✂ ✠ ✝ ✄ ✡ ✠ ✻ ☎ ✍ ✓ ✞ ✁ ✡ ✄ ✛ � ✆✞ ✞ ✂ ✝ ✄ ✡ ☛ ✎ ✄ ✂ ✥ � ✎ ✧ ☎ � ✁ � ✭ ✠ � ☛ ✟ � ☛ ☎ ✍ ✓ ✞ ✁ ✠ ✎ ☛ ✻ ✓ ☛ ✠ ✆ ✿ ✞ ☛ ✻ ✡ ✄ ✩ ✞ ✁ ✂ ✎ ★ ☞ ✰ ✝ ✠ ✿ ✠ ✞ ✩ ✠ ✎ ☛ ✑ ☛ ✤ ✍ ✔ ✓ ✏ ✆ ✿ ☛ ✞ ✿ ✠ ✻ ☛ ✡ ✒ ✝ ✠ ✁ ✠ ✟ ☛ ✧ ✔ ☛ ✎ ✓ ☛ ✍ ✂ ✞ ✄ ❂ ✞ ✆ ✆ – p. 17/43 π, ❚❱❯ ☎ ✷✎ ✒✝✠ ▲❆▼ ✏ ✑✸ ( π i , π j ) − − − ✏ ✑✸ ❄❆❅ − (3 , − 2) ✒✝✠ (1 , − 2) (1 , − 2) | π i | − | π j | = ± 1 . ☎ ✑✸ ✏✑✸

  32. � ✾ ✻ ✎ ✞ ✓ ✎ ✆ ✠ ✗ ✔ ✟ ☛ ✞ ✄ ✡ ✻ ☛ ☛ ✻ ✔ ✓ ✄ ✎ ☛ ✣ ✿ ✆ ✠ ☛ ☎ ✻ ☛ ✎ ✠ ✁ ✞ ✓ ✍ ✞ ✠ ✞ ■ ● ▼ P ❉ ❊ P ❈ ◗ ❉ ❘ ❅ ■ ▼ ❉ ❙ ▼ ◆❖ ❑ ☛ ❇ ✻ ✟ ☛ ✞ ☛ ✻ ❃ ❈❉ ❏ ❊ ❋ ● ❍ ❍ ❋ ❍ ■ ✡ ✞ ✍ ✩ ✻ ✎ ✤ ★ ✞ ✓ ✎ ✆ ☛ ✠ ✠ ✰ ✧ ✭ ✧ ✾ ✥ ✮ ✿ ✧ � ✯ ✪ ✁ ✛ ✆ ✣✤ ✁ ✔ ☞ ✄ ✻ ✞ ✟ ✞ ✑ ✗ ✠ ✭ ✡ ✁ ✞ ✓ ✍ ☎ ✻ ✠ ✄ ✎ ✝ ✂ ✞ ✆✞ ✩ ✠ ✻ ✠ ☛ ✝ ✞ ✟ ☛ ✥ ✆ ✡ ✡ ✄ ☎ ✻ ✻ ☛ ✿ ✆ ✠ ✓ ☛ ✧ – p. 18/43 ❚❱❯ π i + π j = − 1 . π i + π j = +1 ▲❆▼ ❄❆❅ ✒✝✠ , , ρ ( i, j − 1) ρ ( i + 1 , j )    ( π i , π j )

  33. � ✎ ✍ ☎ ✻ ✔ ✸ ☛ ✻ ✎ ✞ ✓ ✎ ✆ ✠ ✾ ✠ ✞ ✆ ✎ ☛ ✻ ☛ ✓ ✠ ✿ ✿ ✻ ✌ ✎ ✸ ✗ ✁ ✓ ✁ ✁ ✻ ✣ ✆ ✡ ✓ ✠ ✝ ✍ ✄ ✎ ✹ ✆ ☛ ✓ ☛ ✓ ✠ ✆ ✎ ☛ ✻ ☛ ✓ ✠ ✿ ✟ ✸ ☛ ✞ ✍ ✆ ✠ ✠ ✞ � ◆❖ ❃ ❇ ❈❉ ❊ ❋ ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ● ☛ ❘ ▼ ❙ ❉ ▼ ■ ❅ ◗ ▼ ■ ❈ P ❊ ❉ P ✟ ✡ ✓ ✝ ✓ ✻ ✹ ✆✞ ☛ ✂ ✄ ✔ ✡ ✻ ✌ ✻ ☎ ✍ ✎ ✻ ✒ ☛ ✝ ✠ ✁ ✠ ✟ ☛ ✓ ✿ ✝ ✠ ✎ ✟ ✠ ✆ ✸ ✠ ☎ ✻ ✆ ✡ ✡ ✄ ✞ ☎ ✻ ☛ ✿ ✆ ✠ ✓ ☛ ☛ ✭ ✎ ✠ ✁ ✞ ✓ ✍ ✻ ✻ ✠ ✡ ✄ ✝ ✂ ✁ ☛ ✆✞ ✔ � ✛ ✆ ✣✤ ✧ ✁ ☞ ✄ ✻ ✞ ✟ ✞ ✑ ✧ ✟ ☛ ✻ ✎ ✞ ✓ ✎ ✆ ✠ ✾ ✗ ✿ ✠ ✝ ✞ ✮ ✯ ✠ ✆ ✠ ✾ ✗ ✟ ☛ ✞ ✄ ✡ ✻ ☛ ✞ ✔ ★ ✓ ✩ ✰ ✧ ✭ ☛ ✻ ✧ ✟ ☛ ✞ ✥ ✩ ☛ ✎ ✞ ✠ ✠ ✻ ✠ ✍ ✞ ✡ ✥ ✄ ✎ ☛ ✞ ✿ ✤ ✆ ✓ ✎ ☛ ✻ ☛ ✎ ✠ ✁ ✞ ✓ ✍ ☎ ✻ ✔ ☛ ✻ ✪ – p. 18/43 ✗✑✸ ❚❱❯ ✏ ✷✿ π i + π j = − 1 . π i + π j = +1 ▲❆▼ ❄❆❅ ✒✝✠ , , ρ ( i, j − 1) ρ ( i + 1 , j )    ✒✝✠ ( π i , π j )

  34. � ☛ ✆ ☛ ❂ ✄ ✡ ✂ ✍ ✎ ✿ ✠ ✸ ✠ ✂ � ✁ � ☛ ✝ � ✠ ✞ ✓ ✎ ✆ ✠ ✾ ✎ ✝ ✠ ✎ ✄ ✁ ✄ ✝ ✎ ☛ ✡ ✠ ✁ ✻ ■ ● ▼ P ❉ ❊ P ❈ ◗ ❉ ❘ ❅ ■ ▼ ❉ ❙ ▼ ◆❖ ❑ � ❇ ☎ ✎ � ✂ � ✄ ❃ ❈❉ ❏ ❊ ❋ ● ❍ ❍ ❋ ❍ ■ ✧ ✎ ☛ ★ ✩ ✥ ✧ ✭ ✧ ✰ ✩ ✤ � ✥ ✮ ✁ ✭ ✦ ✂ ☎ ✭ ✣ ✣ ✤ ✧ ✣ ✆ ✳ ✯ ✤ ✛ ✮ � ✩ ✚ ✣ ✯ ✥ ✦ ✂ ✙ ✢ ✸ ✰ ✥ ☛ ✸ ✿ ✆ ✠ ✓ ☛ ✻ ✎ ✰ ✠ ✁ ✞ ✓ ✍ ☎ ✻ ✔ ✔ ✎ ✆ ✠ ✣ ✩ ✱ ✁ � ✸ ✠ ❂ ✏ ✍ ✄ ✎ ✠ ✓ ✟ – p. 19/43 ❚❱❯ ✗✑✸ ✲ ✄✂ ▲❆▼ − ❄❆❅

  35. � ✝ � ✄ ✄ ✿ ✆ ✠ ✟ ✎ ✠ ✓ � ☛ ☛ ✟ ✠ ✁ ✠ ✝ ✒ ✡ ☛ ✂ ✎ ✡ ✎ ✿ ✧ ✆ ☛ ❂ ✄ ✡ ✂ ✍ ☛ ✠ ☎ ✸ ✠ ✂ � ✁ � ✠ � ✁ � ✄ ✻ ✠ ❊ ■ ❏ ❑ ❉ ◆❖ ● ▼ P ❉ P ❋ ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❍ ❍ ☛ ☎ ✞ ✔ ✠ ✂ � ✁ � ✠ � ✎ ✁ ❍ � ✂ � ✄ ❃ ❇ ❈❉ ❊ ❋ ● ✝ ☛ ✡ ☎ ✰ ✩ ★ ✤ ✥ ✮ ✁ ✭ ✦ ✂ ✭ ✭ ✰ ✤ ✥ ✰ ✣ ✩ ✱ ✁ � ☛ ✧ ✧ ✠ ✣ ✣ ✆ ✳ ✯ ✤ ✛ ✮ ✧ � ✚ ✯ ✥ ✥ ✦ ✂ ✙ ✢ ✩ ✣ � ✣ ✩ ❂ ✸ ✏ ✎ ✍ ☎ ✻ ✔ ✸ ☛ ✻ ✎ ✞ ✓ ✆ ✞ ✠ ✾ ✎ ✠ ✝ ✎ ✄ ✁ ✄ ✝ ✎ ✍ ✓ ✁ ✠ ✄ ✎ ✠ ✓ ✟ ✠ ✆ ✎ ✔ ✸ ✿ ✆ ✻ ✓ ✎ ☛ ✠ ☛ – p. 19/43 ❚❱❯ ✗✑✸ ✲ ✄✂ ▲❆▼ − − − ❄❆❅ ✏ ✷✿ √

  36. � ✠ ✠ ✾ ☛ ✎ ✄ ✍ ✓ ✎ ✡ ✆ ✠ ✟ ✓ ✞ ✏ ☛ ✺ ✠ ✻ ☎ ✍ ✓ ✞ ✁ ✎ ✿ ☛ ✻ ☛ ✓ ✠ ✆ ✄ ☎ ✡ ✄ ✠ ✔ ✞ ☛ ✻ ✡ ☛ � ✡ ✒ ✝ ✠ ✧ ✠ ✂ ✁ ☛ � ✡ ✠ ✿ ✠ ✄ ✄ ✂ � � ✁ ☎ ✎ � ✠ ☛ ✄ ☛ ❍ ❑ ❏ ■ ❍ ❋ ❍ ● ◆❖ ❋ ❊ ❈❉ ❇ ❃ ✞ ❉ ● ✟ ❘ ▼ ❙ ❉ ▼ ■ ❅ ◗ ▼ ■ ❈ P ❊ ❉ P ☛ ☛ ✝ ☛ ✒ ✝ ✠ ✁ ✠ ✟ ✸ ☛ ✔ ✎ ☛ ✆✞ ✞ ✂ ✡ ✸ ✎ ✂ ✞ ☛ ✻ ✡ ✄ ✄ ✝ ✿ ✟ ☎ ✠ ✆ ✿ ✝ ✠ ✟ ✁ ☛ ✱ ✏ ✠ ❂ ✸ � ✁ ✩ ✄ ✣ ✰ ✥ ✤ ✰ ✭ ✍ ✎ ✂ ✆ ✓ ☛ ✻ ☛ ✓ ✠ ✿ ✠ ✸ ✔ ✎ ✆ ✠ ✟ ✓ ☎ ✦ ✁ ✧ ✦ ✥ ✯ ✣ ✚ � ✮ ✙ ✛ ✤ ✯ ✳ ✆ ✣ ✂ ✢ ✭ ✰ ✁ ✮ ✥ ✤ ★ ✩ ✧ ✩ ✭ ✧ ✥ ✩ ✣ � ✣ ✠ ✎ ✞ ✠ ❂ ✄ ✡ ✂ ✓ ✍ ☛ ✎ ✎ ✟ ✠ ✠ ✸ ✂ ✆ � ✆ ✁ ✿ � ✠ � ✁ � ✄ ☎ ✄ ✎ � ☛ ☛ � ✾ ✍ ☎ ✻ ✔ ✸ ☛ ✻ ✎ ✞ ✓ ✎ ✆ ✠ ✝ ✿ ✎ ✠ ✝ ✎ ✄ ✁ ✄ ✝ ✎ ☛ ✠ ✡ ✠ ✝ ✂ – p. 19/43 ❚❱❯ ✗✑✸ ( − 6 , 7) . ✲ ✄✂ ▲❆▼ ( − 5 , 4) − − − ❄❆❅ ✏ ✷✿ √ √

  37. � ☛ ✏ ✎ ✄ ✆ ☛ ✿ ✆ ✠ ✓ ☛ ✻ ✎ ✓ ✠ ✁ ✞ ✓ ✍ ☎ ✻ ☛ ✡ ✄ ✝ ✂ ✓ ✆✞ ✆✞ ✻ ☛ ✸ ✟ ☛ ✞ ✝ ✿ ✞ ✆ ✺ ✠ ✝ ✏ ✄ ✓ ✠ ✎ ✠ ❂ ✝ ✻ ☛ ✎ ✏ ✻ ✿ ✞ ☛ ✍ P ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ ❉ ❊ ❊ P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❋ ❈❉ ✓ ✸ ✎ ✞ ✸ ☛ ✟ ✠ ✁ ✠ ✝ ✒ ✡ ✄ ✿ ❇ ✠ ✝ ✿ ✆ ✠ ✟ ✎ ✠ ✝ ✻ ✟ ❃ ✤ ✞ ✆ ✞ ✶ ✄ ✡ ✺✻ ✄ ✞ ☛ ✆ ✟ ✄ ✎ ✄ ✦ ✻ ❂ ✞ ☛ ✾ ✠ ✻ ✻ ☛ ✄ ✾ ✠ ✁ ✥ ✍ ✛ ✧ ☎ � � ✢ ✲ ✙ ✰ ✣ ✧ ✰ ✦ ✤ ✦ ✢ ✲ ✛ ✤ ✧ ✁ ✧ ✝ ✤ ✭ � ✮ ✆ ☛ ✏ ✠ ✠ ✾ ✠ ✞ ✠ ✿ ✠ ✡ ✄ ✸ ☎ ✺ ✝ ✎ ✻ ✍ ✄ ✂ ✞ ✓ ✎ ✞ ✟ ✏ ✓ ✟ ✡ ✆ ✓ ✁ ✠ ✝ ✍ ✄ ✿ ✆ ✠ ✓ ☛ ✻ ☛ ✎ ✠ ✻ ✞ ✓ ✍ ☎ ✻ ✄ ✸ ✞ ✎ ☛ – p. 20/43 ❚❱❯ ✒✝☛ ▲❆▼ ❄❆❅ ☎ ✑✆ ✏ ✷✿ ★✜✩ ☎ ✑✸

  38. � ✁ ✡ ✄ ✝ ✂ ✞ ✆✞ ☛ ✓ ✎ ✞ ✸ ☛ ✟ ✠ ✠ ✻ ✝ ✒ ✡ ✄ ✸ ✿ ✠ ✝ ✿ ✆ ✠ ✟ ✎ ✠ ☛ ☎ ✻ ✎ ✠ ❂ ✝ ✻ ☛ ✎ ✏ ✻ ✿ ✄ ✆✞ ✓ ✓ ✏ ✄ ✍ ✆ ☛ ✿ ✆ ✠ ✓ ☛ ✻ ☛ ✎ ✠ ✁ ✞ ✓ ✝ ✟ ✠ ● ❇ ❈❉ ❊ ❋ ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ▼ ✌ P ❉ ❊ P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❃ ✔ ❀ ☛ ✡ ✻ ☛ ✞ ✏ ✍ ✌ ✸ ✍ ✞ ☛ ✎ ✠ ✞ ✓ ✎ ✗ ✓ ✞ ✎ ✁ ☎ ☎ ✟ ✎ ✞ ✂ ✔ ✎ ☛ ✆ ✤ ✎ ✓ ✠ ✄ ✞ ☛ ✆ ✟ ✄ ✎ ✞ ✄ ✻ ❂ ✞ ☛ ✾ ✻ ✡ ✻ ☛ ✄ ✾ ☛ ✆ ✍ ✏ ✡ ✠ ✝ ✍ ✄ ✏ ✺✻ ✄ ✠ ✢ ✧ ☎ � � ✢ ✲ ✙ ✰ ✣ ✧ ✰ ✦ ✛ ✤ ✲ ✶ ✛ ✤ ✧ ✁ ✧ ✝ ✤ ✭ � ✮ ✦ ✥ ✦ ✁ ✆ ✿ ✓ ✠ ☛ ✝ ✻ ✍ ✄ ✂ ✞ ✓ ✎ ✞ ✟ ✏ ✓ ✟ ✆ ☎ ✿ ✻ ✝ ✠ ✺ ✆ ✞ ✝ ✍ ✞ ☛ ✟ ✸ ☛ ✞ ✺ ✠ ✸ ✎ ✁ ✍ ☎ ✻ ✄ ✸ ☛ ✠ ✄ ✻ ✎ ✞ ✓ ✞ ✎ ✆ ✠ ✾ ☛ ✠ ✞ ✠ ✻ ✿ ✠ ✡ ✓ – p. 20/43 ❚❱❯ ✒✝☛ ▲❆▼ ❄❆❅ ☎ ✑✆ ✏ ✷✿ ☎ ✷✎ ★✜✩ ☎ ✑✸

  39. � ✄ ✸ ✌ ✿ ✞ ☛ ✎ ✄ ✏ ✻ ✟ ✞ ❂ ✡ ✠ ✿ ☛ ✑ ✔ ✎ ✏ ✿ ☛ ✄ ✟ ✄ ✻ ✒ ✁ ✆ ✄ ✿ ☛ ✞ ✂ ✑ ☛ ✎ ✓ ✞ ✞ ☛ ✕ ✟ ✔ ✎ ✓ ✞ ✎ ✁ ☎ ☎ ✞ ✔ ✞ ✿ ✎ ✑ ✞ ✎ ☛ ✍ ✂ ✞ ✆ ✽ ✌ ✔ ✓ ✿ ✞ ✎ ✁ ☎ ☎ ✗ ✟ ✞ ✂ ✔ ✤ ✔ ☛ ✎ ✟ ✠ ✆ ✿ ☛ ✡ ✠ ✿ ✠ ✔ ✑ ✗ ✞ ✆ ✂ ✔ ✎ ☛ ✆ ✎ ✓ ✌ ✻ ✞ ✿ ☛ ✡ ✠ ✆ ✂ ❇ ❃ ✟ ✡ ✄ ✟ ✆ ☛ ✗ ✟ ✞ ✔ ❊ ✎ ☛ ✆ ✎ ✓ ✌ ✻ ✞ ✿ ☛ ✆ ❈❉ ❋ ✿ ❊ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❉ ● P ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ☛ ✠ ✝ ✎ ✿ ✸ ✔ ✓ ✔ ✍ ✏ ✻ ✞ ✆ ☛ ✻ ☛ ✄✌ ✓ ✹ ✎ ✿ ✆✞ ✄ ✿ ✻ ✌ ✎ ✆ ✂ ✑ ✟ ✻ ✌ ✻ ☎ ✍ ✠ ❂ ✄ ❂ ✻ ✹ ✞ ✍ ✂ ✠ ✁ ✠ ✞ ☎ ✻ ✹ ✟ ✞ ✎ ☛ ☛ ✎ ✎ ✻ ✸ ✄ ✻ ☎ ✍ ✓ ✞ ✁ ✠ ✎ ☛ ☛ ✻ ✓ ✠ ✆ ✿ ✄ ✍ ✝ ✠ ✡ ✏ ✍ ☛ ✎ ☛ ✸ ✓ ✓ ✞ ✂ ✄ ✍ ✻ ✝ ✠ ✺ ☎ ✄ ✞ ✡ ✠ ✿ ✠ ✞ ✠ ✾ ✠ ✆ ✎ ✓ ✆ ✾ ✟ ✛ ✭ ✤ ✝ ✧ ✁ ✧ ✤ ✛ ✲ ✢ ✤ ✦ ✮ ✰ ✧ ✣ ✰ ✙ ✲ ✢ � � ☎ ✧ � ✦ ✄ ✎ ☛ ✻ ✻ ✠ ✾ ☛ ✞ ❂ ✻ ✄ ✞ ✄ ✥ ✟ ✆ ☛ ✞ ✄ ✺✻ ✡ ✄ ✶ ✁ ✦ ✞ ✎ ✏ ✞ ✸ ✄ ✡ ✒ ✝ ✠ ✁ ✠ ✟ ☛ ✸ ✎ ✠ ✓ ☛ ✆✞ ✞ ✂ ✝ ✄ ✡ ☛ ✻ ✓ ✿ ✝ ✓ ✞ ☛ ✞ ✠ ✎ ☛ ✞ ✍ ✸ ✌ ✍ ✏ ☛ ✿ ✻ ✡ ❀ ✟ ✻ ✝ ✠ ✎ ✟ ✠ ✆ ✍ ☎ ✞ ✞ ✁ ✎ ✠ ✓ ✏ ✻ ✝ ✠ ✺ ✆ ✿ ✝ ✝ ✞ ☛ ✟ ✸ ☛ ✞ ✍ ✆ ✠ ✟ ❂ ✠ ✻ ✄ ✠ ✎ ☛ ✻ ☛ ✓ ✠ ✆ ✿ ☛ ✆ ☛ ✎ ✏ ✎ ✏ ✻ ✿ ✄ ✓ ✆✞ ✓ – p. 20/43 ❚❱❯ ✒✝☛ ▲❆▼ ❄❆❅ ☎ ✑✆ ✏ ✷✿ ☎ ✷✎ ★✜✩ ☎ ✑✸ Bergeron 90%

  40. � ✻ ✆ ✿ ✸ ✔ ✓ ✔ ✍ ✏ ✻ ✞ ✆ ☛ ✎ ✄✌ ✂ ✓ ✹ ✎ ✿ ✆✞ ✄ ✿ ✻ ✌ ✎ ✝ ✠ ✿ ✡ ☛ ✍ ✄ ❂ ☛ ✆ ☛ ✿ ✠ ✑ ✻ ✌ ✻ ☎ ✍ ✠ ❂ ✄ ✻ ☛ ✹ ✟ ✞ ✂ ✠ ✁ ✠ ✞ ☎ ✻ ✹ ✟ ✞ ✎ ✆ ✿ ✞ ✟ ✞ ✄ ❂ ✠ ✿ ☛ ✑ ✔ ✎ ✏ ✿ ☛ ✄ ✔ ✻ ✒ ✞ ✎ ✠ ✆ ✿ ☛ ✡ ✠ ✿ ✠ ✔ ✤ ✗ ✟ ✏ ☛ ✞ ✞ ✂ ✑ ☛ ✎ ✓ ✞ ✁ ✞ ✕ ✟ ✔ ✎ ✓ ✎ ✄ ✁ ☎ ☎ ✞ ☛ ✻ ✡ ✄ ✸ ✌ ✿ ✞ ☛ ✎ ✿ ✻ ✞ ✓ ✿ ✞ ✄ ✎ ✟ ✠ ✆ ✿ ✸ ✔ ✓ ☛ ✎ ✏ ✸ ✿ ☛ ✸ ✔ ✎ ✏ ✍ ✓ ✆✞ ✠ ✻ ✠ ✎ ✏ ✌ ✠ ☛ ▼ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❊ ❉ P ● ❃ ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ● ❋ ❊ ❈❉ ❇ ✿ ✻ ✌ ✟ ✌ ✻ ✞ ✿ ☛ ✆ ☛ ✿ ✔ ✏ ✠ ✾ � ✡ ✎ ✄ ✟ ✆ ☛ ✗ ✟ ✞ ✂ ✔ ✎ ☛ ✆ ✎ ✓ ✓ ✆ ☛ ✞ ✡ ✄ ✒ ✝ ✄ ✎ ☛ ✟ ✄ ✎ ✏ ✎ ✎ ✓ ✎ ☛ ✁ ☎ ☎ ✞ ☛ ✻ ✡ ✄ ✗ ✟ ✞ ✂ ✔ ✎ ✟ ✑ ✂ ✎ ✄ ✡ ✠ ✿ ✠ ✞ ✠ ✾ ✠ ✆ ✎ ✓ ✞ ✻ ☎ ☛ ✸ ✄ ✻ ☎ ✍ ✓ ✞ ✁ ✠ ✎ ☛ ✻ ☛ ✸ ✺ ✠ ✍ ✝ ✠ ✔ ✆ ✞ ✿ ✝ ✞ ☛ ✟ ✸ ☛ ✞ ✆ ✠ ✠ ✟ ✓ ✏ ✟ ✞ ✎ ✓ ✞ ✂ ✄ ✍ ✻ ✝ ✓ ✆ ✏ ✲ ✦ ✥ ✦ ✮ � ✭ ✤ ✝ ✧ ✁ ✧ ✤ ✛ ✢ ✶ ✤ ✛ ✦ ✰ ✧ ✣ ✰ ✙ ✲ ✢ � � ☎ ✧ ✁ ✄ ✿ ✠ ✄ ✍ ✝ ✠ ✡ ✏ ✍ ✆ ☛ ✾ ✄ ☛ ✻ ✻ ✾ ✡ ☛ ✞ ❂ ✻ ✄ ✞ ✎ ✄ ✟ ✆ ☛ ✞ ✄ ✺✻ ✻ ✺ ✓ ✞ ✞ ✂ ✔ ✎ ☛ ✆ ✎ ✓ ☛ ✞ ✠ ✎ ☛ ✍ ✗ ✸ ✌ ✍ ✏ ✞ ☛ ✻ ✡ ❀ ✟ ✻ ✝ ✠ ✎ ✟ ☎ ✠ ✑ ✎ ☛ ✆ ✎ ✓ ✌ ✻ ✞ ✿ ☛ ✆ ☛ ✿ ✔ ✞ ☎ ✎ ☛ ✍ ✂ ✞ ✆ ✽ ✌ ✔ ✎ ✓ ✞ ✎ ✁ ✟ ✠ ✆ ✄ ✿ ✞ ✁ ✠ ✎ ☛ ✻ ☛ ✓ ✠ ✆ ✿ ☛ ✆ ✎ ☎ ✎ ✎ ✠ ❂ ✝ ✻ ☛ ✏ ✏ ✻ ✿ ✄ ✆✞ ✓ ✓ ✍ ✓ ✻ ✞ ✝ ✠ ✿ ✸ ✄ ✡ ✒ ✝ ✠ ✁ ✠ ☛ ☛ ✸ ✟ ✂ ✞ ✡ ✄ ✓ ✝ ☛ ✎ ✆✞ – p. 20/43 ❚❱❯ ✒✝☛ ’ ▲❆▼ ✔ ✒✑ d ( π ) = d ( π ′ ) + k ❄❆❅ ✏ ✷✿ ☎ ✑✆ ✏ ✷✿ ☎ ✷✎ ★✜✩ ☎ ✑✸ Bergeron 90%

  41. ☎ ❍ ❑ ❏ ■ ❍ ❋ ❍ ● ◆❖ ❋ ❊ ❈❉ ❇ ❃ ✄ ❉ ● ✂ ❘ ▼ ❙ ❉ ▼ ■ ❅ ◗ ▼ ■ ❈ P ❊ ❉ P ✂ � � ✰ ✳ ✆ ✣ ✧ � ✩ ✮✯ ✎ ✚ ✩ ✳ ✧ ✭ ✧ ✝ ✯ ✤ � ✁ ✮ ✧ ☎ � ✂ � ✁ ✛ ✠ � – p. 21/43 ❚❱❯ . . . ▲❆▼ ❄❆❅ − ⊲

  42. ☎ ◗ ❃ ✄ � ✂ � ✎ ✁ ❇ ❘ ✠ ✂ ✁ ❅ ☎ ■ ■ ❈❉ ▼ ❑ ❉ P ▼ ● ◆❖ ❈ ❉ ❏ ❊ ■ ❍ ❋ ❍ ❍ ● ❋ ✂ ✄ P ✰ ✳ ✆ ✣ ✧ � ✩ ✮✯ ✤ ✚ ✩ ✳ ✧ ✭ ✧ ✝ ✯ ✛ � � ✂ � ✎ ❉ ☎ � ✁ ✠ ✮ � ✁ � ✂ ❙ ▼ ✧ ❊ – p. 21/43 ❚❱❯ . . . ▲❆▼ ❄❆❅ − − − − − − ⊲ ⊲

  43. ☎ � ● ❋ ❊ ❈❉ ❇ ❃ ✄ � ✂ ✁ ❍ � ✎ � ✠ ✁ ☎ ✂ ✄ ✂ ✂ ❍ ❋ ✎ ❈ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ P ❍ ❊ ❉ P ▼ ● ◆❖ ❉ ❑ ❏ ■ � � ✁ � � � ✁ � ✠ � ✁ ✂ ✩ ✧ ✣ ✧ ✮ ✛ ✆ ✤ ☎ ✎ ✳ ✩ ✝ ✠ ✧ ✭ ✁ ✧ ✳ ☎ � ✚ ✮✯ ✂ ✰ ✄ � ✂ ✯ – p. 21/43 ❚❱❯ . . . ▲❆▼ ❄❆❅ − − − − − − − − − ⊲ ⊲ ⊲

  44. ☎ � ☎ P ✂ ❉ ✄ � ✂ � ✁ ✎ ✁ � ✠ ✂ ✁ ❊ ☎ P ✂ ❈ ✄ ▼ � ✂ ❈❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ● ❋ ❊ ❇ ✠ ● ❃ ✄ � ✂ � ✁ � ✎ � � � ◆❖ ✧ ❙ ▼ ✧ ✮ ✛ ✤ ✯ ✳ ✆ ✣ � � ✩ ✰ ✮✯ ✚ ✩ ✳ ✧ ✭ ✧ ✝ ✂ ✁ ✎ ▼ ■ ✁ ◗ ✠ ❘ ✁ ❅ ☎ ■ ✂ ✄ � � ✂ � ✎ ❉ ☎ � ✁ � ✠ ❉ – p. 21/43 ❚❱❯ . . . ▲❆▼ ❄❆❅ − − − − − − − − − − − ⊲ ⊲ ⊲ ⊲

  45. ☎ � � ✁ � ✠ ✎ ✁ ✂ ☎ ✂ ✄ � ✂ ✁ ❃ � ✎ � ✠ � ✁ ☎ ✂ ✄ ✂ ✂ � ✁ ✄ ❇ ✎ P ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❊ ❉ ▼ ❈❉ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ● ❋ ❊ � � � ✤ ✠ ✎ ☎ � ✁ � ✠ � ✁ � ✂ ✧ ✮ ✛ ✯ � ✚ ✝ ✧ ✭ ✧ ✳ ✩ ✮✯ ✳ ✰ ✩ � ✧ ✣ ✆ ✂ � � ✁ ✄ ✁ ☎ ✂ ✄ ✂ ✠ ✂ � ☎ ✎ ✁ – p. 21/43 ❚❱❯ . . . ▲❆▼ ❄❆❅ − − − − − − − − − − − − − − − − ⊲ ⊲ ⊲ ⊲ ⊲

  46. ☎ ✠ � ✂ � ✁ � ✎ � ✠ � ✂ ✄ � ✁ � ✎ � ✁ ✁ ✂ ✠ � ✎ � � ✁ ✂ � ✄ ✂ ☎ ✂ ☎ ✁ ✂ ◗ P ❉ ❊ P ❈ ■ ❘ ● ❅ ■ ▼ ❉ ❙ ▼ ▼ ◆❖ � ● ✄ ❃ ❇ ❈❉ ❊ ❋ ❍ ❉ ❍ ❋ ❍ ■ ❏ ❑ ☎ � ✧ ✄ ✁ ✧ ✤ ✠ ✛ ✁ ✮ ☎ ✭ ✂ ✄ � ✎ ✂ � ✎ ✧ ☎ ✂ � ✁ � ✠ � ✯ � � ✁ ✳ � ✩ ✂ � ✁ � ✎ � ✠ ✚ ✮✯ ✂ ☎ ✝ ✰ ✩ ✂ � ✧ ✣ ✄ ✆ � ✳ ✁ – p. 21/43 ❚❱❯ . . . ▲❆▼ ❄❆❅ − − − − − − − − − − − − − − − − ⊲ ⊲ ⊲ ⊲ ⊲ ⊲

  47. ✁ ✄ ❂ ✄ ✻ ☎ ✺ ✄ ✞ ✠ ☎ ✡ ✻ ☛ ✿ ✆ ✠ ✓ ☛ ✻ ☛ ☛ ✠ ✠ ☛ ✓ ✎ ✞ ✂ ✍ ✗ ✓ ✄ ✟ ✿ ✠ ✁ ✠ ✝ ✒ ✡ ☛ ✔ ✠ ✎ ✁ ✞ P ❑ ❉ ◆❖ ● ▼ P ❉ ❊ ❈ ■ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❏ ❍ ✞ ✞ ✓ ✍ ☎ ✻ ✠ ✡ ✄ ✝ ✂ ✆✞ ❋ ✌ ❃ ❇ ❈❉ ❊ ❋ ● ❍ ❍ ✧ ☛ ✠ ✔ ✠ ✔ ✂ ✠ ✏ ✍ ✄ ✻ ✓ ✿ ✎ ✆ ☛ ✎ ✔ ✂ ✞ ✿ ✆ ✄ ✄ ☎ ✝ ✤ ✭ � ✮ ✦ ✥ ✦ ★ ✁ ★ ✦ ✪ ✫ ✤ ✢ ✧ ✰ ✆ ✗ ✟ ✡ ✎ ✌ ✸ ✏ ✎ ☛ ✻ ✟ ☛ ☛ ✏ ✁ ✗ ✆ ✠ ✝ ✍ ✄ ✟ ✞ ✻ ✍ ✞ ☛ ☛ ✞ ☎ ☎ ✁ ✎ ✞ ✓ ✎ ✔ ✟ ✼ ✞ ✂ ✡ ✻ ✄ ✎ – p. 22/43 ❚❱❯ ✏ ✷✎ Bergeron ▲❆▼ ✬✜✭ ❄❆❅ ★✜✩

  48. ✁ ✞ ✎ ✄ ✡ ✻ ✄ ✞ ✁ ✏ ✓ ✄ ✞ ✸ ✂ ☛ ✆ ✎ ✠ ✡ ✔ ✍ ☛ ☛ ✝ ✎ ✏ ✑ ✠ ✞ ✠ ✿ ✿ ✍ ☛ ✧ ✆ ✠ ✓ ☛ ✻ ☛ ✎ ✠ ✁ ✞ ✓ ☎ ✞ ✻ ✠ ✡ ✄ ✝ ✂ ✞ ✆✞ ✌ ✄ ☎ ✺ ✄ ✠ ✠ ☎ ▼ ❊ ❋ ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● P ❇ ❉ ❊ P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❈❉ ❃ ✡ ✞ ✄ ✸ ✄ ✿ ✞ ✂✆ ☛ ✍ ✍ ☛ ✎ ✞ ✟ ✠ ✞ ✻ ✠ ✍ ✞ ✡ ✠ ✻ ✎ ☛ ✞ ✠ ✟ ☛ ✻ ✿ ✄ ✻ ✓ ✎ ✆ ☛ ✎ ✔ ✂ ✞ ✟ ✗ ✄ ✡ ☛ ✻ ✞ ☎ ☎ ✁ ✎ ✞ ✓ ✎ ✔ ✟ ✼ ✂ ✡ ✔ ✄ ✄ ✫ ✝ ✤ ✭ � ✮ ✦ ✥ ✦ ☎ ★ ★ ✦ ✪ ✤ ✍ ✢ ✧ ✰ ✆ ✁ ✄ ✿ ✆ ✠ ✔ ✂ ✠ ✏ ✻ ✞ ✎ ✠ ✎ ✞ ✂ ☛ ✍ ✺ ✗ ☎ ✓ ✄ ☛ ✟ ✁ ✓ ✻ ✠ ✝ ✒ ✡ ✄ ❂ ☛ ✔ ✠ ☛ ✿ ✄ ☛ ✡ ☛ ✞ ✏ ✠ ✍ ✌ ✸ ✏ ✎ ✞ ☛ ✻ ✟ ✎ ✞ ☛ ✁ ✗ ✆ ✠ ✝ ✍ ✄ ✟ ✞ ✿ ✠ ✠ – p. 22/43 ❚❱❯ ✏ ✷✎ Bergeron ▲❆▼ ✗✷✎ ✬✜✭ ❄❆❅ ★✜✩ Hurdles Merging. Hurdles Cutting Bergeron

  49. ✁ ✠ ✞ ✁ ✄ ✼ ✞ ☛ ✟ ✞ ✞ ☛ ✎ ✻ ✠ ✡ ✞ ✍ ✟ ✑ ✻ ✸ ✠ ✻ ✞ ✆ ☛ ✎ ✑ ✔ ✠ ✎ ✸ ✠ ✍ ✒ ✂ ✁ ☛ ✠ ✠ ✄ ✠ ✎ ✝ ☛ ☛ ✍ ✔ ✡ ✎ ✑ ☛ ✞ ✂ ✸ ✞ ✧ ✓ ✏ ✏ ✠ ✞ ✂✆ ✟ ✞ ✎ ☛ ✍ ✍ ☛ ✞ ✞ ✿ ✄ ✸ ✄ ✡ ✠ ✿ ✠ ✍ ✡ ✞ ❈❉ ❍ ❋ ❍ ❍ ● ❋ ❊ ❇ ❏ ❃ ✟ ✔ ✓ ✔ ✡ ✠ ✿ ■ ❑ ✁ ◗ ▼ ❙ ❉ ▼ ■ ❅ ❘ ■ ❉ ❈ P ❊ ❉ P ▼ ● ◆❖ ✠ ✝ ✍ ✎ ✠ ✁ ✔ ✁ ✁ ☛ ✄ ✠ ✺ ❂ ☛ ✞ ✠ ✿ ✠ ☛ ✞ ✝ ✡ ✔ ✟ ✁ ✔ ✁ ✁ ✞ ✎ ☛ ✄ ☛ ✍ ✠ ✻ ✏ ✆ ✺ ✄ ✓ ✁ ✄ ✄ ✓ ✡ ✂ ✞ ✼ ✟ ✔ ✎ ✞ ✄ ✎ ✁ ☎ ☎ ✞ ☛ ✻ ✡ ✻ ✎ ✗ ✟ ✠ ✻ ✗ ✁ ☛ ✎ ☛ ✻ ☛ ☛ ✎ ✏ ✸ ✌ ✍ ✏ ✞ ✄ ✟ ✍ ★ ✧ ✢ ✤ ✫ ✪ ✦ ★ ☎ ✆ ✦ ✥ ✦ ✮ � ✭ ✤ ✝ ✰ ✁ ✞ ✻ ✂ ✔ ✎ ☛ ✆ ✎ ✓ ✔ ✄ ✄ ✍ ✏ ✠ ✂ ✔ ✠ ✆ ✿ ✝ ✆ ✄ ✻ ✍ ✓ ✞ ✁ ✠ ✎ ☛ ☛ ✻ ✓ ✠ ✆ ✿ ☛ ✻ ✟ ✄ ☎ ✠ ✞ ✄ ✡ ✄ ✎ ✠ ✆ ✿ ✞ ✺ ✡ ☎ ✄ ✌ ✆✞ ✞ ✂ ✝ ✄ ✠ ☎ ✄ ✡ ✞ ✎ ✓ ✗ ✓ ✄ ☛ ✺ ✟ ✠ ✁ ✠ ✝ ✒ ☛ ✍ ☛ ✞ ☎ ✻ ✄ ❂ ✿ ✡ ✞ ✠ ✿ ✠ ✠ ✔ ☛ ✂ – p. 22/43 ❚❱❯ ✏ ✷✎ Bergeron ✗✷✿ ▲❆▼ ✗✷✎ ✬✜✭ Bergeron ❄❆❅ ★✜✩ O ( n 3 ) Hurdles Merging. Hurdles Cutting ✏✑✆✞ Bergeron

  50. ✙ ✍ ✠ ✻ ✻ ☛ ✿ ☛ ✆ ☛ ✒ ☎ ✓ ✠ ✝ ✄ ✁ ✟ ☛ ✞ ✎ ✞ ✸ ☛ ✻ ✎ ✞ ✓ ✎ ✆ ✠ ☎ ✿ ✄ ✁ ✔ ✸ ❂ ✏ ✠ ✎ ✝ ✺ ☛ ✡ ✌ ✻ ☛ ✁ ✔ ✞ ✁ ✠ ✝ ✺ ✞ ✹ ✻ ✑ ☛ ✁ ✁ ✞ ✄ ✿ ✾ ✟ ☛ ● ❈❉ ❊ ❋ ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ▼ ❃ P ❉ ❊ P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❇ ✟ ✄ ✔ ✡ ✻ ✄ ✸ ✠ ✞ ✠ ✿ ✠ ✡ ✄ ✸ ✠ ❂ ✂ ✔ ✠ ✏ ✻ ✓ ✄ ✎ ☛ ✆ ✞ ✻ ✏ ✍ ✔ ✓ ✪ ✓ ✎ ✸ ✯ � ✶ ✄ ❂ ✠ ✍ ☎ ✻ ✠ ✝ ✿ ✌ ☎ ✦ ✺ ✠ ✝ ✍ ✄ ✎ ✠ ✻ ☛ ✁ ✂ ✓ ✒ ✂ ✰ ✍ � ✧ ✰ ✧ ✛ ✤ ✧ ✰ ✩ ✭ ☎ ✆ ★ ✛ ✣✤ ✂ ✁ ✮ ✩ ✁ ✧ ✝ ✤ ✭ � ✣ ✮ ✦ ✥ ✦ ✮ ✠ ✥ ✎ ☛ ☛ ✿ ☛ ✻ ✎ ✗ ✓ ✠ ✝ ✍ ✄ ✓ ✠ ✠ ✽ ✆ ✏ ☎ ✁ ✔ ✍ ☛ ✎ ✔ ✸ � ✿ ✏ ✻ ✻ ✎ ✠ ☛ ✆ ✞ ✻ ✏ ✍ ✔ ✓ ✔ ✸ ✑ ✍ ✿ ✆ ✏ ✠ ✏ ✍ ✄ ✄ ✏ ✟ ✠ ✁ ☛ ✏ ✺ ✞ ✍ ✮ – p. 23/43 ❚❱❯ ▲❆▼ Bergeron. ❄❆❅ ✏✑✆✞ ☎ ✑✸ √

  51. ✙ ✸ ✸ ✔ ✓ ✔ ✍ ✏ ✻ ✞ ✆ ☛ ✎ ✝ ✄ ✠ ✍ ✒ ✡ ✆ ✠ ✂ ✁ ☛ ✸ ✝ ✠ ✑ ✻ ✁ ✞ ✞ ✟ ✞ ✂ ✠ ✁ ✠ ✞ ☎ ✞ ✟ ✎ ✎ ☛ ✍ ✂ ☛ ✆ ✿ ✄ ✓ ✸ ✞ ✠ ✠ ✁ ✞ ✄ ✏ ✟ ✻ ☛ ✸ ✔ ✎ ✸ ✔ ✓ ✔ ✍ ✻ ✞ ✞ ✆ ☛ ✎ ✸ ✠ ✍ ✒ ✂ ✁ ☛ ✪ ☛ ☎ ✏ ✁ ✿ ☛ ☛ ✎ ✔ ✎ ✏ ✟ ✠ ✁ ✿ ✝ ✠ ✺ ✿ ✔ ✆ ✄ ✎ ✏ ✆ ✞ ✄ ✺ ✞ ✄ ❂ ❂ ✔ ✿ ❃ ✟ ☞ ☛ ✎ ✔ ✎ ✏ ✟ ✠ ✁ ✝ ❈❉ ✁ ✠ ✿ ✔ ✆ ✄ ✎ ✏ ✆ ✟ ✞ ✍ ❇ ❊ ✟ ❉ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❊ P ❋ ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ● ✞ ✞ ✠ ✞ ✁ ✏ ✏ ✿ ☛ ☛ ✆ ✄ ✁ ☛ ✟ ✄ ✝ ✻ ✡ ☛ ✂ ✔ ✿ ☛ ✎ ☛ ✻ ☎ ✍ ✠ ✸ ✎ ☛ ✔ ✆ ✄✌ ✒ ✻ ✝ ✠ ✺ ☎ ✝ ✠ ✿ ✸ ✎ ✝ ✠ ✿ ✞ ✠ ✁ ✏ ✿ ✝ ✸ ✝ ✠ ✟ ✁ ✓ ✔ ✍ ✏ ✠ ✆ ✠ ✿ ✍ ✑ ✸ ✔ ✓ ✍ ✄ ✏ ✻ ✞ ✆ ☛ ✎ ✠ ✍ ✒ ✂ ✁ ☛ ✄ ✏ ✠ ✿ ✠ ✔ ✓ ✄ ✍ ✝ ✠ ✓ ✗ ✎ ✻ ☛ ☛ ✟ ☛ ✻ ✠ ✻ ✏ ✍ ✞ ✺ ✏ ☛ ✁ ✠ ✻ ✎ ✆ ✰ ✁ ✩ ✮ ✁ ✣✤ � ✛ ★ ☎ ✭ ✩ ✧ ✝ ✤ ✛ ✧ ✰ ✧ ✮ ✥ ✆ ✣ ✯ � ✮ ✧ ✤ ✄ ✍ ✍ ✝ ✠ ✺ ☎ ✸ ✌ ✿ ✝ ✠ ✻ ☎ ✠ ✭ ❂ ✄ ✶ ✂ ✦ ✰ ✂ ✦ ✥ ✦ ✮ � ✽ ✎ ✏ ✸ ✟ ✄ ✾ ✠ ✆ ✎ ✓ ✞ ✎ ✻ ☛ ✞ ✡ ✎ ✞ ☛ ✟ ✄ ✍ ✝ ✠ ✓ ☎ ✒ ☛ ✄ ✻ ☛ ✠ ✍ ✏ ✻ ✞ ✆ ☛ ✎ ✄ ✓ ✻ ✏ ✂ ✄ ✔ ❂ ✠ ✸ ✄ ✡ ✠ ✿ ✠ ✞ ✠ ✸ ✆ ☎ ✿ ✓ ☛ ✡ ☛ ✺ ✝ ✎ ✠ ✏ ❂ ✸ ✔ ☛ ☛ ✎ ✓ ✏ ✿ � ✸ ✔ ✎ ☛ ✍ ✔ ✁ ✻ ✌ ✁ ✁ ✻ ✻ ✠ ☎ ✁ ✿ ✞ ✿ ✄ ✞ ✁ ✁ ☛ ✑ ✻ ✠ ✹ ✞ ✺ ✝ ✔ ✁ – p. 23/43 ❚❱❯ ✏ ✷✎ ✏ ✷✎ ▲❆▼ Bergeron. Bergeron ❄❆❅ ✏✑✆✞ ☎ ✑✸ ✏✑✆✞ √ √

  52. ✙ ☛ ✿ ✝ ✸ ✝ ✠ ✎ ✸ ✝ ✠ ✁ ✏ ✏ ✿ ☛ ✆ ✁ ☛ ☛ ✎ ☛ ✿ ✔ ✂ ✡ ✄ ✻ ✄ ✞ ✟ ☛ ✁ ✏ ✠ ☎ ✆ ✞ ✟ ✞ ✍ ✞ ✟ ✏ ✔ ✎ ✄ ✆ ✔ ✿ ✠ ✎ ✆ ✞ ✠ ✿ ✠ ✝ ✸ ☛ ✿ ✝ ✄✌ ☎ ✺ ✠ ✝ ✻ ✒ ✻ ✍ ✝ ✸ ✆ ✡ ✒ ✍ ✠ ✝ ✎ ✂ ☛ ✆ ✞ ✻ ✏ ✍ ✠ ✁ ✓ ✿ ✏ ✎ ✔ ✎ ✪ ☛ ✏ ☛ ✞ ✁ ✁ ✠ ✝ ✸ ✔ ✔ ✠ ✠ ✟ ✞ ☎ ✞ ✠ ✁ ✂ ✎ ✞ ✟ ✞ ❂ ✄ ❂ ✞ ☛ ✸ ✞ ✑ ✄ ✻ ✎ ✠ ✠ ✸ ✍ ✓ ✄ ✿ ✆ ☛ ✂ ✁ ✿ ✠ ✠ ✝ ✠ ✁ ✔ ✁ ✁ ☛ ✻ ✌ ✡ ☛ ✺ ✝ ✎ ✏ ✞ ✿ ✍ ☛ ✎ ✔ ✸ � ✏ ❂ ✓ ✎ ☛ ✓ ✔ ✸ ✺ ✹ ✁ ◗ P ❉ ❊ P ❈ ■ ❘ ● ❅ ■ ▼ ❉ ❙ ▼ ▼ ◆❖ ✻ ● ✟ ❃ ❇ ❈❉ ❊ ❋ ❍ ❉ ❍ ❋ ❍ ■ ❏ ❑ ✔ ☎ ✁ ✓ ✟ ✏ ✍ ✔ ✟ ✞ ✞ ✑ ✑ ✠ ☛ ✁ ✂ ✒ ☛ ✞ ✠ ☛ ✠ ✟ ✏ ✎ ✔ ✎ ☞ ✎ ✟ ✆ ✏ ✁ ☛ ✝ ✍ ✸ ✏ ✁ ✞ ✟ ✏ ☎ ✞ ✠ ✠ ☛ ✂ ✞ ✟ ✏ ✽ ✆ ✓ ☎ ☛ ✞ ✝ ✎ ✄ ✡ ✻ ☛ ☛ ✠ ✆ ✂ ✂ ✞ ☛ ✎ ✟ ☛ ✁ ✟ ✿ ☛ ☛ ✻ ✠ ✻ ✏ ✍ ✞ ✺ ✏ ☛ ✁ ✠ ✏ ✻ ✍ ✍ ✔ ✓ ✔ ✸ ✑ ✿ ✄ ✠ ✆ ✠ ✏ ✍ ✄ ☛ ✎ ✻ ☛ ✸ � ✿ ✏ ✓ ✎ ✓ ✎ ✔ ✿ ❂ ✏ ✠ ✎ ✔ ☛ ✗ ✎ ✓ ✠ ✝ ✍ ✄ ✓ ✠ ✍ ✽ ✆ ✏ ☎ ✁ ✔ ✏ ✞ ✺ ☎ ✮ � ✭ ✤ ✝ ✧ ✁ ✩ ✮ ✁ ✣✤ � ✛ ★ ✭ ✥ ✮ ✮ � ✯ ✣ ✆ ✥ ✧ ✩ ✰ ✧ ✛ ✤ ✧ ✰ ✦ ✦ ✆ ☛ ✝ ✍ ✄ ✎ ✠ ✻ ✁ ✺ ✂ ✒ ✍ ✠ ✎ ☛ ✠ ☎ ✂ ✠ ✰ ✦ ✂ ✶ ✄ ❂ ✍ ✸ ☎ ✻ ✠ ✝ ✿ ✌ ✝ ✸ ☛ ✟ ✻ ✏ ✍ ✔ ✓ ✔ ✁ ✆ ☛ ✁ ✂ ✒ ✍ ✠ ✞ ☛ ✎ ❂ ✿ ✠ ✡ ✄ ✸ ✠ ✔ ✎ ✂ ✠ ✏ ✻ ✓ ✄ ✸ ☛ ✞ ✎ ✞ ✺ ✄ ✞ ✆ ✏ ✄ ✺ ✆ ✔ ✿ ✠ ✁ ✝ ✄ ☎ ✆ ✔ ✞ ✻ ✏ ✍ ✔ ✓ ✸ ✞ ✎ ✔ ✸ ☛ ✻ ✟ ☛ ✠ ✡ ✠ ✠ ✄ ✿ ✞ ✿ ✁ ☎ ✻ ✁ ✻ ☛ ✿ ☛ ✆ ✸ ✞ ✁ ☎ ✁ ✌ ✻ ☛ ✁ ✁ ✔ ✠ ☛ ✝ ✺ ✞ ✹ ✻ ✑ ✒ ☛ ✎ ☛ ✞ ✆ ✻ ✠ ☛ ✸ ✞ ✎ ✞ ✾ ✄ ✓ ✟ ✄ ✟ ✄ ✍ ✝ ✠ ✓ ✡ ✻ ✄ ✎ – p. 23/43 ❚❱❯ ✏ ✷✎ ✏✑✸ ✏ ✷✎ ✏✑✸ ▲❆▼ Bergeron. Bergeron ✏✑✆✞ ❄❆❅ ✏✑✆✞ ☎ ✷✎ ☎ ✑✸ ’ ✏✑✆✞ ’ ✽✖☛ √ √ ⊗

  53. ☎ ✠ ✏ ✍ ✄ ✍ ✄ ✎ ✔ ☎ ✗ ✠ ✒ ✄ ✞ ☛ ✎ ✔ ✸ ✁ ✄ ✠ ✎ ✏ ✌ ✭ ✠ ✾ ✂ ✠ ✻ ✞ ❂ ✡ ✻ ☛ ✍ ✿ ✠ ✆ ✠ ✏ ✍ ✄ ✻ ✂ ✍ ✓ P ❏ ❑ ❉ ◆❖ ● ▼ P ❉ ❊ ❈ ❍ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ■ ❋ ✄ ✓ ✻ ✔ ✸ ☛ ✿ ✏ ✓ ✎ ☛ ✔ ❍ ✸ ✟ ❃ ❇ ❈❉ ❊ ❋ ● ❍ ✎ ✆ ✞ ✥ ✥ ✰ ✧ ✥ ✦ ☎ � � ✰ � ✧ ✥ ✦ ✟ ❀ ✻ ☛ ✿ ✎ � ☎ ✎ ✆ ✧ ✂ ✦ ✰ ✰ ✧ ✂ � ✣ ✛ ✭ ✮ ✧ ✰ ✧ ✛ ✤ ✧ ✰ ✦ ✹ ✆ ✠ ✔ ✓ ✎ ☛ ✓ ✔ ✸ ☛ ✁ ✁ ✁ ✿ ✠ ✝ ✺ ✞ ✹ ✻ ✍ ✄ ☛ ✻ ✾ ✏ ✎ ✞ ✂ ✞ ☛ � ✔ ✻ ✎ ✍ ☛ ✂ ✗ ✞ ✆ ✎ ✔ ✸ ✍ – p. 24/43 ❚❱❯ ▲❆▼ ☎✝✆ ❄❆❅ breakpoint BreakPoint ✬✜✭ ☎ ✑✸

  54. ☎ ✿ ✔ ✎ ✓ ☛ ✓ ✏ ✠ ☛ ✞ ✞ ☛ ✻ ✡ ✄ ☛ ✡ ✻ ✁ ✝ ☛ ✔ ✓ ☛ ✎ ✓ ✏ ✿ ✔ ✁ ✑ ✞ ✓ ✎ ❀ ✟ ✟ ✔ ✺ ✠ ✄ ✍ ✠ ✎ ✞ ✄ ✂ ☛ ✞ ✸ ✡ ✌ ✿ ✞ ✁ ✿ ☛ ✞ ✭ ✻ ☛ ✁ ✔ ✔ ✁ ✁ ☛ ☛ ✞ ✍ ✎ ✂ ✓ ✻ ✌ ✡ ❂ ✞ ✻ ✠ ✍ ✎ ✡ ❈❉ ❍ ❋ ❍ ❍ ● ❋ ❊ ❇ ❏ ❃ ✟ ✠ ✏ ❂ ✻ ✌ ✎ ■ ❑ ✹ ◗ ▼ ❙ ❉ ▼ ■ ❅ ❘ ■ ❉ ❈ P ❊ ❉ P ▼ ● ◆❖ ✻ ✎ ☛ ❂ ✻ ✌ ✎ ✞ ✍ ✌ ✞ ✞ ✡ ✻ ✠ ✂ ✠ ✏ ❂ ✏ ✆ ✄ ✻ ✝ ✻ ☛ ✏ ✆ ☛ ✎ ✄ ✍ ✌ ☛ ✎ ✸ ✗ ✁ ✿ ✠ ✎ ✞ ✺✻ ✄ ✄ ✍ ✻ ✔ ✎ ☛ ✞ ✂ ☛ ✂ ✞ ✞ ✆ ✾ ✠ ✎ ✹ ✆ ✎ ✍ ☛ ✓ ✁ ❂ ✁ ✁ ☛ ✸ ✔ ☛ ✗ ✎ ✓ ✏ ✿ � ✸ ✔ ✎ ✿ ✻ ✝ ✆ ✤ ✛ ✧ ✰ ✧ ✮ ✛ ✣ ✰ � ✂ ✧ ✰ ✰ ✦ ✂ ✧ ✧ ✦ ❀ � ✟ ✦ ✥ ✧ ✰ ✥ � ☎ ✭ ✦ ✥ ✧ ✰ ✥ � � ☎ ✠ ✔ ✺ ✄ ✄ ✁ ✸ ✔ ✎ ☛ ✞ ✒ ✏ ✗ ✠ ☎ ✔ ✎ ✄ ✞ ✂ ✍ ✍ ✓ ☛ ✻ ❀ ✟ ✟ ✸ ✔ ☛ ✄ ✎ ✓ ✏ ✿ ☛ ✸ ✔ ✻ ✄ ✍ ✏ ✎ ✞ ✠ ✠ ✂ ✾ ✠ ✆ ✓ ✡ ✞ ✎ ✻ ☛ ✄ ✍ ✻ ✹ ❂ ✻ ✌ ✠ ✠ ✎ ☛ ✻ ✄ ✍ ✻ ✏ ✆ ✍ ✿ ✠ – p. 24/43 ❚❱❯ breakpoints breakpoint ▲❆▼ ☎✝✆ ✒✝✠ ❄❆❅ breakpoint BreakPoint ✬✜✭ breakpoint ☎ ✑✸

  55. ☎ ✻ ✞ ✆ ☛ ✿ ☛ ✞ ☞ ✟ ✠ ✏ ❂ ✌ ✄ ✎ ✻ ✹ ✎ ✝ ☛ ✏ ✆ ☛ ✎ ✄ ❂ ✞ ✻ ☛ ✄ ✍ ✞ ✄ ✡ ✡ ✻ ☛ ✾ ✍ ✄ ✡ ✂ ✒ ✝ ✠ ✁ ✠ ✟ ☛ ✔ ✑ ☛ ✍ ✍ ✌ ☛ ☛ ☛ ✎ ✄ ✍ ✔ ✓ ☛ ✎ ✓ ✏ ✿ ✔ ✏ ✑ ✞ ✓ ✎ ❀ ✟ ✭ ✔ ✁ ✁ ✞ ✆ ❂ ✎ ✻ ✸ ✗ ✁ ✿ ✠ ✎ ✞ ☛ ✻ ✡ ✄ ✌ ✏ ✎ ✞ ✍ ✌ ✞ ❂ ✞ ✻ ✠ ✂ ✠ ✎ ✆ ✔ ✗ ❈❉ ❇ ❃ ✟ ✄ ✍ ✞ ☛ ✠ ✓ ✞ ✎ ❋ ✝ ☛ ✏ ✿ ☛ ✔ ✓ ☛ ✎ ✓ ✏ ❊ ● ☛ ❊ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❉ ❍ P ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ✿ ✔ ✏ ✓ ✻ ✔ ✎ ✄ ✍ ✔ ✓ ☎ ✺ ✓ ✄ ✞ ☛ ☛ ✟ ✻ ✹ ✂ ✝ ✄ ✡ ✻ ✌ ✎ ✎ ✝ ✞ ☛ ☛ ✟ ✸ ✌ ✻ ☎ ✍ ✠ ✿ ❀ ✟ ✡ ✎ ✒ ✝ ✠ ✁ ✠ ✟ ☛ ✗ ✟ ✞ ✎ ✠ ✻ ✟ ✎ ✎ ✝ ✠ ✁ ✔ ✁ ✁ ☛ ✸ ✔ ✓ ☛ ✓ ✞ ✏ ✿ � ✸ ✔ ✎ ✗ ✍ ✞ ✎ ✻ ✺ ✹ ✎ ✻ ✏ ✠ ✆ ✓ ✿ ✍ ✻ ✌ ✡ ❂ ✞ ✠ ✻ ✂ ✾ ✠ ✆ ✎ ✓ ✞ ✎ ✻ ☛ ✄ ✍ ✔ ☛ ✄ ✧ � � ☎ ✭ ✦ ✰ ✧ ✤ ✛ ✧ ✰ ✮ ✰ ✛ ✆ ✣ � ✂ ✧ ✰ ✰ ✦ ✂ ✧ ✥ ✧ ✞ ✿ ✂ ☛ ✍ ✂ ✞ ✆ ✾ ✠ ✎ ✹ ✆ ☛ ✥ ✻ ❀ ✟ ✦ ✥ ✧ ✰ ✥ � � ☎ ✦ ✍ ✠ ✻ ✿ ✎ ✞ ✄ ✂ ✿ ☛ ✞ ✍ ✸ ✌ ✞ ✻ ✁ ✿ ✻ ☛ ✞ ✄ ✺✻ ☛ ✡ ✄ ✠ ✡ ✠ ✻ ✔ ✁ ✁ ☛ ☛ ✞ ✍ ✔ ✎ ✓ ✞ ☛ ☛ ✌ ✝ ✡ ❂ ✞ ✻ ✺ ✠ ✂ ❂ ☛ ✠ ✎ ✄ ✒ ✗ ✓ ✠ ✡ ☎ ☛ ☛ ✔ ✄ ☛ ✍ ✄ ✄ ✍ ✏ ✠ ✡ ✠ ✎ ☛ ✞ ✎ ✻ ✿ ❀ ✟ ✟ ✸ ✔ ✓ ☛ ✎ ✓ ✏ ☛ ✔ ✏ ✸ ✔ ✻ ✄ ✍ ✏ ✂ ✄ ✁ ✸ ✁ – p. 24/43 ❚❱❯ (0 , 3) , (3 , 1) , (1 , 6) , 6 breakpoints breakpoint 0 + 3 + 1 + 6 + 5 − 2 + 4 + 7 ✏ ✷✎ ▲❆▼ ☎✝✆ ✒✝✠ breakpoint ❄❆❅ (4 , 7) 6 breakpoints breakpoint BreakPoint ✬✜✭ breakpoint (5 , − 2) , (2 , 4) ☎ ✑✸

  56. ☎ ✻ ✠ ✻ ✞ ❂ ✞ ✌ ✍ ✞ ✎ ✌ ☛ ✠ ✿ ✏ ✎ ✔ ✻ ☛ ✆ ✺ ✞ ✟ ✂ ✏ ☛ ✎ ✓ ✡ ☛ ✍ ✄ ✡ ✌ ✓ ✔ ✸ ✌ ❂ ✻ ✍ ✄ ✎ ☛ ✆ ✏ ✎ ✌ ✻ ✗ ✿ ✟ ❊ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ P ❉ P ❍ ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❋ ❍ ✏ ✠ ✓ ✎ ☛ ✓ ✔ ✿ ✆ ✠ ✸ ✎ ✍ ● ✔ ❂ ☎ ✻ ✟ ❃ ❇ ❈❉ ❊ ❋ ✭ ☛ ✞ ✥ ✧ ✥ ✦ ☎ � � ✥ ✰ ✧ ✦ ✥ ✟ ❀ ✎ ✓ ✞ ✑ ✔ ✎ ☛ ❂ ✰ � ✻ ✛ ✧ ✂ ✦ ✰ ✰ ✧ ✂ � ✣ ✆ ✮ � ✧ ✰ ✧ ✛ ✤ ✧ ✰ ✦ ✭ ☎ ✞ ✆ ✏ ✓ ✻ ☛ ✿ ☛ ✆ ✠ ✍ ✠ ✞ ☛ ✒ ✄ ✄ ✡ ✍ ✄ ✍ ✞ ☛ ❂ ✞ ☛ ✍ ✡ ✆ ✓ ✔ ✓ ✔ ✍ ☎ ✓ ✌ ☛ ✻ ✎ ✠ ✞ ✎ ✆ ✠ ✾ ✹ ✻ ✍ ✿ – p. 25/43 ❚❱❯ ▲❆▼ breakpoints ❄❆❅ BreakPoint ✬✜✭ √

  57. ☎ ☛ ✎ ☎ ✻ ☛ ✿ ✆ ✹ ✎ ✠ ✾ ✆ ✞ ✂ ✍ ✂ ✿ ✓ ✌ ✎ ✔ ✓ ☛ ✎ ✏ ✞ ✿ � ✻ ✔ ✎ ☛ ✏ ✁ ☛ ✏ ☛ ✿ ✏ ✭ ✔ ✻ ☛ ✆ ✺ ✞ ✟ ✗ ☛ ✿ ✓ ✟ ✎ ✻ ☎ ❂ ✔ ✍ ✠ ✸ ✎ ✠ ✆ ✿ ✔ ✓ ☛ ✻ ✁ ✌ ◆❖ ❃ ❇ ❈❉ ❊ ❋ ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ● ✞ ❘ ▼ ❙ ❉ ▼ ■ ❅ ◗ ▼ ■ ❈ P ❊ ❉ P ✠ ☛ ✁ ✻ ☛ ✄ ✍ ✞ ☛ ✟ ✹ ✎ ✞ ✺ ✝ ✠ ✁ ✔ ✻ ✞ ✻ ✞ ✡ ✄ ✻ ✌ ✡ ❂ ✻ ✓ ✠ ✂ ✾ ✠ ✆ ✎ ✻ ✎ ✎ ✏ ✥ ✦ ✟ ❀ ✎ ✓ ✞ ✑ ✔ ✎ ☛ ✆ ✞ ✻ ✍ ✰ ✻ ✠ ✆ ✎ ✞ ✞ ✎ ☛ ✔ ✌ ✓ ☎ ✍ ✔ ✓ ✧ ✥ ✹ ✧ ✧ ✂ ✦ ✰ ✰ ✧ ✂ � ✣ ✆ ✛ ✮ ✧ ✰ ✛ � ✥ � ☎ ✦ ✥ ✧ ✰ � ✤ � ☎ ✭ ✦ ✰ ✧ ✾ ✓ ✻ ✔ ✄ ✍ ✻ ✌ ✎ ✸ ✓ ☛ ✌ ✡ ✄ ✍ ✍ ✡ ✎ ✆ ☛ ✠ ✍ ✌ ✞ ❂ ✞ ✻ ✂ ✏ ✠ ✏ ❂ ✻ ✌ ✎ ✓ ☛ ✟ ☛ ✓ ☛ ✞ ✠ ✍ ✠ ✆ ✿ ✡ ☛ ✻ ✡ ✄ ✆ ✠ ✿ ✒ ✄ ✍ ✄ ✞ ❂ ✍ ✞ ☛ ☛ ✞ ❂ – p. 25/43 π 1 ❚❱❯ � � d BP ( π 1 , π 2 ) ▲❆▼ 2 breakpoints ❄❆❅ d inv ( π 1 , π 2 ) ≥ ☎ ✑✸ BreakPoint ✬✜✭ ✄ ✒✑ π 2 √ √

  58. ☎ ✆ ✁ ✟ ☞ ✎ ☛ ✎ ✻ ✠ ✂ ✞ ☛ ✺ ✿ ☛ ✻ ☎ ✸ ✌ ☎ ☛ ✏ ✆ ☛ ☎ ✄ ☛ ✍ ✿ ✠ ✔ ✔ ✓ ✄ ✒ ✞ ✎ ✄ ☛ ✞ ✞ ✍ ✞ ✄ ✻ ✡ ✄ ✎ ✠ ✆ ✿ ✁ ✺ ✡ ✓ ✡ ❂ ✞ ✻ ✠ ✂ ✾ ✠ ✆ ✎ ✞ ✻ ✎ ✻ ☛ ✄ ✍ ✞ ✭ ✟ ✻ ✹ ✞ ✌ ✄ ✞ ✎ ☛ ✻ ✡ ✄ ☛ ✍ ✂ ✞ ✆ ✾ ✠ ✸ ✡ ✹ ✺ ✝ ✎ ✓ ✝ ✶ ✠ ✞ ☛ ✻ ✠ ☛ ✝ ✺ ❊ ❈❉ ❇ ❃ ✟ ✝ ✠ ✎ ☛ ✡ ✝ ● ✠ ✁ ✔ ✁ ✁ ☛ ✔ ✡ ✄ ✒ ✔ ❋ ❍ ✠ P ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ ❊ ❍ ❉ P ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ✍ ✻ ☎ ✝ ✻ ✏ ✠ ✎ ✞ ☛ ✿ ☛ ✞ ✎ ☛ ☛ ✁ ✏ ✆ ☛ ✿ ✞ ☛ ✟ ✞ ✄ ✺ ✎ ✞ ✞ ✄ ✆ ☛ ✎ ☛ ✻ ✝ ✠ ✻ ☎ ✍ ✞ ✟ ☛ ✠ ✆ ✿ ✾ ✠ ✆ ✎ ✓ ✞ ✎ ✻ ✺ ☛ ✠ ✻ ✆ ☛ ✿ ☛ ✻ ✡ ✄ ✆ ✠ ✿ ✍ ✹ ✍ ✾ ✠ ✆ ✎ ✓ ✞ ✎ ✻ ☛ ✌ ✓ ✠ ✠ ✍ ☛ ✌ ✡ ✁ ✍ ☛ ✡ ✓ ☛ ✟ ✞ ❂ ✞ ✞ ❂ ☛ ✞ ✍ ✄ ✍ ✡ ✄ ✒ ✓ ☛ ☎ ✔ ✔ ✧ � � ☎ ✭ ✦ ✰ ✧ ✤ ✛ ✧ ✰ ✮ ✰ ✛ ✆ ✣ � ✂ ✧ ✰ ✰ ✦ ✂ ✧ ✥ ✧ ✓ ✓ ✔ ✍ ✏ ✻ ✞ ✆ ☛ ✎ ✔ ✑ ✞ ✎ ✥ ❀ ✟ ✦ ✥ ✧ ✰ ✥ � � ☎ ✦ ✓ ✄ ✸ ✿ ✠ ✎ ✹ ✆ ✿ ☛ ✻ ☎ ✎ ✏ ✁ ✆ ✟ ✻ ☎ ❂ ✔ ✎ ✠ ✎ ✸ ✠ ✆ ✾ ✞ ✔ ✎ ✔ ✁ ✁ ☛ ✻ ✌ ✎ ✔ ✓ ☛ ✓ ✂ ✏ ✿ � ✻ ✔ ✎ ☛ ✞ ✂ ☛ ✍ ✿ ✍ ✓ ✍ ✏ ✎ ✌ ☛ ✻ ✌ ✎ ✞ ☛ ☛ ✻ ✌ ❂ ✞ ❂ ✞ ✏ ✻ ✠ ✆ ✎ ✂ ✞ ✎ ✓ ✏ ✌ ✿ ☛ ✗ ✟ ✺ ✿ ✆ ✻ ☛ ✻ ✍ ✔ ✄ ✎ ✏ ✠ – p. 25/43 Gollan π 1 ’ ❚❱❯ N − 1 breakpoints � � d BP ( π 1 , π 2 ) ▲❆▼ 2 ☎ ✑✸ breakpoints ❄❆❅ d inv ( π 1 , π 2 ) ≥ N − 1 ☎ ✑✸ BreakPoint ✬✜✭ ✄ ✒✑ π 2 ’ √ √ ⊗

  59. � ✍ ✸ ✿ ✆ ✏ ✓ ✔ ☛ ✆ ☛ ✿ ✏ ✓ ✎ ✠ ✡ ✌ ✺ ✟ ✍ ✄ ✍ ✞ ☛ ☛ ✠ ✺ ✁ ✠ ✝ ✒ ✡ ☛ ✞ ✄ ✌ ◗ P ❉ ❊ P ❈ ■ ❘ ● ❅ ■ ▼ ❉ ❙ ▼ ▼ ◆❖ ✡ ❋ ☛ ✟ ❃ ❇ ❈❉ ❊ ● ❉ ❍ ❍ ❋ ❍ ■ ❏ ❑ ✣ ✻ ✡ ✢ ✧ ✛ ✮ ✧ ✰ ✤ � � ✰ ✦ ✰ ✧ ✂ ✁ ✣ ✤ ✆ � ✆ ✱ ✩ ✤ � � ✤ ✧ ✁ ✪ ✮ ✴ ✧ ✂ ✢ ✄ ✽ ✠ ✺ ☛ ✡ ✒ ✝ ✠ ✁ ✠ ✟ ☛ ✎ ✓ ✟ ✞ ✆ ✓ ☛ ✸ ✔ ✎ ✔ ✓ ✌ ✡ ✠ ✠ ✞ ✍ ✠ ✺ ✸ – p. 26/43 ❚❱❯ ▲❆▼ N 2 N ❄❆❅ BreakPoint Graphs ✗✑✸

  60. � ✡ ✟ ✞ ✎ ✻ � ☛ ✒ ✎ ✝ ✠ ✁ ✠ ✟ ☛ ☛ ✞ ✝ ✆ ✹ ✟ ✞ ✎ ✔ ✻ ☛ ✓ ✻ ✌ ✎ ✔ ✓ ☛ ✎ ✿ ✻ ✓ ✺ ✔ ✎ ✻ ✌ ✡ ✄ ✞ ☛ ✠ ✎ ✓ ✻ ✹ ✣ ✞ ✸ ✆ ✔ ✒ ✎ ✄ ✍ ✸ ☛ ✡ ✝ ✺ ✠ ✁ ✠ ✟ ☛ ✟ ✞ ✻ ✎ ✄ ● ❏ ■ ❍ ❋ ❍ ❍ ❋ ❉ ❊ ❈❉ ❇ ❃ ✟ ☛ ✡ ❑ ◆❖ ✝ ❘ ▼ ❙ ❉ ▼ ■ ❅ ◗ ● ■ ❈ P ❊ ❉ P ▼ ✒ ✠ ✠ ✸ ☛ ✟ ✞ ✺ ✆ ☛ ✔ ✠ ✎ ✻ ✌ ✡ ✄ ✺ ✞ ✟ ✁ ✁ ✔ ✠ ✟ ☛ ✿ ✝ ✻ ✎ ✠ ✄ ✍ ✸ ☛ ✡ ✒ ✝ ✎ ✟ ✒ ✸ ☛ ✟ ✞ ✺ ✆ ☛ ✔ ✠ ✎ ✔ ✓ ✌ ✡ ✠ ✍ ✟ ✁ ✓ ✞ ✍ ✻ ✻ ✡ ✄ ✺ ✠ ✠ ✎ ✓ ✸ ☛ ✡ ✒ ✝ ✠ ✠ ✍ ✁ ✢ ✂ ✧ ✴ ✮ ✪ ✤ ✤ � � � ✤ ✩ ✱ ✆ ✧ � ✆ ✰ ✽ ✁ ✂ ✧ ✰ ✦ � ✣ ✢ ✤ ✰ ✧ ✮ ✛ ✧ ✄ ✌ ✞ ✝ ✎ ✄ ✻ ✞ ✺ ✂ ✎ ✞ ✞ ✿ ✄ ✏ ✎ ☛ � ☛ ✍ ☛ ✞ ✌ ✎ ✔ ✓ ☛ ✎ ✓ ✎ ✄ ☛ ✟ ✞ ✎ ✻ � ✠ ✟ ✝ ✡ ✝ ✆ ☛ ✡ ✸ ✡ ✿ ✆ ✄ ✏ ✓ ✔ ✒ ✍ ☛ ✺ ✠ ☛ ✿ ✁ ✏ ✠ ✟ ✓ ☛ ✎ ✠ ✞ ✺ ✌ – p. 26/43 ❚❱❯ ✗✑✸ ✗✑✸ − x ▲❆▼ 2 x − 1 2 x 2 x 2 x − 1 x N 2 N ❄❆❅ BreakPoint Graphs ✗✑✸

  61. � ✎ ✏ ✓ ✞ ☛ ✎ ✡ ✄ ✞ ✠ ✿ ✠ ☛ ❂ ✍ ✂ ☛ ✆ ✽ ✠ ✠ ✾ ✞ ✂✆ ☛ ✎ ✻ ✄ ✠ ✎ ✡ ✄ ✍ ✔ ✓ ✻ ✹ ✟ ✞ ✻ ✠ ✞ ✓ ✄ ✂ ✻ ✌ ✎ ✏ ✆ ☛ ✎ ✄ ✍ ✔ ✻ ✎ ✻ ✁ ✻ ✔ ✎ ✄ ✍ ✸ ☛ ✡ ✒ ✝ ✠ ✠ ✿ ✟ ☛ ✟ ✞ ✺ ✆ ☛ ✸ ✣ ✎ ✻ ✌ ✝ ☛ ✟ ❂ ✠ ✆ ✿ ✻ ✠ ✎ ✓ ✍ ✟ ☛ ✻ ✹ ✞ ✟ ✄ ✠ ✏ ✶ ✟ ☛ ✡ ✒ ✝ ✠ ✁ ✠ ✌ ☛ ✄ ✞ ❇ ❃ ✟ ✟ ✞ ✻ ✠ ✎ ✞ ✄ ✂ ☛ ❊ ✻ ✡ ✄ ☛ ✻ ✄ ✍ ☛ ✁ ☎ ✒ ✝ ❈❉ ❋ ✿ ❊ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❉ ● P ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ✠ ✻ ☎ ✄ ✎ ☛ ✍ ✂ ☛ ✆ ✽ ✟ ☞ ✔ ✓ ✒ ✿ ✞ ✎ ✄ ✍ ✗ ✟ ✞ ✺ ✆ ☛ ✄ ✁ ✠ ✠ ✹ ✄ ✾ ✝ ✆ ✠ ✟ ✻ ✌ ✎ ✏ ✆ ☛ ✎ ✍ ✞ ✔ ✓ ✄ ❂ ✻ ✏ ✓ ✞ ☛ ✎ ✡ ✄ ✡ ✔ ✺ ✄ ✡ ✒ ✝ ✠ ✁ ✠ ✟ ☛ ☛ ✞ ✍ ✍ ✺ ✻ ✌ ✡ ✄ ✺ ✞ ✠ ✎ ✓ ✸ ☛ ✡ ☛ ✌ ✝ ✎ ✄ ✏ ✎ ✞ � ✟ ☛ ✡ ✄ ✺ ✞ ✠ ✓ ✆ ✏ ✿ ☛ ☛ ✍ ✔ ✓ ✏ ✆ ✿ ✸ ✡ ✒ ✠ ✞ ✂ ✤ ✰ ✧ ✮ ✛ ✧ ✣ � ✤ ✧ ✢ ✧ � ✴ ✮ ✪ ✁ ✤ � � � ✤ ✩ ✱ ✆ ✢ ✰ ✁ ✓ ✠ ✟ ☛ ✟ ✞ ✺ ✆ ☛ ✸ ✔ ✎ ✔ ✌ ✦ ✡ ✠ ✍ ✠ ✓ ✠ ✆ ✽ ✁ ✂ ✧ ✰ ✿ ✝ ✎ ✔ ✡ ✒ ✝ ✠ ✁ ✠ ✟ ☛ ✿ ✝ ✻ ✎ � ✄ ✍ ✸ ☛ ✡ ✒ ✝ ✠ ✝ ✠ ✟ ☛ ✻ ✟ ☛ ✠ ✎ ✓ ✻ ✹ ✟ ✞ ✎ ✔ ✻ ✆ ✻ ✎ ✌ ✎ ✔ ✓ ☛ ✎ ✓ ✞ ✎ ☛ ✟ ✞ ☛ ✁ ✌ � ✞ ✎ ✄ ✒ ✍ ✻ ✄ ✞ ✠ ✎ ✞ ✔ ✓ ✻ ☛ ✎ ✓ ✞ ✎ ☛ ✟ ✟ ✹ ✎ ✄ ✂ ✺ ✺ ✆ ☛ ✸ ✔ ✎ ✞ ✻ ✎ ✻ ✻ ☛ ✌ ✡ ✄ ✺ ✞ ✠ ✎ ✓ ✞ – p. 26/43 ❚❱❯ ✗✑✸ ✗✑✸ − x ▲❆▼ 2 x − 1 2 x 2 x 2 x − 1 ☎ ✑✸ x N 2 N ❄❆❅ ✏ ✷✿ ☎ ✷✿ BreakPoint Graphs ✗✑✸ Reality / Black Edges: Desire / Grey Edges: ✒✝☛ ☎ ✑✸

  62. � ✔ ✏ ✎ ✠ ✂✆ ✞ ✾ ✠ ✸ ✎ ✸ ✠ ✿ ✆ ✠ ✟ ✎ ✠ ✝ ✓ ☛ ✼ ✻ ☛ ✔ ✠ ✡ ☛ ✍ ✄ ✎ ☛ ✓ ✺ ✍ ✔ ☛ ✧ ✡ ✡ ✄ ✎ ☛ ✞ ✓ ✎ ✸ ✟ ✠ P ❏ ❑ ❉ ◆❖ ● ▼ P ❉ ❊ ❈ ❍ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ■ ❋ ✠ ✻ ✁ ✠ ✝ ✒ ✡ ☛ ✸ ✄ ✡ ☛ ❍ ✞ ✠ ✠ ❃ ❇ ❈❉ ❊ ❋ ● ❍ ✿ ✎ ✔ ✤ ✧ ✤ � ✣ ✧ ✛ ✮ ✧ ✰ ✢ ✂ ✰ ✦ ✰ ✧ ✂ ✁ ✁ ☞ ✞ ☛ ✢ ✧ ☛ ✣ ✣ ✆ ✳ ✯ ✤ ✛ ✮ ✧ � ✆ ✴ ✱ ✩ ✪ � � � ✤ ✁ ✪ ✮ ✿ � ✆ ✡ ✟ ☛ ✔ ✞ ☛ ✎ ✄ ✻ ❂ ✁ ✸ ☛ ✍ ✞ ✸ ✌ ✿ ✄ ✠ ✠ ✠ ✝ ☛ ❂ ✄ ✞ ✠ ✂ ✍ ☛ ✑ ✸ ✝ ✝ ☛ ✿ ✠ ✒ ☎ ✓ ✡ ✒ ✍ – p. 27/43 7 ❚❱❯ 13 3 −2 π = (0 − 1 + 3 + 5 + 4 + 6 − 2 + 7) 4 π ′ = (0 2 1 5 6 9 10 7 8 11 12 4 3 13) 12 ▲❆▼ 6 11 8 4 7 ❄❆❅ ✏ ✷✿ 10 5 9 6 3 5 BreakPoint 1 −1 2 0 ✄ ✒✑ 0

  63. ✁ ✠ ✎ ✆ ✠ ✾ ✿ ✞ ✻ ✌ ✓ ✎ ✠ ✂✆ ✞ ✾ ✟ ✞ ✔ ✝ ✞ ☛ ✻ ✡ ✄ ✆ ✄ ✞ ✄ ❂ ✏ ✿ ☛ ✓ ✎ ☛ ✎ ✌ ✎ ✻ ✄✌ ✓ ✹ ✿ ✞ ✆✞ ✄ ✿ ✻ ✌ ✞ ✻ ✟ ✻ ✗ ☛ ✸ ✠ ✎ ☛ ✍ ✍ ✆ ✎ ✝ ✠ ✎ ✻ ✌ ✿ ✆ ✓ ❍ ❑ ❏ ■ ❍ ❋ ❍ ● ◆❖ ❋ ❊ ❈❉ ❇ ❃ ✝ ❉ ● ✎ ❘ ▼ ❙ ❉ ▼ ■ ❅ ◗ ▼ ■ ❈ P ❊ ❉ P ✠ ☛ ✟ ✻ ✓ ✞ ☛ ✻ ✡ ✄ ✄ ✻ ❂ ✞ ☛ ✟ ✗ ✎ ☛ ✎ ✍ ✡ ✔ ✆ ✹ ✄ ✒ ✠ ☛ ✠ ✎ ✝ ✄ ✁ ✄ ✎ ✔ ✎ ✔ ✎ ✆ ✄ ✡ ✻ ☛ ✿ ✆ ✠ ✓ ✒ ☎ ✓ ✄ ✞ ✠ ✿ ✏ ✆ ✎ ✓ ✠ ✁ ✟ ✟ ✿ ☛ ✻ ☎ ✏ ✁ ✠ ✠ ✍ ✞ ✁ ✞ ✄ ❂ ☎ ✿ ☛ � ☛ ✠ � ✠ ✎ ✞ ☛ ✟ ✆ ✻ ✗ ✎ ✾ ✠ ✆ ✎ ✓ ✞ ✻ ✿ ☛ ☛ ✞ ✍ ✸ ✌ ✂✆ ✠ ✾ ✄ ✿ ✠ ❂ ✄ ✞ ✆ ✔ ✿ ✆ ✠ ✟ ✎ ✄ ✞ ✏ ✄ ✁ ✍ ☛ ✎ ☛ ✟ ✗ ✿ ✟ ☛ ✄ ✍ ☛ ✁ ✠ ☛ ✏ ✆ ☛ ☛ ✸ ✟ ☛ ✞ ✽ ✝ ✟ ✒ ☛ ✂ ✍ ✞ ✍ ✎ ✞ ✟ ✏ ✎ ✔ ✿ ✎ ❀ ✓ ☛ ✸ ✔ ✡ – p. 28/43 ❚❱❯ ✗✑✸ ▲❆▼ ☎ ✷✎ ❄❆❅ ✏ ✷✿ Pevzner Fermat! Bafna √ √

  64. ✘ ✻ ✔ ✁ ✠ ✝ ✺ ✞ ✹ ✍ ✁ ✄ ☛ ✻ ✎ ✞ ✓ ✎ ✆ ✁ ☛ ✾ ✎ ✞ ✂ ✍ ☛ ✂ ✞ ☛ ✔ ✔ ✻ � ✿ ✏ ✓ ✎ ☛ ✓ ✠ ✂ ✾ ❈ ◆❖ ● ▼ P ❉ ❊ P ■ ❑ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❉ ❏ ✠ ❃ ✻ ✞ ❂ ✡ ✌ ✻ ✟ ❇ ■ ❈❉ ❊ ❋ ● ❍ ❍ ❋ ❍ ☛ ✆ ✌ ✔ ☛ ✎ ✔ ✸ ✿ ✆ ✠ ✂ ✓ ✠ ✏ ✍ ✄ ✻ ✔ ✸ ❂ ✍ ✄ ✎ ✠ ✓ ✞ ✡ ✏ ✍ ✄ ✻ ✞ ✁ ✓ ✎ ✠ ☛ ✿ ✠ ✎ ☎ ✞ ☛ ☛ ✂ ✻ ☛ ✟ ✓ ✝ ✻ ✞ ✠ ✁ ✝ ✍ ✄ ☎ ✻ ☛ ✻ ✾ ✠ ✠ ✿ ✞ ✻ ✄ ✞ ☛ ✸ ✍ ✠ ✟ ✞ ✆ ✠ ✏ ✍ ✄ ✄ ✏ ☎ – p. 29/43 ❚❱❯ ▲❆▼ ☎ ✑✸ ❄❆❅

  65. ☎ ✓ ✄ ✓ ✏ ✎ ✻ ✠ ✎ ✞ ☛ ✺ ✞ ✁ ✝ ✠ ✎ ✞ ☛ ✸ ✻ ✻ ✸ ☛ ✟ ✞ ☛ ✻ ✡ ✄ ✄ ✎ ✓ ✏ ✾ ✠ ✆ ✎ ✓ ✞ ✎ ✏ ✔ ✒ ✞ ✄ ✺ ☎ ✔ ✓ ✄ ✞ ✏ ✎ ✄ ✍ ✗ ✟ ✞ ✭ ✠ ✟ ✟ ✠ ✻ ✎ ✞ ☛ ✿ ☛ ✑ ✸ ✌ ☎ ✁ ✍ ✠ ✿ ❀ ✟ ✸ ✝ ✠ ✞ ❂ ✝ ❇ ❋ ❍ ❍ ● ❋ ❊ ❈❉ ❃ ■ ✠ ✗ ❂ ☛ ✁ ✔ ✶ ✟ ❍ ❏ ✄✌ ■ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ❈ ❑ P ❊ ❉ P ▼ ● ◆❖ ❉ ✻ ✓ ✞ ✁ ✟ ✻ ✌ ✎ ✸ ✝ ✗ ✿ ✟ ✝ ✠ ✎ ✞ ✆ ✠ ✾ ☛ ✏ ✁ ☎ ✻ ✒ ☛ ✎ ✄ ✍ ✠ ✏ ❂ ✌ ✌ ✎ ✏ ✆ ☛ ✎ ✄ ✍ ✻ ✎ ✎ ☛ ☛ ✔ ✠ ✆ ✿ ✸ ✔ ✎ ✍ ✠ ✓ ✄ ✁ ☎ ✎ ✠ ✿ ☛ ✂ ✏ ✎ ✻ ✎ ✠ ✿ ✍ ✸ ☛ ✞ ✄ ✞ ✍ ✾ ☛ ✞ ❂ ✸ ✔ ✻ ✄ ✠ ✓ ✏ ✆ ✤ ✛ ✧ ✰ ✧ ✮ ✛ ✣ ✰ � ✂ ✧ ✰ ✰ ✦ ✂ ✧ ✧ ✦ ✞ ☛ ✠ ✻ ✄ ✍ ✏ ✡ ✞ ✓ ✘ ✭ ✦ ✥ ✧ ✰ ✥ � � ☎ ✠ ✆ ✍ ✔ ✻ ✹ ✞ ✺ ✝ ✠ ✁ ✁ ✄ ✁ ☛ ✔ ✓ ☛ ✎ ✄ ✏ ✍ ☛ � ✻ ✄ ✟ ✻ ✌ ✡ ❂ ✞ ✠ ✻ ✂ ✾ ✠ ✆ ✎ ✓ ✞ ✎ ✿ ✓ ✻ ✝ ☎ ✔ ✍ ✝ ✠ ✂ ✞ ✻ ✓ ☛ ☛ ✻ ☛ ✁ ✠ ✟ ✏ ✄ ✻ ✄ ✻ ✌ ✎ ☛ ✞ ✂ ☛ ✍ ✂ ✞ ✆ ☎ ✾ ✎ ✞ ✟ ✠ – p. 29/43 ❚❱❯ d inv ( π, I ) ≥ N + 1 − c ( π ) ▲❆▼ ☎ ✑✸ ☎ ✑✸ N + 1 ❄❆❅ Cycle Decomposition. I ✬✜✭ ✒✝✠

  66. ☎ ✗ ✾ ✏ ✎ ✝ ☛ ✠ ☎ ✻ ✠ ✼ ✠ ❂ ✞ ☛ ✁ ✔ ✶ ✟ ✻ ✄✌ ✓ ☎ ✒ ☛ ✎ ✆ ✂ ✍ ✄ ✎ ✓ ✁ ✏ ✎ ✺ ✞ ✾ ✓ ✞ ✎ ✟ ✍ ✆ ☛ ✸ ✌ ✍ ✏ ✞ ☛ ✻ ✡ ✄ ☛ ✄ ✠ ✸ ✡ ✾ ☛ ✞ ❂ ✔ ✞ ☛ ✟ ✞ ☛ ✻ ✄ ✆ ✸ ✄ ✓ ✏ ✾ ✠ ✆ ✎ ✓ ✞ ✭ ✻ ✠ ✞ ✏ ✁ ❂ ✻ ✌ ✎ ✏ ✆ ☛ ✎ ✄ ✍ ✻ ✌ ✟ ✎ ✏ ✟ ✻ ✌ ✎ ✸ ✝ ✗ ✁ ✿ ✝ ✠ ✞ ✿ ✸ ✂ ❃ ✟ ✠ ✝ ✠ ✎ ✔ ✆ ✄ ✁ ☛ ✄ ❈❉ ✍ ✞ ☛ ✻ ✡ ✄ ✔ ✓ ✞ ✁ ✟ ✏ ❇ ❊ ☛ ❉ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❊ P ❋ ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ● ✿ ✔ ✄ ✎ ✂ ✠ ✁ ✠ ✞ ☎ ✸ ✞ ✄ ✓ ✹ ✿ ✟ ✆✞ ✄ ✿ ✸ ✄ ✆ ✄ ✎ ✏ ✓ ✓ ✆✞ ✞ ✹ ☛ ✞ ✞ ✻ ✞ ✿ ✓ ✞ ☛ ✟ ☎ ✆✞ ✟ ☛ ☛ ✻ ✻ ✡ ✄ ✻ ✌ ✻ ☎ ✍ ✠ ❂ ✄ ❂ ☛ ✎ ✄ ✻ ✍ ✏ ✠ ✆ ✠ ✿ ✍ ✸ ☛ ✞ ✄ ✞ ✄ ✾ ☛ ✞ ❂ ✸ ✔ ✻ ✄ ✍ ✏ ✠ ✂ ✄ ✏ ✠ ✄ ✆ ✓ ✌ ✎ ✞ ✟ ✠ ☎ ✻ ☛ ✻ ☎ ✍ ✟ ✝ ✠ ✂ ✞ ✻ ✝ ✓ ☛ ✻ ☛ ✁ ✠ ✔ ✆ ✂ ✰ ✰ ✥ � � ☎ ✭ ✦ ✰ ✧ ✤ ✛ ✧ ✧ ✥ ✮ ✛ ✆ ✣ � ✂ ✧ ✰ ✰ ✦ ✂ ✧ ✧ ✦ ✿ ☛ ✸ ✔ ✎ ☛ ✍ ✓ ✄ ✁ ☎ ✎ ✠ ✿ ✠ ✘ ✎ ✓ ✞ ✠ ✻ ✄ ✍ ✏ ✡ ✞ ✓ ☛ ✞ ✾ ✍ ✟ ✍ ✠ ✿ ❀ ✟ ✸ ✝ ✠ ✁ ✟ ✏ ✞ ✻ ✄ ✺ ☎ ✔ ✓ ✄ ✒ ✞ ✎ ☛ ✍ ✗ ☎ ✌ ✞ ✠ ✏ ✎ ✻ ✠ ✎ ✓ ✞ ✺ ✞ ✁ ✝ ✎ ✸ ✞ ☛ ✎ ✻ ✏ ✠ ✎ ✞ ☛ ✿ ☛ ✑ ✟ ✄ ✎ ✞ ✞ ✎ ✻ ✏ ✠ ☛ ✄ ✍ ✻ ✹ ✺ ✎ ✝ ✠ ✁ ✔ ✁ ✁ ✓ ☛ ✔ ✓ ✓ ✆ ✎ ✡ ✎ ✝ ☛ ✂ ✎ ✄ ✟ ✻ ✌ ✞ ❂ ✠ ✞ ✻ ☛ ✠ ✂ ✎ ✔ ✻ ✾ � ✿ ☛ – p. 29/43 ❚❱❯ d inv ( π, I ) ≥ N + 1 − c ( π ) ▲❆▼ ☎ ✑✸ ☎ ✑✸ ☎ ✑✸ N + 1 ❄❆❅ Cycle Decomposition. I ✏ ✷✎ ✬✜✭ ✒✝✠ √

  67. ☎ ☛ ✻ ✌ ✻ ✄ ✡ ✻ ✞ ✍ ☛ ✟ ✆✞ ☎ ✟ ☛ ☎ ✠ ✓ ✁ ✄ ✞ ✸ ☎ ✞ ✠ ✠ ❂ ✂ ✞ ✟ ✹ ✻ ❂ ✄ ✞ ✿ ✹ ✎ ✂ ☛ ✁ ✄ ✆ ✔ ✠ ✍ ✝ ✠ ✟ ✆ ✏ ✁ ✄ ✞ ✞ ✏ ✻ ✞ ☛ ✔ ☛ ✿ ✟ ☛ ✁ ✞ ✓ ✔ ✄ ✡ ✻ ✓ ✎ ✝ ✆ ✠ ☛ ✝ ✎ ✏ ✾ ✞ ✻ ✂ ✍ ☛ ✄ ✡ ✻ ☎ ✠ ✞ ✻ ✎ ☛ ✒ ✭ ✓ ✄✌ ✟ ✼ ✶ ✔ ✁ ☛ ❂ ✗ ✠ ☛ ✏ ✿ ✎ ✿ ✄ ✆✞ ✓ ✓ ✏ ✄ ✞ ✆ ✄ ✸ ✿ ✄ ✆✞ ✸ ✎ ✍ ✎ ✌ ✸ ☛ ✆ ✟ ✄ ✞ ✓ ✓ ✾ ✞ ✺ ✎ ✏ ✁ ☛ ✎ ✍ ✟ ✄ ☛ ✟ ✏ ✍ ✔ ✞ ✓ ✿ ✠ ✞ ✠ ✝ ✸ ✎ ☛ ✆✞ ✗ ✄ ✓ ✄ ☛ ✝ ✎ ✑ ✍ ✺ ✆ ✄ ✞ ☛ ✡ ✏ ✠ ✓ ✝ ◗ P ❉ ❊ P ❈ ■ ❘ ● ❅ ■ ▼ ❉ ❙ ▼ ▼ ◆❖ ✍ ❋ ✠ ✟ ❃ ❇ ❈❉ ❊ ● ❉ ❍ ❍ ❋ ❍ ■ ❏ ❑ ✍ ✠ ✞ ✔ ✗ ☛ ✿ ✞ ✻ ✎ ✓ ✆✞ ✔ ✂ ✞ ☛ ✎ ✠ ☎ ✟ ✆ ❂ ✑ ☛ ✟ ✏ ✍ ✔ ✄ ☛ ✻ ☎ ✺ ✠ ✝ ✍ ✄ ✽ ✏ ✓ ✻ ✠ ✟ ✄ ✞ ✍ ☎ ✠ ✽ ✝ ✻ ☛ ✾ ✎ ✞ ✆ ✟ ☎ ✸ ✁ ✔ ✍ ☛ ✎ ✔ � ✸ ✿ ✏ ✓ ✎ ☛ ✓ ✔ ✄ ☎ ✠ ✞ ✻ ☛ ✻ ☎ ✠ ✟ ✎ ✄ ✌ ✾ ✆ ✞ ✂ ✍ ☎ ✍ ✂ ✻ ✄ ✏ ✟ ✠ ✁ ☛ ☛ ✝ ✓ ✝ ✻ ✞ ✂ ✠ ☛ ✞ ✍ ✏ ✺ ✞ ✹ ✻ ✍ ✄ ☛ ✠ ✻ ✮ ✎ ✛ ✆ ✞ ✝ ✁ ☛ ✓ ✎ ✔ ✻ � ✿ ✏ ✎ ✔ ☛ ✓ ✔ ☛ ✁ ✁ ✄ ✏ ✎ ✠ ✞ ✡ ✏ ✍ ✄ ✻ ✞ ☛ ✓ ✎ ✠ ☛ ✿ ✠ ✓ ✧ ☎ � ✧ ✰ ✦ ✭ ☎ � ✥ ✘ ✰ ✧ ✥ ✦ ✧ ✰ ✎ ✁ ✠ ✄ ❂ ✞ ☛ ✾ ✞ ✻ ✞ ✔ ☛ ✸ ✍ ✿ ✠ ✆ ✸ ✻ ✄ ✿ ✓ ✍ ☛ ✎ ✔ ✸ ✆ ✄ ✠ ✔ ✂ ✠ ✏ ✍ ✓ ✆ ✛ ✻ ✏ ✓ ✄ ✸ ✄ ✡ ☛ ✾ ✞ ✟ ☛ ✞ ✔ ❂ ✰ ✠ ☛ ✄ ✎ ✠ ✻ ✎ ✏ ✓ ✸ ✆ ☛ ✻ ✎ ✞ ✓ ✎ ✞ ✾ ✞ ✎ ✟ ✁ ✌ ✻ ✍ ✄ ☛ ✟ ✆ ✏ ✎ ✌ ✻ ❂ ✏ ✧ ✠ ✁ ✆ ✞ ✎ ✠ ✝ ✿ ✗ ✂ ✦ ✝ ✸ ✎ ✌ ✻ ✓ ✺ ✠ ✎ ✎ ✞ ✟ ✗ ✍ ✄ ✞ ✎ ✒ ✄ ✓ ✔ ✂ ☎ ✠ ✝ ✺ ✞ ✾ ✣ ✂ � ✠ ✻ ❂ ☛ ✡ ✌ ✻ ✟ ✄ ✎ ✧ ✄ ✞ ✻ ✿ ☛ ✞ ✎ ✠ ✏ ✎ ✑ ☛ ✞ ✎ ✠ ✝ ✁ ☛ ✸ ✞ ✝ ✰ ✟ ✏ ✟ ✁ ✠ ✸ ✌ ✟ ❀ ✿ ✠ ✍ ☎ ✻ ✤ – p. 29/43 ❚❱❯ d inv ( π, I ) ≥ N + 1 − c ( π ) ▲❆▼ ☎ ✑✸ ☎ ✑✸ ☎ ✑✸ N + 1 ❄❆❅ Cycle Decomposition. I ✏ ✑✸ ✏ ✷✎ ✬✜✭ ✒✝✠ ’ ’ ✽✖☛ √ ⊗

  68. � ✄ ☛ ✾ ✠ ✆ ✄ ✎ ✞ ✟ ✗ ✿ ✆ ❂ ✡ ✿ ✎ ✌ ✓ ✔ ✠ ✻ ✠ ✍ ✞ ✞ ✄ ✄ ✠ � ✄ ✆✞ ✎ ✎ ✏ ✿ ✁ ✗ ✸ ✟ ✆ ✁ ✝ ✾ ✹ ✻ ✓ ✎ ✠ ✂✆ ✞ ✾ ✠ ✡ ✎ ✄ ❉ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ P ❊ ● P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❍ ❋ ☛ ✁ ✞ ✍ ✿ ✆ ✠ ✓ ☛ ✻ ☛ ✎ ✠ ✞ ❊ ✓ ✍ ☎ ✻ ✔ ☛ ☞ ✟ ❃ ❇ ❈❉ ✞ ✿ ✡ ✰ ✧ ✂ ✢ ✧ ✤ � ✣ ✧ ✛ ✮ ✧ ✤ ✮ ✢ � ✰ ✦ ✰ ✧ ✂ ✁ ✕ ✞ ✄ ✴ ✪ ✟ ✴ ✤ � ★ ✁ ✩ ✭ � ✣✤ ✥ ✮ ✩ ✥ ✁ ✯ � ✣ ✆ ✱ ✩ ✪ ✂ � � � ✤ ☛ ☛ ✍ ☎ ✠ ✓ ☛ ✻ ☛ ✎ ✠ ✁ ✞ ✓ ✍ ✻ ✿ ✔ ☛ ☞ ✑ ☛ ✻ ✿ ✄ ✆✞ ✟ ✁ ✗ ✆ ✞ ☞ ✓ ✄ ✿ ✞ ✒ ✝ ✍ ✡ ☛ ✸ ✔ ☛ ✞ ✡ ☛ ✠ ✻ ✠ ✎ ✄ ✍ – p. 30/43 ❚❱❯ ▲❆▼ ✒✝✠ Desire / Grey ❄❆❅ unoriented BreakPoint. oriented ✔ ✂✁

  69. � ✻ ✍ ✓ ✞ ✁ ✠ ✎ ☛ ☛ ✻ ✓ ✠ ✆ ✿ ❅ ✍ ✞ ☛ ☎ ✠ ✄ ☎ ✡ ✍ ☛ ✟ ✞ ✄ ✺ ✆✞ ✸ ✄ ✿ ✻ ✄ ❂ ✻ ☛ ✑ ✎ ✡ ✿ ☛ ☎ ✍ ✓ ✞ ✁ ✠ ✎ ✻ ✔ ☛ ✓ ✠ ✆ ✿ � ✍ ✞ ✻ ☛ ✞ ✟ ✍ ✠ ✻ ✠ ✸ ✠ ✁ ✏ ■ ✟ ✸ ☛ ✻ ❀ ✟ ✟ ☞ ☛ ✆ ✎ ✍ ❘ ❃ ✟ ✸ ✠ ✻ ☎ ✓ ❈❉ ✞ ✁ ✠ ✎ ☛ ✻ ☛ ✓ ❇ ❊ ✆ ◗ P ❊ ❉ P ▼ ● ◆❖ ❉ ❋ ❑ ❏ ■ ❍ ❋ ❍ ❍ ● ✠ ✿ ✠ ✍ ✗ ✍ ✟ ☛ ✔ ✻ ☎ ✓ ✶ ✞ ✁ ✠ ✎ ☛ ✻ ☛ ✓ ✟ ✞ ✞ ✻ ☛ ✎ ✄ ✡ ✞ ✍ ✠ ✠ ☛ ✞ ✟ ✞ ✎ ✄ ✆ ✠ ✾ ☛ ✄ ■ ✞ ✄ ✓ ✗ ✍ ✟ ☛ ☛ ✕ ✞ ✁ ✂ ✧ ✰ ✦ ✰ � ✢ ✿ ✒ ✰ ✻ ☛ ✎ ✄ ✡ ✞ ✍ ✠ ✠ ✝ ☞ ▼ ☛ ✔ ✸ ☛ ✡ ✍ ✤ ✧ ✿ ✮ ✆ ✣ � ✯ ✥ ✴ ✩ ✥ ✩ ✣✤ � ✭ ✩ ✁ ★ � ✤ ✱ ✪ ✮ ✂ ✛ ✧ ✣ � ✤ ✧ ✢ ✧ ✂ ✴ ✮ ✪ ✁ ✤ � � � ✞ ✆ ✡ ✠ ✾ ☛ ✞ ❂ ✄ ✁ ▼ ✾ ✆ ✞ ✂✆ ✠ ✎ ✓ ✻ ✹ ✾ ✠ ✄ ✆ ✌ ✞ ✍ ✠ ✻ ✠ ✔ ✓ ✎ ✎ ✿ ✡ ✆ ✄ ✿ ✗ ✟ ✞ ✝ ✠ ✠ ✍ ✑ ☞ ❙ ☛ ✔ ✻ ☎ ✓ ✻ ✞ ✁ ✠ ✎ ☛ ✻ ☛ ✓ ☛ ✿ ✟ ✆✞ ✸ ❉ ✗ ✁ ✿ ✏ ✎ ✎ ✄ ✄ ✿ ✞ ✄ ✡ ✄ ✁ ✟ ✆✞ ❈ – p. 30/43 ❚❱❯ ▲❆▼ ✒✝✠ Desire / Grey ❄❆❅ unoriented ✔ ✂✁ BreakPoint. oriented ✔ ✂✁

  70. � ☎ ✎ ☛ ✓ ✹ ✎ ✓ ✞ ✻ ✝ ✓ ✔ ✻ ✍ ✝ ✓ ✞ ✁ ✠ ✎ ☛ ✻ ☛ ✓ ✠ ✆ ✿ ✍ ✠ ✂✆ ✞ ✁ ✠ ✁ ✟ ✏ ✟ ✸ ✠ ✻ ☎ ✍ ✓ ✞ ✠ ✞ ✎ ☛ ✻ ☛ ✓ ✠ ✆ ✿ ✍ ☛ ✝ ✠ ✾ ☛ ✕ ☞ ✻ ✆ ✠ ✾ ☛ ✞ ✶ ✟ ✗ ✍ ✟ ☛ ✔ ☎ ✎ ✍ ✓ ✞ ✁ ✠ ✎ ☛ ✻ ☛ ✓ ✠ � ✿ ✄ ✞ ✟ ✓ ✸ ✠ ✻ ☎ ✍ ✓ ✞ ✁ ✠ ✎ ☛ ✻ ☛ ✠ ✟ ✆ ✿ ✞ ☛ ✎ ✄ ✡ ✞ ✍ ✠ ✻ ✠ ✞ ✸ ✄ ✡ ✍ ✟ ✠ ✞ ❂ ✏ ✿ ✍ ✄ ✍ ✠ ✻ ☎ ✝ ❇ ✄ ✎ ☛ ✎ ✓ ✠ ✆ ✿ ✞ ☛ ✻ ✡ ✄ ❃ ❈❉ ✎ ❉ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❊ P ❊ ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ● ❋ ✄ ✡ ✍ ✎ ❂ ✸ ✄ ✒ ✝ ✠ ✁ ✏ ✟ ☛ ✸ ✞ ✏ ✠ ✿ ☛ ☛ ✞ ✍ ✄ ✓ ✞ ✄ ✎ ✿ ✡ ✿ ✏ ✟ ✄ ✡ ✑ ☞ ☛ ✠ ✞ ❂ ✏ ✿ ✍ ✄ ✞ ☛ ✻ ✄ ☛ ✄ ✎ ✡ ❀ ✠ ✸ ✄ ✡ ✆ ✠ ✂ ✔ ✎ ☛ ✆ ✍ ✓ ✔ ✻ ☎ ✍ ✓ ✞ ✁ ✠ ✎ ☛ ✻ ☛ ✠ � ✆ ✿ ✞ ☛ ✎ ✄ ✡ ✞ ✍ ✠ ✻ ✠ ☛ ☞ ✣✤ ✆✞ ✝ ✆ ✠ ✟ ✸ ☛ ✗ ✁ ✿ ✏ ✎ ✎ ✄ ✑ ✿ ✞ ✄ ✡ ✄ ✁ ✟ ✆✞ ✄ ✿ ✻ ☛ ☞ ☛ ✹ ✮ ✛ ✧ ✣ � ✤ ✧ ✩ ✢ ✂ ✴ ✧ ✴ ✪ ✮ ✁ ✤ � � � ✂ ✪ ✩ ✱ ✥ ✆ ✣ ✮ ✧ ✔ ✍ ✸ ☛ ✡ ✥ ✍ ✝ ✒ ✞ ✿ ✄ ✓ ✗ ✟ ✰ ☛ ☛ ✞ ✕ ✁ ✂ ✧ ✰ ✦ ✰ � ✢ ✤ ✾ ✻ ✯ ✠ ✿ ✤ ✍ ✞ ☛ ✎ ✄ ✡ ✞ ✍ ✠ ✻ ✸ ✠ ✠ ✁ ✟ ✏ ✟ ✸ ☛ ✻ ❀ ✟ ✟ ☞ ✆ ✓ ☛ ☛ ✟ ✞ ✄ ✺ ☎ ✆✞ ✄ ✿ ✻ ✄ ❂ ✻ ✑ ☛ ✸ ✠ ✻ ☎ ✍ ✓ ✞ ✁ ✠ ✎ ☛ ✻ � ✔ ✓ ✠ ✿ ✡ ✆ ✄ ✿ ✗ ✟ ✞ ✎ ✩ ✄ ✆ ✾ ✌ ☛ ✞ ❂ ✄ ✁ ✭ ✠ ✾ ✞ ✂✆ ✠ ✎ ✎ ✓ ✻ ✠ ☎ ✍ ✓ ✞ ★ ✁ ✠ ✎ ☛ ✻ ☛ ✓ ✆ ✔ ✿ ✁ ✍ ✞ ☛ ✎ ✄ ✡ ✞ ✍ ✠ ✻ ✠ � – p. 30/43 ❚❱❯ (protected non-hurdle) ▲❆▼ ✒✝✠ Desire / Grey ❄❆❅ unoriented ✔ ✂✁ hurdle ✔ ✂✁ BreakPoint. oriented ✔ ✂✁ ✔ ✂✁ ✔ ✂✁

  71. � ☎ ✝ ✞ ✓ ☞ ✔ ☎ ✂ ✕ ✆ ☞ ✖ ✡ ✔ ✝ ✗ ✂ ✥ ✖ ✥ ✩ ✮ ✏ ☞ ✞ ☞ ✎ ☎ ✁✪ ✖ ☎✫ ✬ ✓ ✎ ✭ ✆ ✡ ✛ ✆ ✔ ✕ ❈ ◆❖ ● ▼ P ❉ ❊ P ■ ❑ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❉ ❏ ✘ ❃ ✂ ✡ ✔ ✕ ✙ ✔ ✆ ❇ ■ ❈❉ ❊ ❋ ● ❍ ❍ ❋ ❍ ☎✥ ★ ✏ ✕ ✞ ✓ ☞ ✔ ☎ ✂ ☎ ☞ ☎ ✖ ✡ ✔ ✝ ✗ ✕ ✏ ✝ ✡ ✘ ☎ � � ✳ ✤ ✧ ✁ � ✆✝ ✝ ✞ ✟ ✆ ✎ ✂ ☎ ✡ ✔ ✂ ✂ ✧ ✡ ✝ ✎ ☎✥ ✦ ☎ ✡ ✓ ✡ ☎ ✚ ✝ ☎ ✕ ✡ ✦ ✟ ✤ ✘ ✓ ✚ ✁ ✣ ✔ ✘ ☞ ✕ ✙ ✔ ☎ ✕ ☞ ✞ ✘ – p. 31/43 ❚❱❯ ✛✢✜ ✏✒✑ ▲❆▼ ❄❆❅ ✠☛✡ ✏✒✑ ☞✍✌ ✥✍✮ ✠☛✡ ✁✄✂

  72. � ✕ ☞ ❉ ✓ ✚ ✘ ☞ ☎ ✣ ✔ ✙ ✕ ✔ ✡ ✂ ✘ ✁ ✔ ☎ ✝ ✚ ☎ ✓ ✧ ✡ ✦ ✞ ☎✥ ✎ ✝ ✘ ✤ ✡ ✘ ✏ ✕ ✮ ✡ ☎ ✂ ✎ ✆ P ✟ ✡ ✞ ✆✝ ❈ ✝ ☞ ✂ ✝ ☎ ✂ ✕ ✗ ✝ ✔ ✡ ✖ ☞ ☎ ✝ ✂ ☎ ✔ ☞ ✓ ✞ ❊ ☎ ✡ ✝ ✆ ✔ ✡ ✂ ✘ ✔ ✥ ✂ ✙ ✗ ✝ ✔ ✡ ✖ ☞ ✕ ✔ ☎ ❍ ❑ ❏ ■ ❍ ❋ ❍ ● ✆ ❋ ❊ ❈❉ ❇ ● ❃ ▼ ✕ ✂ ✦ ☞ � ✩ ☎ ✎ ☞ ✞ ✏ ✂ ★ ☎✥ ✕ ✏ ✕ ✟ ✁ ✄ ☎ ✥ ✔ ☞ ✓ ✞ P ✝ ✆ ☎ ✖ ✖ ✛ � ✁ ✆ � ✗ ✘ ◆❖ ✔ ☞ ✕ ☎ ✔ ✙ ✕ ✡ ✚ ✂ ✘ ✔ ✏ ✂ ✗ ✘ ✓ ✔ ✝ ✧ ✡ ☎ ✦ ☎✥ ✎ ✘ ■ ✤ ✡ ✞ ✁ ✣ ✘ ☞ ✝ ✡ ☎ ▼ ❉ ❙ ✟ ✞ ✆✝ ☎ � ✎ ✁ ✧ ✤ ✳ � � ✆ ✂ ✖ ☞ ☞ ✕ ☎ ✂ ☎ ✔ ✓ ☎ ✞ ▼ ✝ ☎ ✡ ✝ ✡ ✓ ✚ ✆ ✔ ✔ ✥ ✆ ✂ ✗ ✝ ✡ ✂ ✖ ☞ ✕ ☎ ✂ ☎ ✘ ✡ ☞ ✞ ✕ ☎ ✕ ✔ ☞ ✓ ☎ ✔ ■ � ◗ ✆ ✔ ✙ ✕ ✔ ✓ ✝ ✕ ☞ ✞ ☞ ✏ ★ ☎✥ ✏ ☎ ✕ ✟ ✦ ✡ ✕ ☎ ✎ ✩ ✞ ✛ ❘ ✝ ✥ ✆ ✖ ✖ ✡ ✁✪ ✆ ✭ ✎ ✓ ✬ ☎✫ ❅ ❉ – p. 31/43 ❚❱❯ ✛✢✜ ✏✒✑ ▲❆▼ ✏✒✑ ❄❆❅ ✠☛✡ ✥✍✮ ☞✍✌ ✠☛✡ ✏✒✑ ✏✒✑ ☞✍✌ ✥✍✮ ✠☛✡ ✛✢✜ ✏✒✑ ✁✄✂ ✁✄✂

  73. � ✦ ✁ ✠ ☎ ✡ ☎ ✂ ✎ ✆ ★ ☎✥ ✕ ✏ ✕ ✟ ✡ ☎ ☎ ✤ ✘ ✝ ✎ ☎✥ ✦ ✡ ✕ ✧ ✓ ☎ ✚ ✝ ☎ ✂ ✕ ✞ ✞ ✡ ✕ ✟ ✞ ☎ ✆ ✕ ☞ ✞ ✄ ✠ ☎ ✡ ☞ ☞ ☞ ✔ ✂ ☎ ✞ ✆ ✓ ✛ ☞ ✟ ✘ ✝ ✆ ✂ ☎ ✞ ✡ ✁ ✠ ✆ ✥ ✆ ✔ ✙ ✕ ✔ ✡ ✂ ✘ ✔ ☞ ✁ ✠ ✡ ✭ ☎ ✖ ✡ ✔ ✕ ✮ ✔ ☎ ✆ ✎ ✞ ✬ ✥ ☎✫ ✬ ✓ ✦ ✡ ✣ ✂ ✦ ☞ ✓ ✘ ✜ ✙ ✕ ✬ ☞ ✘ ✚ ✓ ☞ ✘ ✂ ✕ ☎ ✂ ✂ ✔ ✆ ✗ ✂ ☎ ✞ ☞ ✆ ✓ ✎ ✞ ☎ ✕ ✥ ✁ ✘ ✦ ✡ ✆ ✦ ✄ ✂ ☎ ✘ ☞ ✞ ✄ ✜ ✘ ✓ ☞ ✆ ✕ ✂ ✎ ✥ ✂ ☎ ✔ ✘ ☎ ✝ ✂ ✕ ✄ ✔ ☞ ✘ ✔ ☞ ✬ ▼ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❊ ❉ P ● ✮ ❍ ❃ ❇ ❈❉ ❊ ❋ ● ❍ ◆❖ ❋ ❍ ■ ❏ ❑ ❉ ✓ ✫ ☞ ✔ ✘ ✓ ☞ ✦ ☎ ✡ ✝ ✟ ✞ ☎ ☎ ✥ ✡ ✆ ✂ ✄ ✗ ✦ ✘ ✦ ✖ ✡ ✦ ✥ ✁ ✦ ☎ ✕ ✆ ✘ ✤ ✜ ✞ ☞ ☎ ✘ ✜ ✄ ✞ ☞ ✘ ✂ ☞ ✄ ✦ ✆ ✡ ✔ ✕ ✓ ✦ ☞ ✡ ✘ ☎ ✂ ✄ ✦ ✆ ✔ ☎ ✕ ☞ ✔ ✆ ✝ ✡ ✂ ✙ ✔ ✁✪ ✞ ✝ ✥ ✆ ✖ ✖ ✛ ✡ ✆ ✭ ✎ ✓ ✬ ☎✫ ✩ ☞ ✕ ✕ ✡ ✦ ✟ ✕ ✏ ☎✥ ☎ ★ ✏ ☞ ✞ ☞ ✎ ✓ ✔ ✝ ✔ ✆ ✕ ☎ ✕ ✔ ☎ ✓ ✞ ☎ � ✆ ✔ ✙ ✕ ✡ ☎ ✔ ✂ ☎ ✕ ☞ ✖ ✡ ✝ ✂ ✗ ✂ ✆ ✥ ✔ ✘ ☎ ✚ ✝ ✝ ✡ ✖ ☞ ✕ ☎ ✂ ☎ ✔ ☞ ✓ ✞ ✝ ☎ ✡ ✡ ✝ � � � ✳ ✤ ✧ ✁ ☎ ☎ ✆✝ ✞ ✟ ✆ ✎ ✂ ✔ ✗ ☎ ✘ ✓ ✧ ✡ ☎ ✦ ☎✥ ✎ ✝ ✘ ✤ ✡ ✞ ✁ ✣ ☞ ✂ ✕ ✏ ✔ ✘ ✂ ✡ ✔ ✙ ✓ ✔ ☎ ✕ ☞ ✘ ✚ ✘ ☞ ✗ ☞ � ✁ ✆ � ☎ ✄ ✂ ✁ � ✩ ☎ ✎ ☞ ✞ ✏ ✖ ★ ☎✥ ✕ ✏ ✕ ✟ ✦ ✡ ✂ ☎ ✝ ✚ ☎ ✓ ✛ ✖ ✡ ✗ ✡ � ✆ ✔ ✙ ✕ ✔ ✡ ✂ ✘ ✔ ✥ ✆ ✂ ✝ ✆ ✔ ✡ ✖ ☞ ✕ ☎ ✂ ☎ ✔ ☞ ✓ ✞ ✝ ✥ ✧ ✕ ☎ ✓ ✂ ✗ ✝ ✔ ✡ ✖ ✦ ✕ ☎ ✂ ☎ ✔ ☞ ✞ ✔ ✝ ☎ ✡ ✝ ✡ ☎ ✂ ✎ ✆ ✟ ✞ ✆✝ ✝ ☞ ✏ ☞ ✘ ✕ ✣ ✁ ✞ ✡ ✓ ✚ ✤ ✘ ☞ ☎ ☞ ✘ ✔ ✙ ✝ ✎ ✕ ✔ ☎✥ ✡ ✂ ✘ – p. 31/43 ❚❱❯ w ✛✢✜ w v ✏✒✑ ▲❆▼ ✏✒✑ u ❄❆❅ ✠☛✡ ✥✍✮ v ☞✍✌ ✛✢✜ ✠☛✡ ✠☛✡ ✏✒✑ ✏✒✑ ✗✢✝ u ☞✍✌ ✥✍✮ ✠☛✡ ✛✢✜ ✏✒✑ ✁✄✂ ✁✄✂

  74. � ✡ ✡ ✥ ☎ ☎ ✞ ✟ ✝ ☎ ✔ ✦ ☞ ✓ ✘ ✜ ✄ ✞ ☞ ✆ ✂ ☎ ✡ ✠ ✁ ☞ ✦ ✘ ✦ ✖ ✦ ✁ ✗ ✥ ✦ ☎ ✕ ✆ ✘ ✤ ✘ ✂ ✡ ✫ ✄ ✦ ✆ ✡ ✔ ✕ ☞ ✬ ☎ ✓ ✎ ✥ ✂ ☎ ✔ ✘ ✂ ✂ ✘ ✄ ✝ ✦ ✆ ✡ ✔ ✕ ☞ ✔ ✆ ✡ ☞ ☎ ✦ ☞ ✓ ✘ ✜ ✄ ✞ ✥ ✕ ✂ ☎ ✎ ☎✥ ✦ ☎ ✡ ✧ ✓ ✚ ✘ ✝ ☎ ✕ ✡ ✦ ✟ ✕ ✏ ✝ ✤ ☎✥ ✕ ✦ ☞ ✓ ✘ ✜ ✙ ✮ ☞ ✡ ✘ ✚ ✓ ☞ ✘ ✣ ✁ ✞ ✕ ★ ✟ ✡ ✂ ☎ ✞ ✟ ☞ ✞ ☞ ☎ ✝ ✠ ✄ ✞ ☞ ✕ ✆ ☎ ✞ ✆ ✘ ✆ ☎ ✎ ✂ ☎ ✡ ☎ ✠ ✁ ✂ ✕ ☞ ☞ ✂ ☎ ✞ ✆ ✓ ✛ ✔ ☎ ✕ ✬ ✆ ✆✝ ✞ ✟ ☎ ✜ ☞ ✓ ✕ ☎ ✡ ✦ ✛ ✦ ☎ ✖ ✆ ✎ ✂ ✔ ☎ ✗ ✆✝ ✞ ✟ ☞ ✞ ✓ ☞ ✦ ✂ ✘ ✞ ✕ ✆ ✞ ✖ ✗ ☞ ✕ ✡ ☞ ❈ ◆❖ ● ▼ P ❉ ❊ P ■ ❑ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❉ ❏ ☎ ❃ ✞ ✖ ✟ ✆✝ ✞ ✟ ☞ ✮ ❇ ■ ❈❉ ❊ ❋ ● ❍ ❍ ❋ ❍ ✝ ✂ ✄ ✟ ✕ ☞ ✮ � ☎ ✆✝ ✞ ☞ ✡ ✦ ☎ ✖ ✆ ✎ ✕ ☎ ✡ ✔ ✆ ✞ ✓ ✔ ☞ ✘ ✝ ✆ ✦ ☞ ✘ ✦ ✜ ✄ ✞ ☞ ✘ ☎ ✂ ✄ ✘ ✆ ✗ ✞ ✜ ☎ ✝ ✆ ✗ ✂ ☎ ✓ ✆ ☞ ✔ ✕ ☎ ✕ ✆ ✘ ✝ ✎ ✖ ✓ ☎ ✑ ✆✝ ✞ ✟ ☞ ✂ ☎ ✌ ✂ ☎ ✟ ✕ ☎ ✂ ✕ ☞ ✆ � ✂ ✂ ✕ ✔ ☎ ✂ ☎ ✕ ☞ ✖ ✡ ✝ ☞ ✗ ✂ ✆ ✥ ✔ ✘ ✂ ✔ ✓ ✔ ✎ ✎ ☎ ✩ ✁✪ ☎✫ ✬ ✓ ✭ ✞ ✆ ✡ ✛ ✖ ✖ ✆ ✥ ✝ ✡ ✕ ✞ ☎ ✆ ✎ ✂ ☎ ✡ ☞ ✡ ✝ ✞ ✞ ✓ ☞ ✔ ☎ ✂ ☎ ✟ ✆✝ ✙ ✔ ✔ ✆ � ☎ ✞ ✓ ☞ ✕ ✝ ☎ ✕ ✆ ✘ ✝ ✗ ✂ ☞ ☞ ☞ ☞ ☞ ✓ ☞ ✔ ☎ ✂ ☎ ✕ ✖ ✝ ✡ ✔ ✝ ✗ ✂ ✏ ✔ ✞ ☎ ✂ ☎ � � ✳ ✤ ✧ ✁ � ✆✝ ✡ ✞ ✟ ✆ ✎ ✂ ☎ ✡ ✝ ✘ ✡ ✏ ☎ ☎ ✡ ✧ ✓ ☎ ✚ ✝ ✕ ☎✥ ✡ ✦ ✟ ✕ ✏ ✕ ☎✥ ★ ✦ ✎ ✔ ✚ ✕ ✙ ✔ ☎ ✕ ☞ ✘ ✓ ✝ ☞ ✘ ✣ ✁ ✞ ✡ ✤ ✘ ✕ ✝ ✖ ☎ ✔ ✕ ✙ ✔ ✆ � ✡ ✔ ✂ ✘ ✂ ✡ ✔ ✕ ✙ ✔ ☎ ✡ ✘ ✆ ✡ ✞ ✓ ☞ ✔ ☎ ✂ ☎ ☞ ✔ ✖ ✡ ✔ ✝ ✗ ✂ ✆ ✥ ✖ ✞ ✥ ✂ ☎ ✡ ☎ ✂ ✔ ✆ ✗ ☎ ✥ ✂ ✞ ✆ ✓ ✎ ✞ ☎ ✕ ✦ ✆ ✬ ✠ ✥ ☎✫ ✬ ✓ ✎ ✭ ✆ ✡ ✁ ✔ ☞ ✔ ✘ ✂ ✡ ✔ ✕ ✙ ✝ ✕ ✆ ✘ ✘ ✣ ✁ ✞ ✡ ✤ ✝ ✓ ✎ ☎✥ ✖ ☎ ✡ ✧ ✓ ☞ ✚ ✚ ✂ ✔ ✝ ✗ ✂ ✏ ✔ ✘ ✡ ✘ ✔ ✕ ✙ ✔ ☎ ✕ ☞ ☎ ✦ ✝ ✁ ✏ ☞ ☎ ★ ✎ ☎✥ ☎ ✩ ✕ ✏ � ✕ ✂ ✞ ✟ ✄ ☎ � ✦ ✆ ✁ ✡ � ✛ ✖ ✕ ☞ – p. 31/43 ❚❱❯ w ✛✢✜ w ✠☛✡ v ✏✒✑ ☞✍✌ ▲❆▼ ✠☛✡ super-hurdle ✏✒✑ u ❄❆❅ ✠☛✡ ✥✍✮ v ✠☛✡ ☞✍✌ ✛✢✜ ✠☛✡ ✠☛✡ (simple hurdle) ✏✒✑ ✠☛✡ ✏✒✑ ✗✢✝ ✏✒✑ u ☞✍✌ ✥✍✮ ✠☛✡ ✠☛✡ ✠☛✡ ✛✢✜ ✏✒✑ ✁✄✂ ✁✄✂ ✁✄✂

  75. � ✞ ✎ ✠ ✝ ✿ ☛ ✆ ☛ ✿ ✻ ✄ ✌ ✂✆ ✞ ✾ ✠ ✝ ✄ ✡ ✻ ✸ ✓ ✞ ✄ ✧ ✆ ✄ ✞ ✸ ✓ ✝ ✻ ✟ ✹ ✎ ✞ ✟ ✓ ✝ ✻ ✞ ✓ ✎ ☛ ✠ ✁ P ❑ ❉ ◆❖ ● ▼ P ❉ ❊ ❈ ■ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❏ ❍ ✁ ✄ ✆✝ ✞ ✁ ✟ ✄ ✆✠ ✞ ✁ ✆✡ ✠ ❋ ✞ ❃ ❇ ❈❉ ❊ ❋ ● ❍ ❍ ✞ ✎ ❀ ✩ ✄ ✎ ✴ ✞ ✒ ✳ ✄ ✓ ✔ ✔ � � ✎ ✂ ✁ ☎ ✄ ✍ ✞ ✟ ☛ ✓ ✎ ✌ ✿ ✌ ✸ ✍ ✸ ✵ ✁ ❂ ✭ ✡ ✻ ✄ ✎ ☛ ✁ � ✠ ✰ ✁ ✻ ☛ ✞ ✠ ✧ ✧ ✄ ✮ ✛ ✴ ✯ ✳ ✧ ✣ ✡ � ✸ ✺ ☛ ✻ ✎ ✡ ✓ ✎ ✠ ✠ ✞ ✠ ✸ ✮ ✂✆ ✞ ✾ – p. 32/43 17 ❚❱❯ 2 1 15 16 = [4 7 8 11 12 9 10 13 14 5 6 15] ▲❆▼ 6 5 14 13 ❄❆❅ = [8 11 12 9 10 13] π 10 = [16 1 2 17 0 3] 9 12 BreakPoint 11 8 ☎ ✑✸ 7 4 3 ✂☎✄ 0 � � �

  76. ✆ ✎ ❙ ✻ ✌ ✎ ✸ ☛ ✡ ✆ ✄✌ ✒ ✸ ✔ ☛ ☛ ✍ ✓ ✄ ✁ ☎ ✎ ✠ ✿ ☛ ✠ ✸ ✄ ✟ ✞ ✏ ❑ ◗ ■ ❈ P ❊ ❉ P ▼ ● ◆❖ ■ ❉ ❏ ❉ ■ ❍ ❋ ❍ ❍ ● ❋ ❊ ❈❉ ❇ ▼ ❃ ✎ ✟ ❅ ✍ ☛ ✎ ✝ ✄ ✁ ✄ ✎ ✸ ☛ ✻ ☎ ✔ ✏ ✠ ✟ ☛ ✞ ☛ ✎ ✄ ✡ ✞ ✞ ✄ ✆ ✺ ✡ ✸ ✞ ✠ ✎ ✻ ☛ ✍ ✔ ✓ ✠ ✎ ✓ ✄ ✍ ✝ ✓ ✠ ✞ ✎ ✾ ☛ ✻ ☛ ✞ ✂ ▼ ✍ ✓ ✆✞ ❘ – p. 33/43 Pevzner. ❚❱❯ Hannenhalli ▲❆▼ ✏✑✸ ❄❆❅ ✕✖☛

  77. � ✄ ✄ ✞ ✿ ✄ ✆✞ ✎ ✎ ✏ ✿ ✁ ✗ ✸ ✍ ☎ ✿ ✠ ❂ ✞ ✹ ✻ ✟ ☛ ✞ ✟ ✞ ✒ ✺ ✆✞ ☎ ✠ ✣ ✠ ✝ ✺ ✡ ☛ ☎ ✺ ✄ ✞ ✾ ✆ ✠ ✄ ☛ ✻ ✟ ☛ ✞ ✍ ✏ ✻ ✠ ☛ ✻ ✿ ✄ ✻ ✁ P ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ ❉ ❊ ❊ P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❋ ❈❉ ☛ ✝ ☛ ✿ ✏ ☛ ✝ ✎ ✞ ✄ ✡ ✻ ☛ ✞ ✿ ❇ ✄ ✆ ✁ ✄ ✍ ✿ ✏ ❂ ✞ ✠ ✟ ❃ ✔ ✁ ✁ ✻ ✎ ☛ ✡ ✠ ✸ ✠ ✆✞ ✓ ✍ ✂ ✞ ☛ ☛ ✄ ✾ ✎ ✞ ✓ ✠ ✝ ✍ ✄ ✓ ✎ ✠ ✓ ✝ ✁ ✍ ✄ ✩ ✚ ✣✤ ✧ ✁ ✸ ✺ ✆ ✄ ✞ ✞ ✡ ✎ ✄ ☛ ✞ ☛ ✟ ✏ ✍ ✔ ☎ ✻ ☛ ✸ ✎ ☛ ✔ ☛ ✄ ✻ ✌ ✻ ✟ ☛ ✞ � ☛ ✁ ☎ ✍ ✁ ☛ ✠ ✞ ✿ ✏ ✻ ✑ ✿ ✌ ✸ ✍ ✞ ☛ ✸ ✎ ✡ ✓ ✿ ✠ ✎ ☎ ✆ ✆ ✄ ✎ ✁ ✄ ✍ ✠ ✏ ✟ ☛ ✎ ✔ ✞ ✸ ✒ ✄✌ ✎ ☛ – p. 33/43 Pevzner. ❚❱❯ (fortress) Hannenhalli ▲❆▼ ✏ ✑✆✞ ✏✑✸ ❄❆❅ ✒✝✠ ✕✖☛

  78. � ✻ ✸ ✄ ✁ ✠ ✟ ✓ ✏ ❂ ✠ ✎ ☎ ✁ ✿ ✞ ✠ ✞ ☛ ✻ ✓ ✔ ✄ ✔ ✓ ✆✞ ✺ ☎ ✍ ✞ ☛ ✟ ✓ ✻ ✔ ✍ ✏ ✻ ✞ ✆ ☛ ✎ ✡ ✎ ✞ ✞ ✄ ✿ ✝ ✞ ☛ ✻ ✡ ✄ ✎ ✁ ✝ ☛ ✏ ✿ ☛ ☛ ✻ ✣ ✆ ✄ ✝ ✎ ☛ ✸ ✞ ✄ ✓ ☎ ✒ ☛ ✄ ✍ ✍ ✞ ✁ ✟ ✠ ✞ ❂ ✏ ✿ ✎ ✂ ✒ ✂ ❍ ❍ ● ❋ ❊ ❈❉ ❇ ❃ ☛ ❍ ✠ ✆ ✾ ✞ ✄ ✺ ☎ ✆✞ ❋ ■ ✿ ■ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ❈ ❏ P ❊ ❉ P ▼ ● ◆❖ ❉ ❑ ✄ ☛ ✍ ✄ ✞ ☎ ☛ ✡ ✍ ☛ ✟ ✡ ✒ ✁ ✄ ✞ ✄ ✺ ☎ ✻ ✄ ❂ ✠ ✠ ✻ ✺ ☛ ✡ ✠ ✿ ✠ ✔ ☛ ✡ ✝ ✂ ✠ ✁ ✔ ✁ ✁ ☛ ✗ ✟ ✞ ✄ ☎ ✞ ✞ ☛ ✠ ✆ ✄ ✎ ✏ ✟ ✎ ✠ ✻ ☛ ✍ ✔ ✓ ✠ ✎ ✓ ✿ ✎ ✍ ✒ ✌ ✎ ✸ ☛ ✡ ✟ ✆ ✄✌ ✸ ☎ ✔ ✎ ☛ ✺ ✍ ✓ ✄ ✁ ✄ ✝ ✸ ☎ ✄ ✁ ✄ ✎ ✸ ☛ ✻ ✔ ✝ ✍ ✏ ✟ ☛ ✞ ✞ ☛ ✎ ✞ ✎ ✠ ✂ ✓ ✞ ✎ ✾ ☛ ✻ ☛ ✞ ✆ ☛ ✍ ✓ ✆✞ ✠ ✸ ✠ ✡ ✄ ✻ ✟ ✡ ✻ ✞ ✄ ✺ ☎ ✆✞ ✄ ✿ ☛ ✄ ✠ ✻ ✏ ✍ ✞ ☛ ✟ ✻ ✿ ✆✞ ✚ ✿ ✞ ☛ ✟ ✻ ✹ ✞ ❂ ✠ ✍ ✎ ✄ ✸ ✩ ✗ ✁ ✿ ✏ ✎ ☛ ✠ ☛ ✠ ✌ ✿ ✑ ✻ ✏ ✿ ✞ ✁ ✍ ✄ ✍ ☎ ✁ ☛ � ✁ ✞ ✸ ✞ ✣✤ ☛ ✠ ✧ ✆ ✾ ✞ ✄ ✺ ☎ ✡ ☛ ✺ ✝ ✠ ✁ ✔ ✁ ✁ ☛ ✄ – p. 33/43 Pevzner. ❚❱❯ (fortress) Hannenhalli ▲❆▼ ✏ ✑✆✞ ✏✑✸ ❄❆❅ ✒✝✠ ☎✝✆ ☎ ✑✸ ✏ ✑✆✞ ✗✑✸ ✕✖☛

  79. ✁ ✗ ✎ ☛ ✝ ✎ ✠ ✎ ✞ ✟ ✍ ✔ ✄ ✎ ✞ ✒ ✄ ✓ ✔ ❂ ✡ ✻ ✎ ✄ ✍ ✎ ☛ ✓ ✔ ✍ ✞ ☛ ✸ ✄ ✏ ✎ ✞ ✒ ✄ ✓ ✔ ✸ ☛ ✿ ✻ ✎ ✏ P ❑ ❉ ◆❖ ● ▼ P ❉ ❊ ❈ ■ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❏ ❍ ☛ ✓ ✞ ☛ ✿ ✏ ✎ ✔ ✓ ✺ ☎ ✔ ❋ ✠ ❃ ❇ ❈❉ ❊ ❋ ● ❍ ❍ ✩ ✓ ✿ ✠ ✝ ✸ ✠ ✆✞ ✓ ✍ ✠ ✿ ✝ ✎ ☎ ✺ ✠ ✝ ✍ ✄ ☛ ✹ ✠ ✄ ✄ ✵ � ✯ ✬ ✣ ✦ ✮ ✧ ✰ ✭ ✍ ✢ ✧ ✤ ✁ ✏ ✍ ✾✌ ✻ ☛ ✓ ❂ ✞ ✎ ✂ ✞ ✓ ✿ ✆ ☛ ✂ ✍ ☛ ✞ ✔ ✟ ✏ ✎ ✔ ✎ ☛ ✸ ✔ ✔ ✟ ☛ ✸ ☛ ✞ ☛ ✎ ✞ ✸ ☛ ✟ ✍ ✓ ✄ ✿ ✞ ✒ ✝ ✍ ✡ – p. 34/43 π d inv ( π, I ) = ( N + 1) − cycles ( π ) + hurdles ( π ) + ( fortress ) ❚❱❯ Pevzner. ☎ ✑✸ ▲❆▼ Hannenhalli ❄❆❅ ✏ ✑✸ √

  80. ✁ ✄ ✓ ✔ ✠ ✟ ✁ ✍ ✌ ✸ ✎ ✠ ✒ ✹ ✺ ✆ ✔ ✍ ☛ ✑ ✓ ✎ ✔ ✻ ✿ ✆ ✌ ☎ ✓ ✏ ✄ ✩ ✎ ✠ ✎ ✞ ✟ ✗ ✍ ✄ ✎ ✞ ✒ ✓ ✔ ✔ ❂ ✡ ✻ ✄ ✎ ☛ ✞ ☛ ✿ ✏ ✎ ✎ ✎ ✎ P ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ ❉ ❊ ❊ P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❋ ❈❉ ✝ ✄ ✿ ✔ ✍ ✠ ✆ ✾ ✗ ✎ ✠ ✝ ☎ ✺ ✞ ❇ ✎ ✔ ✻ ☎ ✟ ✾ ✆ ☛ ✓ ✔ ✠ ❃ ☛ ✝ ✻ ✂ ☎ ✺ ✠ ✝ ✍ ✄ ❂ ✹ ✓ ✄ ✞ ✞ ✠ ☛ ✎ ✞ ✸ ☛ ✟ ✍ ✔ ✓ ✄ ✿ ✝ ✿ ✒ ✍ ✧ ✤ ✵ ✰ ✁ ✧ ✏ ✍ ✾✌ ✻ ☛ ✄ ✣ ✎ ✠ ✝ ✸ ✠ ✆✞ ✮ ✓ ✦ ✍ ✠ ✞ ✝ ✭ ✄ ✏ ✓ ✎ ☛ ✓ ✔ ✍ ✞ ☛ ✸ ✍ ✎ ☛ ✞ ✒ ✄ ✓ ✔ ✸ � ☛ ✿ ✏ ✎ ✿ ✔ ✍ ✂ ✡ ☛ ✸ ✔ ✟ ☛ ✞ ✓ ✿ ✆ ☛ ✍ ✔ ☛ ✎ ✞ ✟ ✬ ✯ ✏ ✎ ✔ ✎ ☛ ✸ ✢ – p. 34/43 d inv ( π, I ) = breakpoints ( π ) − cycles ( π ) + hurdles ( π ) + ( fortress ) π d inv ( π, I ) = ( N + 1) − cycles ( π ) + hurdles ( π ) + ( fortress ) ❚❱❯ Pevzner. ☎ ✑✸ ▲❆▼ Hannenhalli ❄❆❅ ✏ ✑✸ √ ◮

  81. ✁ ✂ ☎ ✡ ☞ ✡ ☎ ✦ ✡ ☎ ✕ ✆ ✭ ✎ ☎ ✝ ✜ ✆✝ ✥ ✘ ☞ ✎ ☞ ✞ ☞ ✥ ✘ ☞ ✕ ✦ ✗ ✥ ✫ ✓ ✞ ✂ ☎ ✕ ✟ ☞ ✂ ✟ ✝ ✂ ✘ ☞ ✓ ✥ ✛ ✞ ✘ ★ ☎✥ ✎ ✝ ✘ ✤ ✡ ✞ ✆ ✣ ✔ ✘ ✚ ✚ ✁ ✮ ★ ☎✥ ✕ ✏ ✕ ✟ ✦ ✡ ✕ ☎ ✝ ☎ ☎ ✓ ✞ ✣ ✡ ☎ ✦ ★ ☎✥ ✎ ✝ ✘ ✤ ✩ ✡ ✦ ☞ ✏ ✂ ✦ ✂ ✘ ☞ ✗ ✠ ✂ ✘ ✔ ★ ☎✥ ✕ ✕ ✝ ✟ ✦ ✡ ✕ ☎ ✝ ✚ ☎ ✓ ✞ ✣ ✥ ✗ ☎ ✝ ✆ ☎ ☎ ★ ☎✥ ✕ ✏ ✕ ✟ ✦ ✡ ✕ ☎ ✝ ✚ ✓ ❃ ✞ ✣ ✥ ✗ ✝ ✦ ☎ ✥ ✡ ✕ ✆ ✝ ★ ✮ ❇ ✕ P ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❊ ❉ ▼ ❈❉ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ● ❋ ❊ ☞ ✗ ☎ ✂ ✦ ✁ � ☎ ✕ ✆ ✝ ✥ ✡ ✆ ✔ ✗ ✠ ✘ ✬ ✔ ✆ ☞ ✝ ✘ ✤ ✡ ✞ ✆ ✛ ✦ ✡ ✕ ✂ ✂ ✦ ✂ ☎ ✞ ✣ ✆ ✂ ✛ ✞ ✄ ✠ ☎ ✖ ✏ ✠ ✆ ✦ ✝ ✘ ☞ ✫ ✗ ✥ ✡ ✕ ✂ ✆ ✠ ✡ ☎ ✞ ✡ ✣ ✏ ✎ ✓ ✏ ✿ ☛ ✔ ✔ ✸ ☛ ✎ ✔ ✎ ✟ ✓ ✞ ✎ ☛ ✍ ✂ ☛ ✆ ✿ ✓ ✞ ☛ ✟ ✔ ☛ ✔ ☛ ✏ ✍ ✗ ✟ ✂ ✎ ✠ ✎ ✝ ☛ ✎ ✻ ✔ ✎ ✿ ✍ ☛ ✸ ✔ ✓ ✄ ✒ ✞ ✎ ✄ ✍ ✸ ☛ ✞ ✸ ✡ ✎ ✏ ✆✞ ✠ ✸ ✝ ✠ ✎ ✄ ✍ ☛ ✻ ✾✌ ✍ ✁ ✍ ✤ ✧ ✢ ✭ ✵ ✰ ✧ ✮ ✦ ✣ ✬ ✯ � ✓ ✠ ✍ ✞ ✝ ✒ ✞ ✿ ✄ ✓ ✍ ✟ ☛ ✸ ✞ ✎ ☛ ✂ ✿ ✞ ✄ ✓ ✹ ❂ ✄ ✍ ✝ ✠ ✺ ☎ ✝ ✠ ✄ ✞ ✞ ✔ ✡ ✆ ✦ ✚ ✗ ✛ ✓ ☞ ✜ ☎ � ✠ ✓ ☎ ☛ ✆ ✾ ✟ ☎ ✻ ✔ ✎ ✞ ✒ ✺ ☎ ✝ ✕ ✡ ✎ ☞ ✙ ✝ ✦ ☎ ✂ ✬ ✕ ✟ ✝ ✔ ✡ ✓ ✟ ✔ ✦ ✡ ✕ ✆ ✓ ☞ ✜ ☎ ☞ ✚ ✎ ✖ ✆ ✠ ✄ ✗ ✔ ✎ ✸ ✾ ✍ ✁ ✟ ✠ ✔ ✓ ☎ ✺ ✓ ✎ ✒ ✏ ✿ ☛ ✞ ☛ ✎ ✄ ✻ ✡ ❂ ✔ ✓ ✄ ✠ ✌ ✏ ✆ ✄ ✔ ✝ ✎ ✓ ✎ ✿ ✔ ✑ ✍ ✿ ✌ ✠ ☛ ✍ ✔ ✎ ✆ ✆ ✹ ✻ – p. 34/43 d inv ( π, I ) = breakpoints ( π ) − cycles ( π ) + hurdles ( π ) + ( fortress ) π d inv ( π, I ) = ( N + 1) − cycles ( π ) + hurdles ( π ) + ( fortress ) ❚❱❯ Pevzner. ☎ ✑✸ ✟✢✝ ▲❆▼ Hannenhalli ✟✢✝ ❄❆❅ ✏ ✑✸ breakpoint ✠☛✡ ✁ ✢✝ ✠☛✡ √ ◮ ⊚

  82. ✙ ❂ ✻ ✹ ✞ ✺ ✝ ✠ ✁ ✔ ✁ ✁ ☛ ✠ ✏ ✔ ❃ ✓ ☛ ✎ ✓ ✏ ✿ ☛ ✻ ✔ ✎ ☛ ✞ ✄ ✟ ❇ ✪ P ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❊ ❉ ▼ ❈❉ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ● ❋ ❊ ☎ ✡ ✆ ✂ ✄ ✟ ✍ ✝ ✠ � ✟ ✯ ✤ ✧ ✢ ✣ ✩ ✁ ✰ ✆ ✧ ✮ ✲ ✣ ✵ ✯ � ✧ ✰ ✧ ✿ ☎ ✥ ☛ ☛ ✄ ✍ ✞ ✄ ✡ ✡ ✆ ✮ ☛ ✟ ✸ ✍ ✠ ✝ ✠ ✿ ✔ ✓ ☎ ✺ ✓ ☛ ✞ ✍ ✻ ✮ – p. 35/43 ❚❱❯ Hannenhalli ▲❆▼ ✒✝✠ ❄❆❅ Pevzner. ❀✖✺

  83. ✙ ✟ ☛ ✡ ✄ ✒ ✝ ✄ ✎ ☛ ✠ ☛ ✻ ✏ ✆ ✺ ✄ ✓ ✞ ☛ ✎ ✻ ✿ ✡ ✝ ✡ ✡ ✆ ✌ ✻ ✂ ✻ ☛ ✠ ✏ ✎ ✏ ✍ ✓ ✆✞ ✠ ✻ ✠ ✎ ✄ ✡ ✝ ☛ ✄ ✠ ✻ ✏ ✆ ✺ ✄ ✓ ✁ ✆ ✠ ✟ ✏ ✄ ✞ ☛ ✪ ✻ ✠ ☛ ✆ ✂ ✆ ✠ ✁ ✠ ✿ ✝ ✝ ✠ ✡ ✝ ✔ ✠ ✆ ✾ ✻ ✄ ✞ ✺ ✏ ✗ ✠ ✍ ✡ ✟ ❍ ❍ ● ❋ ❊ ❈❉ ❇ ❃ ☛ ❍ ✡ ✺ ✝ ✠ ✁ ✔ ✁ ✁ ☛ ❋ ■ ✔ ■ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ❈ ❏ P ❊ ❉ P ▼ ● ◆❖ ❉ ❑ ✻ ✎ ✄ ✠ ✻ ✌ ✎ ✸ ✠ ❂ ✡ ✄ ✎ ✍ ✞ ☛ ✟ ✸ ✗ ✁ ✿ ✠ ✎ ✄ ✿ ✓ ☛ ✞ ☛ ✎ ✻ ✠ ✡ ✡ ✻ ✾ ✠ ✍ ✄ ✝ ✠ ✿ ✻ ✌ ✡ ❂ ✡ ✎ ✂ ☛ ✎ ☛ ✞ ✄ ☎ ✡ ✆ ✟ ✄ ✻ ✍ ✞ ✄ ✡ ✡ ✆ ☛ ✟ ✸ ✔ ☛ ✍ ✁ ✹ ✠ ✺ ✝ ✠ ✁ ✔ ✁ ☛ ✿ ✠ ✏ ❂ ✔ ✓ ☛ ✎ ✓ ✏ ☛ ✝ ✟ � ✩ ✰ ✧ ✮ ✲ ✣ ✵ ✯ ✧ ✢ ✰ ✧ ✮ ✥ ✆ ✣ ✯ � ✮ ✂ ✧ ✠ ✍ ✿ ✔ ✓ ☎ ✺ ✓ ☛ ✞ ✻ ✤ ✠ ☎ ✁ ✿ ✄ ✍ ✝ ✠ ✟ ✻ ✞ � ✏ ✠ ✼ ✟ ✔ ✓ ☛ ✎ ✓ ✿ ✁ ☛ ✻ ✔ ✎ ✿ ✍ ✝ ✠ ✿ ✗ ✡ ✏ ✁ ✠ ✿ ✝ ✻ ✌ ✁ ✟ ✟ ✸ ✻ ✌ ✎ ✞ ☛ ✟ ✻ ✌ ✎ ✓ ✄ ✂ ✂✆ ✆ ✠ ✂ ✄ ✞ ✻ ✗ ☛ ☛ ❂ ✠ ✠ ✓ ✏ ✍ ✏ ✻ ✍ ✄ ✄ ✠ ✁ ✠ ✿ ✝ ☛ ✻ ✍ ✿ ✏ ✠ ✠ ✆ ✠ ✍ ☎ ✿ – p. 35/43 ’ ❚❱❯ O (1) Hannenhalli ✒✝✠ ▲❆▼ ✒✝✠ ❄❆❅ breakpoints O ( n ) . ✏✑✸ Pevzner. ✒✝✠ ✏ ✷✿ ❀✖✺

  84. ✙ ✿ ✾ ✍ ✄ ✝ ✠ ✿ ✻ ✌ ✡ ❂ ✠ ✍ ✻ ✄ ✻ ✌ ✎ ✸ ✠ ❂ ✡ ✄ ✠ ✎ ☛ ✡ ☛ ✁ ☎ ✍ ✠ ✿ ❀ ✟ ☛ ✡ ✺ ✝ ✠ ✔ ✡ ✁ ✁ ☛ ✻ ✔ ✎ ✓ ✞ ☛ ✎ ✻ ✠ ✞ ✟ ✌ ✟ ✿ ☛ ✻ ☛ ✡ ✄ ✒ ✝ ✄ ✎ ☛ ✠ ✎ ✻ ✏ ✆ ✺ ✄ ✓ ✞ ☛ ✎ ✄ ✪ ✡ ✏ ✠ ✸ ✆ ✗ ✁ ✿ ✠ ✎ ✄ ✍ ✝ ✠ ✡ ✡ ✌ ✻ ✻ ✂ ✻ ☛ ✝ ✠ ✎ ✏ ✍ ✓ ✆✞ ✠ ✻ ✸ ✠ ✎ ❇ ❃ ✌ ✔ ✝ ✠ ✾ ✞ ✂✆ ✝ ✠ ✸ ❊ ✄ ✓ ✹ ✎ ✓ ✞ ✻ ✝ ✓ ✟ ✞ ❈❉ ❋ ✟ ❊ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❉ ● P ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❍ ✎ ✄ ✑ ✔ ✠ ✂ ✗ ✂✆ ✠ ✓ ✏ ✽ ✓ ☛ ✍ ✎ ☛ ✹ ✆ ✄ ✠ ✎ ✞ ☛ ✎ ✄ ✒ ✡ ✎ ✆ ✍ ✻ ✂ ✝ ✓ ✸ ✞ ✎ ✄ ✍ ✝ ✠ ✓ ✡ ✠ ✿ ✁ ✠ ✿ ✝ ☛ ✻ ✄ ✍ ✏ ✠ ✆ ✠ ✂ ✡ ✁ ✏ ✁ ✁ ☛ ✠ ✏ ❂ ✔ ✓ ☛ ✎ ✓ ✿ ✁ ☛ ✻ ✔ ✎ ☛ ✞ ✄ ☎ ✡ ✆ ✟ ✔ ✠ ✄ ✻ ✂✆ ✠ ✓ ✏ ✿ ✄ ✍ ✏ ✠ ❂ ☛ ✞ ✝ ✄ ✻ ☎ ✍ ✠ ✿ � ✟ ✻ ✹ ✞ ✺ ☛ ✍ ✂ ✵ ✟ ✤ ✧ ✢ ✂ ✩ ✰ ✧ ✮ ✲ ✣ ✯ ✝ � ✧ ✰ ✧ ✮ ✥ ✆ ✣ ✯ � ✮ ✠ ✍ ✞ ✿ ✄ ✡ ✡ ✆ ☛ ✟ ✸ ☛ ✍ ✝ ✠ ✔ ✄ ✓ ☎ ✺ ✓ ☛ ✞ ✍ ✻ ✠ ☎ ✁ ✿ ✗ ✠ ✠ ✻ ✓ ☛ ✁ ✠ ✟ ✏ ✄ ✞ ☛ ✎ ✠ ✺ ✡ ✡ ✂ ✠ ✁ ✠ ✿ ✆ ✻ ✌ ✁ ✄ ✆ ✏ ✻ ✿ ✝ ✝ ✠ ✡ ✆ ✝ ✠ ✆ ✾ ✄ ✏ ✞ ✺ ✏ ✗ ✔ ✆ ✆ ☛ ✄ ✠ ✻ ✟ ✝ ✟ ✠ ✔ ✎ ✻ ✍ ✝ ✠ ✓ ✡ ✂ ✠ ✁ ✿ ☛ ✝ ☛ ✻ ✄ ✍ ✏ ✠ ✆ ✠ ✿ ✍ ☛ ✻ ✄ ✿ ✻ ✏ ✎ ✁ ✿ ✌ ✠ ✼ ✟ ✟ ✸ ☛ ✔ ✞ ✓ ✎ ☛ ✎ ✌ ✓ ✗ – p. 35/43 ’ ❚❱❯ O (1) Hannenhalli ✒✝✠ breakpoint ▲❆▼ ✒✝✠ ❄❆❅ breakpoints ☎ ✑✸ O ( n ) . ✏✑✸ Pevzner. ✒✝✠ ✏ ✷✿ ❀✖✺

  85. � ❙ ■ ◗ ❘ ❅ ■ ▼ ❉ ▼ P ✣ ✁✂ ✁ ✁ ✧ ✭ ✲ ❈ ❊ ● ❍ ▼ ❉ ❑ ❏ ■ ❍ ❋ ❍ ❉ ● ❋ ❊ ❈❉ ❇ P ❃ ✁ ☎ � ✦ ✁ ✯ ✤ ✞ � � ✧ ✥ ✲ ✭ ✧ ✮ ✣ ✁ ✁ ✩ ✭ � ◆❖ ★ ✢ ✣✤ ✩ ✱ ✩ ✣ ✦ ★ � ✧ ✳ ✆ ✤ � ✮ ✪ ✲ ✄✂ [ Empodia, Emiris & Diochnos, IOI ’04 ] The ancient mathematician and philosopher Pythagoras believed that reality is mathematical in nature. Present-day biologists study properties of orderings of the sequence I = (0 , 1 , 2 , . . . , N, N + 1) where 0 is the first element in the ordering, N + 1 is the last element in the ordering, and there is no element E such that E + 1 directly follows E in the ordering. Such orderings of I are called biosequences, and consecutive subsequences of them are called segments. A segment of a biosequence is called a framed interval if it includes all integers whose values are between the value of the first element, which must be the smallest element in the segment, and the last element, which must be the largest. A framed interval is called empodio if it does not contain any shorter framed intervals. As an example, consider the biosequence (0 , 3 , 5 , 4 , 6 , 2 , 1 , 7) . The whole biosequence is a framed interval. However, it contains another framed interval (3 , 5 , 4 , 6) and therefore it is not an empodio. The framed interval (3 , 5 , 4 , 6) does not contain a shorter framed interval, so it is an empodio. It is also the only empodio in that biosequence. You are to write a program that, given a biosequence, finds all empodia (plural for empodio) in that biosequence. ❄❆❅ ▲❆▼ ❚❱❯ – p. 36/43

  86. � ✠ ✻ ✄ ❂ ✸ ✄ ✡ ✠ ✿ ✸ ✿ ✞ ✎ ✓ ✞ ☛ ✟ ☛ ✍ ✝ ✞ ✁ ✞ ☎ ✒ ✟ ✞ ✺ ✠ ❂ ☛ ❂ ✆ ✠ ✏ ❂ ✄ ✓ ✻ ✝ ✠ ✺ ✔ ☎ ✄ ☛ ✠ ✁ ✠ ✟ ☛ ✞ ✁ ✟ ✎ ✒ ☛ ✍ ✗ ✎ ✓ ☛ ✞ ✿ ✝ ✡ ✏ ✻ ✆ ✿ ✠ ✎ ✓ ✞ ☛ ✎ ✹ ✄ ✎ ✄ ✁ ✄ ✍ ✝ ✠ ✿ ✸ ✓ ✞ ✻ ✟ ❍ ❍ ● ❋ ❊ ❈❉ ❇ ❃ ☞ ❍ ☛ ✸ ✄ ✻ ☎ ✍ ✂ ✔ ❋ ■ ☛ ■ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ❈ ❏ P ❊ ❉ P ▼ ● ◆❖ ❉ ❑ ✻ ✞ ✸ ✒ ✞ ❂ ✠ ✏ ❂ ✸ ☛ ✡ ✝ ❂ ✠ ✁ ✠ ✟ ☛ ✸ ✔ ✎ ☛ ✠ ☛ ✠ ✎ ✻ ✠ ✡ ✞ ✍ ✠ ✻ ✡ ✺ ✠ ✍ ✒ ✆✞ ☛ ✡ ✠ ✟ ✞ ☛ ❂ ☎ ✝ ✞ ☛ ✎ ✄ ✻ ✡ ✶ ✠ ✠ ✻ ✁ ✏ ✟ ☛ ☛ ✎ ✄ ✍ ☎ ☛ ✠ ✎ ✡ ✍ ✠ ✿ ✍ ✄ ✝ ✠ ✍ ✸ ✓ ✆✞ ✠ ✸ ✠ ✓ ✄ ✍ ✞ ✝ ✓ ✝ ✍ ✍ ✝ ✁ ✁ ✸ ✔ ✎ ☛ ✔ ✞ ✁ ☎ ✏ ✆ ✿ ✠ ✎ ✏ ✿ ✞ ✗ ✻ ✆ ✔ ✎ ☛ ✆ ☛ ✿ ☛ ✄ ❂ ✍ ✏ ✠ ✆ ✠ ✿ ✍ ✸ ☛ ✠ ❂ ✿ ✠ ✻ ✄ ❂ ✠ ✡ ✠ ✿ ✠ ✎ ✄ ☛ ✍ ✔ ✎ ✓ ✞ ✞ ❂ ✠ ✿ ✆✞ ☎ ✆ ✌ ✞ ✓ ✞ ☛ ✁ ✿ ☛ ✄ ☎ ✎ ✏ ✆ ✟ ✞ ✍ ✞ ✄ ✺ ✞ ✻ ✍ ✟ ✒ ✝ ✠ ✁ ✠ ✟ ☛ ✗ ✞ ☛ ✺ ✆ ☛ ✻ ✔ ✎ ✓ ✌ ✻ ✌ ✡ ✟ ☛ ✞ ✓ ✞ ☛ ✁ ✿ ☛ ✻ ☎ ☛ ✞ ✄ ✞ ✡ ✻ – p. 37/43 ❚❱❯ (framed interval) reduced ▲❆▼ i, i + 1 ❄❆❅ E, E + 1 ✏✑✸ ☎ ✑✸ ✒✝☛

  87. � ✠ ❍ ● ❋ ❊ ❈❉ ❇ ❃ ✟ ✸ ✎ ❋ ☛ ✍ ✗ ✁ ☎ ✠ ✆ ✿ ✝ ❍ ❍ ✎ ❈ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ P ■ ❊ ❉ P ▼ ● ◆❖ ❉ ❑ ❏ ✁ ✠ ✏ ✟ ✿ ✍ ☛ ✎ ✔ ✎ ✍ ✏ ✠ ✠ ✁ ✿ ✝ ✁ ✠ ✿ ✄ ✍ ✆ ✟ ✠ ☛ ✓ ✆✞ ✠ ✻ ✠ ✎ ✏ ✿ ☛ ✒ ✓ ✄ ✍ ✞ ✂ ✞ ✄ ✎ ✸ – p. 38/43 ❚❱❯ ▲❆▼ ✏ ✷✿ O ( N 3 ) ❄❆❅ ✏✑✆✞

  88. � ✏ ❂ ✡ ✠ ✝ ❂ ✄ ✻ ✍ ✿ ✠ ✆ ✠ ✻ ✿ ✿ ✠ ✎ ☎ ✻ ☛ ✡ ✓ ✟ ✠ ✻ ✎ ✠ ✍ ✞ ✝ ✚ ✻ ✿ ☛ ✆ ☛ ✎ ✔ ✆ ✗ ✓ ✠ ✍ ✄ ✄ ✿ ✌ ✸ ✎ ☛ ✞ ✟ ✆ ☛ ✄ ✻ ☛ ✓ ✍ P ● ❍ ❍ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ ❉ ❊ ❊ P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❋ ❈❉ ✎ ✁ ✠ ✄ ✓ ✌ ✎ ✄ ✆✞ ✟ ✏ ✄ ✻ ✞ ✁ ❇ ✠ ✝ ✄ ✍ ✿ ✠ ❂ ✡ ✠ ✝ ✟ ❃ ✠ ☛ ✒ ✠ ✍ ✠ ✸ ✍ ✄ ✿ ✠ ✁ ✝ ✿ ✁ ✟ ✂ ✏ ✎ ✔ ✎ ☛ ✿ ✆ ✠ ✟ ✎ ✄ ✞ ✒ ✁ ✄ ✰ ✥ ✯ ✤ ✂ � ✣ ✩ � ★ ✲ ✮ ✧ ✩ � ✂ � ★ ✪ ✮ � ✴ ✧ ✳ ✧ ✂ ✁ ✍ ✞ ✓ ✏ ✠ ✸ ✟ ☛ ✝ ✕ ✠ ✿ ✠ ✆ ✠ ✍ ☛ ✄ ✻ ☛ ✟ ✓ ☛ ✎ ☛ ✓ ✟ ✄ ✎ ✍ ✞ ✆✞ ☛ ✿ ✂ ✏ ✁ ☛ ✎ ✠ ✻ ✠ ✄ ✓ ✗ ✍ ✏ ✍ ✎ ✠ ✝ ✿ ✆ ✠ ☎ ✁ ✝ – p. 38/43 ❚❱❯ ☎✝✆ O ( N 2 ) ▲❆▼ ✏ ✷✿ O ( N 3 ) ✏✑✆✞ ❄❆❅ ✏✑✸ ✏✑✸ ✏✑✆✞

  89. � ✠ ☛ ✞ ✂ ☞ ☛ ☛ ✎ ✔ ✎ ✏ ✟ ✁ ✔ ✿ ✝ ✁ ✠ ✿ ✗ ✎ ✔ ✍ ✝ ✒ ✎ ✻ ✿ ✍ ✠ ✓ ✏ ✿ ✑ ✻ ✌ ✡ ❂ ✠ ✿ ✄ ✟ ✻ ✌ ✎ ✔ ✓ ✔ ✆ ✍ ☛ ✎ ☛ ✞ ✄ ✞ ✍ ✓ ✌ ✎ ✿ ✡ ✆ ✄ ✿ ✔ ✻ ☎ ✝ ❂ ✄ ✎ ✓ ✝ ✠ ✁ ✿ ☛ ✻ ✚ ✎ ✔ ✄ ✻ ☛ ✔ ✎ ✄ ✍ ✸ ✠ ✍ ✒ ✂ ✁ ☛ ✁ ✻ ✠ ✟ ✏ ✄ ✞ ☛ ✎ ✄ ✻ ✡ ☛ ✾ ✍ ✁ ✎ ✹ ❈❉ ❇ ❃ ✠ ✾ ✞ ✂✆ ✠ ✎ ✓ ✻ ✓ ❋ ✌ ✎ ✓ ✞ ✻ ✝ ✓ ✻ ✹ ✟ ✞ ❊ ● ✟ ❊ ▼ ❙ ❉ ▼ ■ ❅ ❘ ◗ ■ ❈ P ❉ ❍ P ▼ ● ◆❖ ❉ ❑ ❏ ■ ❍ ❋ ❍ ✎ ✄ ✁ ✠ ❂ ✓ ✠ ✆ ✿ ✏ ✟ ✞ ✎ ✠ ✞ ✿ ✠ ✻ ✠ ✎ ☛ ✞ ✂ ✞ ☛ ✟ ✻ ✠ ✞ ✆✞ ✻ ✎ ✝ ✓ ✻ ✌ ✻ ☎ ✍ ✓ ✞ ✁ ✠ ☛ ✓ ✻ ☛ ✓ ✠ ✆ ✿ ✍ ✻ ✌ ✎ ✏ ✍ ✗ ✔ ✝ ✍ ✍ ✗ ✁ ☎ ✠ ✆ ✿ ✝ ✠ ✎ ✏ ✓ ✎ ✆✞ ✠ ✻ ✠ ✎ ✏ ✿ ☛ ☛ ✓ ✄ ☛ ✠ ✞ ☛ ✍ ☛ ✠ ✓ ✞ ✝ ✄ ✟ ✓ ☛ ✎ ✟ ✸ ☛ ✻ ✄ ✍ ✏ ✠ ✆ ✠ ✿ ✕ ✟ ✍ ✞ ☛ ✰ ✧ ✳ ✧ ✴ � ✮ ✪ ★ � ✂ ✩ ✧ ✁ ✮ ✲ ★ � ✩ ✣ � ✂ ✤ ✯ ✥ ✂ � ✄ ✠ ✎ ✟ ✠ ✆ ✿ ☛ ✎ ✔ ✎ ✏ ✟ ✁ ✁ ✿ ✝ ✁ ✠ ✿ ✄ ✍ ✸ ✠ ✍ ✒ ✂ ✄ ✝ ✁ ✎ ✻ ✄ ✏ ✟ ✆✞ ✄ ✎ ✌ ✓ ✄ ✠ ✓ ✁ ✞ ☛ ✎ ✻ ✠ ✟ ✓ ✡ ✂ ✻ ☎ ✞ ✁ ✠ ✏ ✄ ✓ ✞ ☛ ✟ ✸ ✌ ✍ ✏ ✔ ✍ ✟ ✠ � ✟ ✝ ✠ ✡ ❂ ✠ ✿ ✍ ✄ ✝ ✎ ☛ ✿ ☛ ☛ ☛ ✆ ✆ ☛ ✻ ✎ ✟ ✞ ✎ ✻ ✸ ✌ ✿ ✄ ✍ ✝ ✔ ✆ ✠ ✄ ✿ ✗ ✻ ✏ ✠ ✆ ✒ ✍ ✠ ✿ ✍ ✠ ☛ ✄ ✄ ❂ ✝ ✠ ✡ ❂ ✻ ✠ ✿ ✍ ✓ – p. 38/43 ❚❱❯ O ( N ) ☎✝✆ O ( N 2 ) ▲❆▼ ✏ ✷✿ O ( N 3 ) ✏✑✆✞ ❄❆❅ BreakPoint. ✔ ✂✁ ✏✑✸ ✏✑✸ ✏✑✆✞ ☎ ✷✎ ✏✑✆✞

  90. � ✞ ✟ ❀ ✎ ✓ ✞ ✑ ✠ ✞ ✟ ☛ ✄ ✸ ✡ ✺ ☛ ✻ ❂ ✹ ✓ ✄ ✞ ✁ ✏ ✟ ✝ ✔ ✁ � ✂✆ ✞ ✾✌ ✻ ✑ ✎ ✠ ✝ ✸ ✄ ✠ ✂ ✏ ✍ ✄ ✻ ✠ ✝ ✸ ✂✆ ✞ ✾ ✓ ✎ ✆ ❉ ❋ ❍ ■ ❏ ❑ ❉ ◆❖ ● ▼ P ❊ ❍ P ❈ ■ ◗ ❘ ❅ ■ ▼ ❉ ❙ ▼ ❍ ● ✠ ✠ ✝ ✿ ✆ ✠ ☎ ✁ ✗ ✍ ☛ ✎ ✸ ❋ ✓ ✄ ✺ ✆ ✏ ✻ ✠ ❃ ❇ ❈❉ ❊ ✡ ☛ ✠ ✆ ✿ ✆ ✹ ✎ ✄ ✸ ☛ ✿ ✄ ✞ ✄ ✁ ✸ ✄ ✿ ✡ ✁ ✝ ✓ ✔ ✸ ✎ ✂ ✞ ✩ ✿ � ✴ ★ ✪ ✥ ✦ ✰ ✩ ✪ � ✣ ✩ ★ ✭ ✲ ✮ ✧ ✰ ✩ ✂ ✥ ✯ ✫ ✣ � ✝ ✠ ✆ ✂ ☛ ✻ ✠ ✓ ☛ ✻ ✄ ✁ ✆ ☛ ✔ ✁ ✁ ✄ ✡ ✠ ✟ ☛ ✞ ✍ ✞ ✓ ✏ ✞ ✆ ☎ ✁ ✗ ✍ ☛ ✎ ✠ ✸ ✺ ✔ ✔ ✟ ✓ ✎ ✞ ☛ ✍ ✠ ✿ ✠ ✞ ✠ ☛ – p. 39/43 O ( N ) . ❚❱❯ Overlap ▲❆▼ ❄❆❅ Hannenhalli O ( Nα ( N )) , Berman ✏✷✿

  91. � ☛ ✞ ✄ ✓ ✹ ❂ ✻ ✺ ✏ ✡ ✄ ✞ ☛ ✟ ✞ ✁ ✓ ✠ ✗ ✓ ✸ ✠ ✎ ☛ ✍ ✁ ✔ ☎ ✠ ✆ ✿ ✝ ✠ ✎ ✠ ✻ ✺ ✹ ✓ ✏ ✁ ✞ ✄ ✓ ❂ ✎ ✻ ☛ ✺ ✡ � ✞ ✔ ✠ ✏ ✎ ✆ ✺ ✄ ✓ ✸ ✠ ☛ ✝ ✍ ✗ ✁ ☎ ✠ ✆ ✿ ✄ ✆ ✟ ● ❏ ■ ❍ ❋ ❍ ❍ ❋ ❉ ❊ ❈❉ ❇ ❃ ✸ ✠ ❑ ◆❖ ☎ ❘ ▼ ❙ ❉ ▼ ■ ❅ ◗ ● ■ ❈ P ❊ ❉ P ▼ ✁ ✎ ✏ ✠ ✓ ☛ ✞ ✄ ✒ ✗ ☎ ☛ ✔ ✎ ✄ ✍ ✠ ✻ ✺ ✆ ✞ ✁ ☛ ✟ ✻ ✹ ✟ ✞ ✟ ✠ ✔ ✻ ✌ ✻ ✄ ✍ ✏ ☛ ✄ ✠ ✍ ✁ ✝ ✓ ✔ ✸ ✎ ✠ ✝ ✿ ✆ ✠ ☎ ✁ ✗ ☛ ✿ ✞ ✠ ✞ ✠ ✿ ✞ ✍ ✓ ✎ ✰ ✔ ✆ ✺ ✸ ✠ ✡ ✄ ✓ ✥ ✩ ✭ � ✣ ✫ ✯ ✂ ✁ ✩ ✩ ✰ ✧ ✮ ✲ ✣ ✞ ✸ ✿ ✄ ✆ ✞ ✩ ✄ ✪ ☛ ✿ ✸ ✄ ✎ ✹ ✆ � ✏ ☛ � ✄ ✠ ✻ ✄ ✍ ✏ ✂ ✁ ✸ ✸ ✝ ✠ ✎ ✑ ✻ ✝ ★ ✞ ❀ ✠ ✴ ✑ ✞ ✓ ✎ ✟ ✂✆ ✟ ✸ ✝ ✠ ✾ ✞ ✾✌ ✂✆ ✻ ✁ ✞ ☛ ✟ ✠ ✡ ✄ ☛ ✞ ✂ ✆ ✄ ✻ ☛ ✓ ✍ ☛ ☛ ✔ ✡ ✪ ✆ ✠ ✥ ✂ ✎ ✞ ✦ ☛ ✟ ✔ ✁ ✁ ★ – p. 39/43 O ( N ) . (happy cliques) ❚❱❯ Overlap ▲❆▼ ❄❆❅ Hannenhalli Tarjan O ( Nα ( N )) , Berman Kaplan, Shamir O ( N ) ✏✷✿

  92. � ✺ ✎ ✄ ✍ ✠ ✻ ✏ ✆ ✄ ☎ ✓ ✸ ✠ ✎ ☛ ✍ ✗ ✁ ✔ ✠ ✠ ✏ ✟ ✔ ✻ ✌ ✻ ✄ ✍ ✠ ✗ ✆ ☛ ✺ ✓ ☛ ✞ ✄ ✒ ☎ ✆ ✞ ✓ ✠ ✠ ✻ ✏ ✆ ✺ ✄ ✸ ✟ ✠ ✎ ☛ ✍ ✗ � ☎ ✠ ✞ ☛ ✿ ✞ ✝ ✠ ✎ ✔ ✓ ✏ ✁ ✄ ✞ ✓ ✹ ❂ ✻ ☛ ✺ ✡ ✄ ✁ ✟ ✿ ❈❉ ❍ ❋ ❍ ❍ ● ❋ ❊ ❇ ❏ ❃ ✠ ✻ ✏ ✆ ✺ ✄ ✓ ■ ❑ ✠ ◗ ▼ ❙ ❉ ▼ ■ ❅ ❘ ■ ❉ ❈ P ❊ ❉ P ▼ ● ◆❖ ✸ ✎ ✹ ✠ ✺ ✡ ✄ ✞ ☛ ✟ ✞ ✸ ✻ ✠ ✁ ☎ ✎ ✞ ☛ ✟ ✻ ☛ ❂ ☛ ✝ ✍ ✗ ✁ ☎ ✠ ✆ ✿ ✠ ✹ ✎ ✔ ✓ ✏ ✁ ✞ ✄ ✓ ✆ ✁ ✝ ✿ ☛ ✍ ✗ ✁ ☎ ✠ ✆ ✝ ✠ ✠ ✎ ✸ ✔ ✓ ✝ ✁ ✡ ✎ ✸ ✄ ✠ ✻ ✠ ✓ ✻ ☛ ✓ ✏ ✞ ✺ ✠ ✿ ✠ ✍ ✞ ✓ ✔ ✆ ✿ ✸ ✆ ✣ ✰ ✧ ✮ ✲ ★ � ✩ � ✂ ✪ ✩ ✰ ✦ ✥ ✪ ★ ✴ ✩ ✥ ✄ ✹ ✆ ✞ ✄ ✿ ☛ ✸ ✄ ✎ ✆ ✯ ✿ ✞ ✁ ✩ ✭ � ✣ ✫ ✄ ☛ ✂ ✝ ✞ ✓ ✎ ❀ ✟ ✟ ✸ ✠ ✠ ✾ ✞ ✂✆ ☛ ✝ ✠ ✻ ✑ ✞ ✍ ✓ ✎ ✔ ✓ ✏ ✁ ✞ ✄ ✹ ✟ ❂ ✻ ☛ ✺ ✡ ✄ ✞ ☛ ✄ ✸ ✏ ☛ ✎ ☛ ✟ ✔ ✁ ✁ ✞ ✞ ✂ ✍ ✞ ☛ ✟ ✠ ✡ ✄ ✁ ✂ ✔ ✠ ☛ ✄ ✁ ✸ ✝ ✠ ✎ ✑ ✻ ✾✌ ✞ ✆ ✂✆ ✡ – p. 39/43 O ( N ) . (happy cliques) ❚❱❯ Overlap ▲❆▼ ❄❆❅ Hannenhalli Tarjan O ( Nα ( N )) , Yan Berman Kaplan, Shamir Bader, Moret O ( N ) ✏✷✿ O ( N )

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