ts Prs r - - PDF document

❘❡♣❡❛ts ❛♥❞ P❛❧✐♥❞r♦♠❡s✿ ❛♥ ❖✈❡r✈✐❡✇

❙❛r❛ ❍✳ ●❡✐③❤❛❧s ❆❞✈✐s♦r✿ ❉✐♥❛ ❙♦❦♦❧ ❚❤❡ ●r❛❞✉❛t❡ ❈❡♥t❡r ✸✵ ❏✉❧② ✷✵✶✹

❆❜str❛❝t ❲✐t❤ ❛ ❧♦♥❣ t❡①t str✐♥❣ ❧✐❦❡ ❉◆❆✱ r❡♣❡❛ts ❛♥❞ ♣❛❧✐♥❞r♦♠❡s ❛r❡ ♥♦t ❡❛s✐❧② s♣♦tt❡❞✳ ❨❡t ✜♥❞✐♥❣ s✉❝❤ s✉❜str✐♥❣s ✐s ✐♠♣♦rt❛♥t❀ ❢♦r ✐♥st❛♥❝❡✱ r❡♣❡❛ts ✐♥ ❉◆❆ ❛r❡ ✐♥❞✐❝❛t♦rs ♦❢ ❝❡rt❛✐♥ ❤❡r❡❞✐t❛r② ❞✐s♦r❞❡rs ❛♥❞ ❛r❡ ✉s❡❞ ❛s ❣❡♥❡t✐❝ ♠❛r❦❡rs✳ ❲❡ ❞✐s❝✉ss r❡♣❡❛ts ❛♥❞ t❤❡♥ ♣❛❧✐♥❞r♦♠❡s ❛♥❞ t❤❡♥ ✇❡ r❡❧❛t❡ t❤❡ t✇♦✳ ■♥ ♦✉r ❞✐s❝✉ss✐♦♥ ♦❢ r❡♣❡❛ts✱ ✇❡ ✜rst ❞❡✜♥❡ ❛♥ ❡①❛❝t r❡♣❡❛t ❛♥❞ t❤❡♥ ✜✈❡ ❞❡✜♥✐t✐♦♥s ♦❢ ❛♣♣r♦①✐♠❛t❡ r❡♣❡❛ts✳ ❲❡ ♠❡♥t✐♦♥ ❛❧❣♦r✐t❤♠s t❤❛t s❡❛r❝❤ ❛ t❡①t str✐♥❣ ❢♦r s✉❜str✐♥❣s t❤❛t s❛t✐s❢② t❤❡s❡ s✐① ❞❡✜♥✐t✐♦♥s✳ ■♥ ❛❞❞✐t✐♦♥✱ ✇❡ ❝❛t❡❣♦r✐③❡ t❤❡ ✜✈❡ ❛♣♣r♦①✐♠❛t❡ r❡♣❡❛ts ✐♥ ✜✈❡ ❞✐✛❡r❡♥t ✇❛②s✳ ❲❤❡♥ ✇❡ ❧♦♦❦ ❛t ♣❛❧✐♥❞r♦♠❡s✱ ✇❡ ❧♦♦❦ ❛t ▼❛♥❛❝❤❡r✬s ❛❧❣♦r✐t❤♠ t♦ ✜♥❞ t❤❡ ❧♦♥❣❡st ❡①❛❝t ♣❛❧✐♥❞r♦♠❡ ✐♥ ❛ str✐♥❣ ❛♥❞ ❛❧s♦ ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ✜♥❞s t❤❡ ❧♦♥❣❡st ❛♣♣r♦①✐♠❛t❡ ♣❛❧✐♥❞r♦♠❡ ✐♥ ❝♦♠♣r❡ss❡❞ ❞❛t❛✳

✶

❈♦♥t❡♥ts

✶ ■♥tr♦❞✉❝t♦r② ❈♦♥❝❡♣ts ✸ ✶✳✶ ❑▼P ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❙✉✣① ❚r❡❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷✳✶ ❈♦♥str✉❝t✐♥❣ ❛♥❞ ❯s✐♥❣ ❛ ❙✉✣① ❚r❡❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷✳✷

• ❡♥❡r❛❧✐③❡❞ ❙✉✣① ❚r❡❡

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷✳✸ ❙♦❧✈✐♥❣ ▲❈P ✇✐t❤ ❛ ❙✉✣① ❚r❡❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✷ ❚❛♥❞❡♠ ❘❡♣❡❛ts ✻ ✷✳✶ ❊①❛❝t ❚❛♥❞❡♠ ❘❡♣❡❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✷✳✷ ❆♣♣r♦①✐♠❛t❡ ❚❛♥❞❡♠ ❘❡♣❡❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✸ ❘❡❧❛t❡❞ ❘❡s❡❛r❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸✳✶ ◆❡st❡❞ ❚❛♥❞❡♠ ❘❡♣❡❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸✳✷ ❆♣♣r♦①✐♠❛t❡ ❙❡❡❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✸✳✸ P❡r✐♦❞ ❘❡❝♦✈❡r② Pr♦❜❧❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✸ P❛❧✐♥❞r♦♠❡s ✶✾ ✸✳✶ ▲✐♥❡❛r✲t✐♠❡ ❆❧❣♦r✐t❤♠ ★✶ ❢♦r ✭❊①❛❝t✮ P❛❧✐♥❞r♦♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✸✳✷ ▲✐♥❡❛r✲t✐♠❡ ❆❧❣♦r✐t❤♠ ★✷ ❢♦r ✭❊①❛❝t✮ P❛❧✐♥❞r♦♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✸✳✸ ❱❛r✐❛t✐♦♥s ♦❢ P❛❧✐♥❞r♦♠❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✸✳✶ P❛❧st❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✸✳✷

• ❛♣♣❡❞ P❛❧✐♥❞r♦♠❡

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✸✳✸ ❆♣♣r♦①✐♠❛t❡ ●❛♣♣❡❞ P❛❧✐♥❞r♦♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✸✳✹ ❆♣♣r♦①✐♠❛t❡ P❛❧✐♥❞r♦♠❡s ✐♥ ❘▲❊✲❝♦♠♣r❡ss❡❞ ❉❛t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✹ ❘❡s❡❛r❝❤ ❉✐r❡❝t✐♦♥s ✷✹ ✷

✶ ■♥tr♦❞✉❝t♦r② ❈♦♥❝❡♣ts

✶✳✶ ❑▼P

❚❤❡ ♣❛tt❡r♥ ♠❛t❝❤✐♥❣ ♣r♦❜❧❡♠ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿ ▲❡t Σ ❜❡ ❛♥ ❛❧♣❤❛❜❡t✳ ■◆P❯❚✿ str✐♥❣ t❡①t✱ ♦❢ ❧❡♥❣t❤ n✿ T = t1t2 . . . tn❀ str✐♥❣ ♣❛tt❡r♥✱ ♦❢ ❧❡♥❣t❤ m✿ P = p1p2 . . . pm❀ ti, pi ∈ Σ✳ ❖❯❚P❯❚✿ ❛❧❧ ♣♦s✐t✐♦♥s i ✐♥ T✱ ✇❤❡r❡ t❤❡r❡ ✐s ❛♥ ♦❝❝✉rr❡♥❝❡ ♦❢ P✱ ✐✳❡✳✱ T[i+k] = P[k], 1 ≤ k ≤ m✳ ❆ ♥❛✐✈❡ s♦❧✉t✐♦♥ ✈✐❡✇s ❡❛❝❤ t❡①t ♣♦s✐t✐♦♥ ❛s ❛ ♣♦ss✐❜❧❡ ♣❛tt❡r♥ st❛rt✳ ❚❤❡ ❝♦♠♣❛r✐s♦♥ ♦❢ t❤❡ ❝❤❛r❛❝t❡r ❛t ❡❛❝❤ t❡①t ♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ ♣❛tt❡r♥ ❝♦sts O(nm) t✐♠❡✳ ❆ ❜❡tt❡r s♦❧✉t✐♦♥ ✐s ♣r❡s❡♥t❡❞ ❜② ❑♥✉t❤✱ ▼♦rr✐s✱ ❛♥❞ Pr❛tt ❬✶✽❪✳ ❲❡ ✇✐❧❧ r❡❢❡r t♦ ✐t ❛s ❑▼P✳ ❑▼P ❤❛s t✇♦ st❛❣❡s✳ ❚❤❡ ✜rst ✐s t❤❡ ♣r❡♣r♦❝❡ss✐♥❣ st❛❣❡✱ ✇❤✐❝❤ t❛❦❡s O(m) t✐♠❡✳ ■♥ t❤✐s st❛❣❡✱ ❛ ❑▼P ❛✉t♦♠❛t♦♥ ♦❢ P✱ s✉❝❤ ❛s t❤❡ ♦♥❡ ✐♥ ❋✐❣✉r❡ ✶ ❢♦r ❛❜❛❜❝✱ ✐s ❝♦♥str✉❝t❡❞✳ ❚❤❡r❡ ❛r❡ m + 1 ♥♦❞❡s✱ ✇✐t❤ t❤❡ m ❝❤❛r❛❝t❡rs ♦❢ P ❜❡t✇❡❡♥ t❤❡ ♥♦❞❡s✳ ❊❛❝❤ ♥♦❞❡ ❤❛s t✇♦ ❛r❝s✳ ❚❤❡ ✜rst ♠♦✈❡s ❢♦r✇❛r❞ ✭t♦ t❤❡ r✐❣❤t✮ ❛♥❞ ✐s ❝❛❧❧❡❞ ❛ s✉❝❝❡ss ❛r❝✳ ❚❤❡ ♦t❤❡r ❛r❝✱ ❦♥♦✇♥ ❛s ❛ ❢❛✐❧✉r❡ ❛r❝✱ ❧❡❛❞s t♦ t❤❡ ♥♦❞❡ ❜❡❢♦r❡ t❤❡ ❧♦♥❣❡st ❜♦r❞❡r ✭♦r ❜♦r❞❡r✮ ♦❢ t❤❡ s✉❜str✐♥❣ ✉♥t✐❧ t❤❛t ♣♦✐♥t✳ ❚❤❡ ❜♦r❞❡r ♦❢ ❳ ✐s t❤❡ ❧♦♥❣❡st s✉❜str✐♥❣ t❤❛t ✐s ❜♦t❤ ❛ ♣r♦♣❡r ♣r❡✜① ❛♥❞ ♣r♦♣❡r s✉✣① ♦❢ ❳ ✭❢♦r ❡①❛♠♣❧❡✱ t❤❡ ❜♦r❞❡r ❢♦r ❛❜❛❜ ✐s ❛❜✱ t❤❡ ❜♦r❞❡r ❢♦r ❛❜❛❜❛ ✐s ❛❜❛✱ ❛♥❞ t❤❡ ❜♦r❞❡r ❢♦r ❛❜❛❜❝ ✐s t❤❡ ❡♠♣t② str✐♥❣✮✳ ■♥ t❤❡ s❡❝♦♥❞ st❛❣❡✱ ✐♥ O(n) t✐♠❡✱ t❤❡ ❑▼P ❛❧❣♦r✐t❤♠ ❣♦❡s t❤r♦✉❣❤ t❤❡ ❝❤❛r❛❝t❡rs ♦❢ T ✐♥ ♦r❞❡r t♦ ❧♦♦❦ ❢♦r ♦❝❝✉rr❡♥❝❡s ♦❢ P✳ ■❢ ❛ ❝❤❛r❛❝t❡r ✐♥ T ♠❛t❝❤❡s t❤❡ ✜rst ❝❤❛r❛❝t❡r ♦❢ P✱ t❤❡ ❛❧❣♦r✐t❤♠ st❛rts ❛t t❤❡ ✜rst ♥♦❞❡ ♦❢ t❤❡ ❛✉t♦♠❛t♦♥✳ ❋♦r ❡❛❝❤ ❝❤❛r❛❝t❡r ♠❛t❝❤✱ t❤❡ ❛✉t♦♠❛t♦♥ ❢♦❧❧♦✇s ❛ s✉❝❝❡ss ❛r❝✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ❛ ♠✐s♠❛t❝❤✱ ❤♦✇❡✈❡r✱ t❤❡ ❛✉t♦♠❛t♦♥ ❢♦❧❧♦✇s ❛ ❢❛✐❧✉r❡ ❛r❝✳ ■❢ t❤❡ ❛❧❣♦r✐t❤♠ r❡❛❝❤❡s t❤❡ ❧❛st ♥♦❞❡✱ t❤❡♥✱ ②❡s✱ P ✐s ✐♥ T✳ ❑▼P✬s ✉s❡ ♦❢ t❤❡ ❜♦r❞❡r ✐s ✇❤❛t ❡♥❛❜❧❡s t❤❡ s❡❝♦♥❞ st❛❣❡ t♦ s❦✐♣ ♣♦s✐t✐♦♥s t❤❛t ♥❡❡❞ t♦ ❜❡ ❝❤❡❝❦❡❞✳ ❋♦r ✐♥st❛♥❝❡✱ ✉s✐♥❣ ✜❣✉r❡ ✶✱ ✐❢ T ❤❛s t❤❡ ❝❤❛r❛❝t❡rs ❛❜❛❜✱ t❤❡♥ t❤❡ ❛✉t♦♠❛t♦♥ ✐s ✉♣ t♦ ♥♦❞❡ ✹✳ ■❢ ❛ ✐s ♥❡①t s❡❡♥✱ t❤❡♥ ✐♥st❡❛❞ ♦❢ st❛rt✐♥❣ ❜❛❝❦ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ P✱ ✇❡ ❝❛♥ st❛rt ❛s ✐❢ ❛❜ ✭✇❤✐❝❤ ✐s t❤❡ ❜♦r❞❡r ♦❢ ❛❜❛❜✮ ✇❛s s❡❡♥✳ ❚❤❡ ❛❧❣♦r✐t❤♠ s❛✈❡s ♦♥ r✉♥t✐♠❡ ❜② ❛tt❡♠♣t✐♥❣ ❢✉rt❤❡r s✉❝❝❡ss❢✉❧ ♠❛t❝❤❡s st❛rt✐♥❣ ❛t t❤❡ ❜♦r❞❡r ❛♥❞ ✐t ♥❡❡❞ ♥♦t st❛rt ❜❛❝❦ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ t♦ ❛tt❡♠♣t ❢✉rt❤❡r s✉❝❝❡ss❢✉❧ ♠❛t❝❤❡s✳ ❇❡❝❛✉s❡ ♦❢ t❤❛t✱ t❤❡ s❡❝♦♥❞ st❛❣❡ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ r✉♥s ✐♥ t✐♠❡ ❧✐♥❡❛r t♦ t❤❡ ❧❡♥❣t❤ ♦❢ T✱ ✇❤✐❝❤ ✐s O(n)✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t ❢♦r ❡✈❡r② ❢❛✐❧✉r❡ ❛r❝ ❢♦❧❧♦✇❡❞ ❜❛❝❦✱ ✇❡ ♠✉st ❤❛✈❡ ✜rst ❤❛❞ ❛ s✉❝❝❡ss ❛♥❞ t❛❦❡♥ ❛ ❢♦r✇❛r❞ ❛r❝✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❛t ❛♥② ❣✐✈❡♥ ♣♦✐♥t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❢❛✐❧✉r❡s ❝❛♥♥♦t ❡①❝❡❡❞ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ s✉❝❝❡ss❡s✳ ❲✐t❤ t❤❡ t✇♦ st❛❣❡s✱ t❤❡ r✉♥t✐♠❡ ✐s O(m + n)❀ s✐♥❝❡ m ≤ n✱ t❤❡ ♦✈❡r❛❧❧ r✉♥t✐♠❡ ✐s O(n)✳ ✸

❋✐❣✉r❡ ✶✿ ❑▼P ❛✉t♦♠❛t♦♥ ❢♦r P ♦❢ ❛❜❛❜❝

✶✳✷ ❙✉✣① ❚r❡❡

❚❤❡ ✐♥❞❡①✐♥❣ ♣r♦❜❧❡♠ ✐s s✐♠✐❧❛r t♦ t❤❡ ♣❛tt❡r♥ ♠❛t❝❤✐♥❣ ♣r♦❜❧❡♠✱ ❡①❝❡♣t t❤❛t ✐t s❡❛r❝❤❡s ❢♦r ❛♥② ♥✉♠❜❡r ♦❢ ♣❛tt❡r♥ str✐♥❣s ✐♥ ♦♥❡ t❡①t str✐♥❣✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ✇❛♥t q✉❡r✐❡s t♦ ❜❡ ❛♥s✇❡r❡❞ ✐♥ t✐♠❡ ♣r♦♣♦rt✐♦♥❛❧ t♦ t❤❡ ✭s❤♦rt✮ Ps✱ r❛t❤❡r t❤❛♥ t❤❡ ✭❧♦♥❣✮ T✳ ❚♦ s♦❧✈❡ t❤❡ ✐♥❞❡①✐♥❣ ♣r♦❜❧❡♠✱ t❤❡r❡ ❛r❡ t✇♦ st❛❣❡s✳ ❚❤❡ ✜rst ✐s ♣r❡♣r♦❝❡ss✲ ✐♥❣✱ ✇❤✐❝❤ ✐s t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❡✐t❤❡r t❤❡ s✉✣① tr❡❡ ✭❙❚✮ ♦r s✉✣① ❛rr❛② ✭❙❆✮✳ ❚❤❡s❡ t✇♦ ❛r❡ ❡q✉✐✈❛❧❡♥t ❛♥❞ ❡✐t❤❡r ♦♥❡ ❝❛♥ ❡❛s✐❧② ❜❡ ❝♦♥✈❡rt❡❞ t♦ t❤❡ ♦t❤❡r✳ ❲❡ ✇✐❧❧ ❢♦❝✉s ♦♥ t❤❡ ❙❚✳ ❚❤❡ s❡❝♦♥❞ st❛❣❡ ♦❢ t❤❡ ✐♥❞❡①✐♥❣ ♣r♦❜❧❡♠ ✐s t❤❛t ♦❢ t❤❡ r✉♥♥✐♥❣ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠ ♦♥ t❤❡ ❙❚ ✭♦r ❙❆✮✱ ❡✈❡r② t✐♠❡ t❤❛t t❤❡r❡ ✐s ❛♥♦t❤❡r P t♦ ❧♦♦❦ ❢♦r✳ ✶✳✷✳✶ ❈♦♥str✉❝t✐♥❣ ❛♥❞ ❯s✐♥❣ ❛ ❙✉✣① ❚r❡❡ ❆♥ ❙❚ ✐s ❛ tr✐❡ ❝♦♥t❛✐♥✐♥❣ ❛❧❧ s✉✣①❡s ♦❢ T✳ ❯♥❧❡ss ✐t ✐s ❦♥♦✇♥ t❤❛t t❤❡ ❧❛st ❝❤❛r❛❝t❡r ♦❢ t❤❡ str✐♥❣ ✐s ✉♥✐q✉❡✱ ❛ s♣❡❝✐❛❧ ❝❤❛r❛❝t❡r ✭❢r❡q✉❡♥t❧② ✩ ♦r ★✮ ✐s ❛♣♣❡♥❞❡❞ t♦ t❤❡ str✐♥❣ ❜❡❢♦r❡ t❤❡ ❝♦♥str✉❝t✐♦♥✱ ❛s ❛ ❣✉❛r❛♥t❡❡ t❤❛t ❛ s✉✣① ♦❢ t❤❡ str✐♥❣ ❝❛♥♥♦t ❜❡ ❛ s✉❜str✐♥❣ ♦❢ ❛♥♦t❤❡r s✉✣①✳ ❆ s❛♠♣❧❡ ❙❚✱ ❢♦r T ♦❢ ①❛❜①❛❝✱ ✐s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✷ ✭t❛❦❡♥ ❢r♦♠ ❬✶✹❪✱ s❡❝t✐♦♥ ✺✳✸✮✳ ❚❤❡ ♥❛✐✈❡ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ ❝♦♥str✉❝t✐♦♥ r❡q✉✐r❡s q✉❛❞r❛t✐❝ t✐♠❡ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s✐③❡ ♦❢ t❤❡ ✐♥♣✉t✳ O(n)✲t✐♠❡ ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ❙❚✱ ✇❤❡♥ t❤❡ ❛❧♣❤❛❜❡t s✐③❡ ✐s ❝♦♥st❛♥t✱ ✇❛s ✜rst ✐♥tr♦❞✉❝❡❞ ❜② ❲❡✐♥❡r ❬✸✽❪✳ ❚❤❡ ❝♦♥str✉❝t✐♦♥ ✇❛s s✐♠♣❧✐✜❡❞ ❜② ▼❝❈r❡✐❣❤t ❬✷✽❪✳ ❲❡✐♥❡r✬s ❛❧❣♦r✐t❤♠ s❝❛♥s T ❢r♦♠ r✐❣❤t t♦ ❧❡❢t ❛♥❞ ❜❡❣✐♥s ✇✐t❤ t❤❡ s❤♦rt❡st s✉✣①✱ ✇❤✐❧❡ ▼❝❈r❡✐❣❤t✬s s❝❛♥s T ❢r♦♠ ❧❡❢t t♦ r✐❣❤t ❛♥❞ ❜❡❣✐♥s ✇✐t❤ t❤❡ ❧♦♥❣❡st s✉✣①✳ ▲❛t❡r✱ ❯❦❦♦♥❡♥ ❬✸✻❪ ♣r♦✈✐❞❡❞ ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ♣❡r❢♦r♠s ♦♥❧✐♥❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❙❚ ✐♥ ❧✐♥❡❛r t✐♠❡✳ ❋♦r ♠♦r❡ ❞❡t❛✐❧s ♦♥ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ t❤❡ ❧✐♥❡❛r✲t✐♠❡ ❙❚ ❝♦♥str✉❝t✐♦♥ ❛❧❣♦r✐t❤♠s ❛♥❞ t❤❡ ❞✐st✐♥❝t✐♦♥s ❜❡t✇❡❡♥ t❤❡♠✱ s❡❡ ❬✶✸❪✳ ❖♥❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ tr✐❝❦ t❤❛t ❤❡❧♣s ❛❝❤✐❡✈❡ ❧✐♥❡❛r t✐♠❡ ❛♥❞ s♣❛❝❡ ❜♦✉♥❞s ✐♥ ❙❚ ❝♦♥str✉❝t✐♦♥ ❛❧❣♦r✐t❤♠s ✐s t❤❡ ✉s❡ ♦❢ s✉✣① ❧✐♥❦s✱ s✐♥❝❡ t❤❡② ❛❧❧♦✇ t❤❡ ❛❧❣♦r✐t❤♠ t♦ ♠♦✈❡ q✉✐❝❦❧② t♦ ❛ ❞✐st❛♥t ♣❛rt ♦❢ t❤❡ tr❡❡✳ ❆ s✉✣① ❧✐♥❦ ✐s ❛ ♣♦✐♥t❡r ❢r♦♠ ❛♥ ✐♥t❡r♥❛❧ ♥♦❞❡ ❧❛❜❡❧❡❞ ①❙ t♦ ❛♥♦t❤❡r ✐♥t❡r♥❛❧ ♥♦❞❡ ❧❛❜❡❧❡❞ ❙✱ ✹

❋✐❣✉r❡ ✷✿ ❙❚ ❢♦r T ♦❢ ①❛❜①❛❝ ✇❤❡r❡ ① ✐s ❛♥ ❛r❜✐tr❛r② ❝❤❛r❛❝t❡r ❛♥❞ ❙ ✐s ❛ ♣♦ss✐❜❧② ❡♠♣t② s✉❜str✐♥❣✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ ❋✐❣✉r❡ ✷✱ t❤❡r❡ ✇✐❧❧ ❜❡ ❛ s✉✣① ❧✐♥❦ ❢r♦♠ ①❛ t♦ ❛❀ t❤❡r❡ ✇✐❧❧ ❛❧s♦ ❜❡ ♦♥❡ ❢r♦♠ ❛❝ t♦ ❝✳ ❲❤❡♥ t❤❡ ❛❧♣❤❛❜❡t s✐③❡ ✐s ♥♦t ❝♦♥st❛♥t✱ t❤❡ ❝♦♥str✉❝t✐♦♥ ✇❛s ❧♦♥❣ t❤♦✉❣❤t t♦ ❤❛✈❡ ❛ ❧♦✇❡r ❜♦✉♥❞ ♦❢ O(n log n)✳ ❬✶✶❪✱ ❤♦✇❡✈❡r✱ s❤♦✇s t❤❛t t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛♥ ❙❚ ❢♦r ❛♥ ✐♥t❡❣❡r ❛❧♣❤❛❜❡t ✭✇❤❡r❡ ✈❛❧✉❡s ❛r❡ ❞r❛✇♥ ❢r♦♠ ❛ r❛♥❣❡ ♦❢ ✐♥t❡❣❡rs✮ ❝❛♥ ❜❡ ♣❡r❢♦r♠❡❞ ✐♥ O(n)❀ ❜✉t ✇❤❡♥ t❤❡ ❛❧♣❤❛❜❡t ✐s ✉♥❜♦✉♥❞❡❞✱ ✐t ✐s st✐❧❧ O(n log n)✳ ❋♦❧❧♦✇✐♥❣ ❛♥ ❙❚✬s ❝♦♥str✉❝t✐♦♥✱ q✉❡r✐❡s ❛s t♦ ✇❤❡t❤❡r P ✐s ✐♥ T ✇✐❧❧ ❜❡ ❛♥s✇❡r❡❞ ✐♥ O(m) t✐♠❡✳ ❚♦ ❞♦ s♦✱ t❤❡ ❛❧❣♦r✐t❤♠ st❛rts ❛t t❤❡ r♦♦t ❛♥❞ ✉s❡s t❤❡ ❝❤❛r❛❝t❡rs ♦❢ P t♦ ❢♦❧❧♦✇ t❤❡ ♣❛t❤s ♦❢ t❤❡ ❙❚✳ ■❢ ✐t ❣❡ts st✉❝❦✱ t❤❡♥ P ✐s ♥♦t ✐♥ T✱ ❜✉t ✐❢ ✐t ❞♦❡s ♥♦t ❣❡t st✉❝❦✱ t❤❡♥ P ✐s ✐♥ T✳ ❋♦r ✐♥st❛♥❝❡✱ P ❂ ①❛❜ ✐s ✐♥ t❤❡ ❙❚ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✷✱ ✇❤✐❧❡ P ❂ ①❛❛ ✐s ♥♦t✱ ❜❡❝❛✉s❡✱ ✇❤❡♥ st❛rt✐♥❣ ❛t t❤❡ r♦♦t✱ t❤❡ ❛❧❣♦r✐t❤♠ ❢♦❧❧♦✇s t❤❡ ♣❛t❤ ♦❢ ①✱ ❛✱ ❜✉t ❣❡ts st✉❝❦ ✇✐t❤ t❤❡ ♥❡①t ❛✳ ✶✳✷✳✷

• ❡♥❡r❛❧✐③❡❞ ❙✉✣① ❚r❡❡

❆♥ ❙❚ ❝❛♥ ❜❡ ✉s❡❞ t♦ ✐♥❞❡① s❡✈❡r❛❧ t❡①ts✳ ❖♥❡ ♦♣t✐♦♥ ✐s t♦ ❝♦♥❝❛t❡♥❛t❡ t❤❡ Ts✱ s❡♣❛r❛t❡❞ ❜② ✉♥✐q✉❡ ❝❤❛r❛❝t❡rs ✲ ❢♦r ❡①❛♠♣❧❡✱ ①❛❜①❛$1❜❛❜①❜❛$2 ❢♦r ①❛❜①❛ ❛♥❞ ❜❛❜①❜❛✳ ■♥ s❡❝t✐♦♥ ✻✳✹✱ ❬✶✹❪ ✇r✐t❡s t❤❛t t❤❡ ♣r♦❜❧❡♠ ✇✐t❤ t❤❛t ♦♣t✐♦♥ ✐s t❤❛t tr❡❡ ✇✐❧❧ ❝♦♥t❛✐♥ s✉❜str✐♥❣s ✭♦❢ t❤❡ ❝♦♥❝❛t❡♥❛t❡❞ str✐♥❣✮ t❤❛t s♣❛♥ ♠♦r❡ t❤❛♥ ♦♥❡ ♦❢ t❤❡ ♦r✐❣✐♥❛❧ str✐♥❣s✳ ❋♦r ✐♥st❛♥❝❡✱ ✐♥ t❤❡ ♣r❡✈✐♦✉s❧②✲♠❡♥t✐♦♥❡❞ ❡①❛♠♣❧❡✱ t❤❡ ❙❚ ✇✐❧❧ ❝♦♥t❛✐♥ ❛$1❜❛❜①❜❛$2✱ ❡✈❡♥ t❤♦✉❣❤ t❤❛t s✉❜str✐♥❣ ✐s ♥♦t ♦❢ ✐♥t❡r❡st✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ ✐s t❤❛t ❛ ❣❡♥❡r❛❧✐③❡❞ s✉✣① tr❡❡✱ ♦r ●❙❚✱ ❝❛♥ ❜❡ ❝♦♥str✉❝t❡❞✳ ❆ s❛♠♣❧❡ ●❙❚ ✭❝♦♣✐❡❞ ❢r♦♠ ❬✶✹❪✱ s❡❝t✐♦♥ ✻✳✺✮ ✐s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✸✳ ■t ✐s ❢♦r T1 ♦❢ ①❛❜①❛✩ ❛♥❞ ❢♦r T2 ♦❢ ❜❛❜①❜❛✩✳ ❚❤❡ ♥✉♠❜❡rs ❛t t❤❡ ❧❡❛✈❡s ✐♥❞✐❝❛t❡ ✇❤✐❝❤ t❡①t ✐t ✐s✱ ❢♦❧❧♦✇❡❞ ❜② t❤❡ st❛rt✐♥❣ ♣♦s✐t✐♦♥ ♦❢ t❤❡ s✉✣① ✐♥ t❤❛t t❡①t✳ ✶✳✷✳✸ ❙♦❧✈✐♥❣ ▲❈P ✇✐t❤ ❛ ❙✉✣① ❚r❡❡ ■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ✐♥❞❡①✐♥❣ ♣r♦❜❧❡♠✱ ❙❚s ❛r❡ ✉s❡❢✉❧ ❢♦r ❛ ✈❛r✐❡t② ♦❢ str✐♥❣ ♣r♦❜✲ ❧❡♠s✳ ❚❤❡ ✜rst t❤❛t ✇❡ ✇✐❧❧ ❞❡s❝r✐❜❡ ✐s t❤❡ ❧♦♥❣❡st ❝♦♠♠♦♥ ♣r❡✜①✱ ♦r ▲❈P✳ ▲❈P ✺

❋✐❣✉r❡ ✸✿ ●❙❚ ❢♦r T1 ❂ ①❛❜①❛✩ ❛♥❞ T2 ❂ ❜❛❜①❜❛✩ ✭✈✱ ✇✮ ❞❡♥♦t❡s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❧♦♥❣❡st ❝♦♠♠♦♥ ♣r❡✜① ❜❡t✇❡❡♥ t✇♦ str✐♥❣s✱ ✈ ❛♥❞ ✇✳ ❚❤❡r❡ ✐s ❛ ♣r♦❜❧❡♠ r❡❧❛t✐♥❣ t♦ tr❡❡s✱ ❦♥♦✇♥ ❛s t❤❡ ❧♦✇❡st ❝♦♠♠♦♥ ❛♥❝❡st♦r ✭♦r ▲❈❆✮ ♣r♦❜❧❡♠✳ ❚❤❡ ▲❈❆ ♦❢ t✇♦ ♥♦❞❡s v ❛♥❞ w ✐♥ ❛♥ ❙❚ ✐s t❤❡ ❧♦✇❡st ♥♦❞❡ ✐♥ t❤❡ tr❡❡ t❤❛t ❤❛s ❜♦t❤ v ❛♥❞ w ❛s ❞❡s❝❡♥❞❛♥ts✳ ❬✶✺❪ ❞❡♠♦♥str❛t❡❞ t❤❛t ❛ tr❡❡ ✇✐t❤ n ♥♦❞❡s ❝❛♥ ❜❡ ♣r❡♣r♦❝❡ss❡❞ ✐♥ O(n) t✐♠❡❀ ❢♦❧❧♦✇✐♥❣ t❤❛t ♣r❡♣r♦❝❡ss✐♥❣✱ ✐t ❛♥s✇❡rs ▲❈❆ q✉❡r✐❡s ✐♥ ❝♦♥st❛♥t t✐♠❡✳ ❚❤❡ s♦❧✉t✐♦♥ t♦ ❛♥ ▲❈P ♣r♦❜❧❡♠ ✐s t❤❡ s❛♠❡ ❛s t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ ▲❈❆ ♣r♦❜❧❡♠ ♦♥ ❛♥ ❙❚✳ ❋♦r ❡①❛♠♣❧❡✱ ✉s✐♥❣ ❋✐❣✉r❡ ✹ ✭t❛❦❡♥ ❢r♦♠ ❬✷❪✮✱ ✇❤✐❝❤ ✐s ❛♥ ❙❚ ❢♦r ❚ ❂ ❛❜❛❜❜❛✩✿ ▲❈❆ ✭✻✱ ✾✮ ❂ ✶✵❀ ▲❈P ✭❜❛❜❜❛✩✱ ❜❜❛✩✮ ❂ ❜ ▲❈❆ ✭✶✱ ✷✮ ❂ ✸❀ ▲❈P ✭❛❜❛❜❜❛✩✱ ❛❜❜❛✩✮ ❂ ❛❜ ▲❈❆ ✭✺✱ ✾✮ ❂ r♦♦t❀ ▲❈P ✭❛✩✱ ❜❜❛✩✮ ❂ ❡♠♣t② str✐♥❣ ❆s ❞❡♠♦♥str❛t❡❞✱ ❛♥ ❙❚ s♦❧✈❡s t❤❡ ▲❈P ♣r♦❜❧❡♠ ✐♥ O(n) t✐♠❡✳ ▲❛t❡r✱ s✉❝❤ ❛s ✇❤❡♥ ✇❡ ❞✐s❝✉ss ♣❛❧✐♥❞r♦♠❡s✱ ✇❡ ✇✐❧❧ ❞❡s❝r✐❜❡ ♦t❤❡r ✉s❡s ♦❢ ❛♥ ❙❚✳

✷ ❚❛♥❞❡♠ ❘❡♣❡❛ts

❉◆❆ ♦❢t❡♥ ❝♦♥t❛✐♥s ❝♦♥s❡❝✉t✐✈❡ ❝♦♣✐❡s ♦❢ ❛ ♣❛tt❡r♥✱ ❡✳❣✳ ❝❛❣❝❛❣❝❛❣✱ ❦♥♦✇♥ ❛s ❛ t❛♥❞❡♠ r❡♣❡❛t ✭❚❘✮✳ ■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❚❘s ✐s ✉s❡❢✉❧✱ ❛s ❚❘s ❛r❡ ❧✐♥❦❡❞ t♦ ♦✈❡r t❤✐rt② ❤❡r❡❞✐t❛r② ❞✐s♦r❞❡rs ✐♥ ❤✉♠❛♥s✳ ■♥ ❛❞❞✐t✐♦♥✱ ❚❘s ❛r❡ ✉s❡❞ ❛s ❣❡♥❡t✐❝ ♠❛r❦❡rs ✐♥ ❉◆❆ ✜♥❣❡r♣r✐♥t✐♥❣✱ ♠❛♣♣✐♥❣ ❣❡♥❡s✱ ❝♦♠♣❛r❛t✐✈❡ ❣❡♥♦♠✐❝s✱ ✻

❋✐❣✉r❡ ✹✿ ❙❚ ❢♦r ❚ ♦❢ ❛❜❛❜❜❛✩ ❛♥❞ ❡✈♦❧✉t✐♦♥ st✉❞✐❡s✳ ❋✉rt❤❡r♠♦r❡✱ t❤❡② ❛r❡ ✉s❡❞ ✐♥ ♣♦♣✉❧❛t✐♦♥ st✉❞✐❡s ❛♥❞ ❝♦♥s❡r✈❛t✐♦♥ ❜✐♦❧♦❣②✳ ❉✉❡ t♦ t❤❡ s✐❣♥✐✜❝❛♥❝❡ ♦❢ ❚❘s✱ t❤❡r❡ ❤❛s ❜❡❡♥ ♠✉❝❤ ❡✛♦rt ❞❡✈♦t❡❞ t♦ ❞❡✲ s✐❣♥✐♥❣ ❛❧❣♦r✐t❤♠s ❛♥❞ s♦❢t✇❛r❡ t♦ ❧♦❝❛t❡ t❤❡♠✳

✷✳✶ ❊①❛❝t ❚❛♥❞❡♠ ❘❡♣❡❛t

❆ str✐♥❣ r = a1 . . . an ✐s ♣❡r✐♦❞✐❝ ✐❢ t❤❡r❡ ❡①✐sts s♦♠❡ p s✉❝❤ t❤❛t ai = ai+p ❢♦r ❛❧❧ i✱ ✇❤❡♥ 1 ≤ i, i + p ≤ n✳ r ✐s ♥♦t❛t❡❞ ❛s uhu′✱ t♦ ✐♥❞✐❝❛t❡ t❤❛t u ✐s ✇r✐tt❡♥ h t✐♠❡s✱ ❢♦❧❧♦✇❡❞ ❜② ❛ ✭♣♦ss✐❜❧② ♥✉❧❧✮ ♣r❡✜① u′ ♦❢ u✳ ❚❤❡r❡ ♠❛② ❜❡ ♠✉❧t✐♣❧❡ ♣❡r✐♦❞s ❢♦r ❛ str✐♥❣ r❀ ❢♦r ❛❧❧ ♦❢ t❤❡♠✱ ✇❡ s❛② t❤❛t h ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♦❝❝✉rr❡♥❝❡s ♦❢ u✱ ❛♥❞ ✇❡ r❡❢❡r t♦ ❜♦t❤ u ❛♥❞ |u| ❛s ♣❡r✐♦❞s ♦❢ r✳ ❛❜❝❛❜❝ ❛♥❞ ❛❜❝❛❜❝❛❜❝ ❤❛✈❡ t❤❡ s❛♠❡ ♣❡r✐♦❞ ♦❢ ❛❜❝❀ t❤❡ ❢♦r♠❡r ❤❛s t✇♦ ♦❝❝✉rr❡♥❝❡s✱ ✇❤✐❧❡ t❤❡ ❧❛tt❡r ❤❛s t❤r❡❡✳ ❛❜❝❛❜❝❛❜❝❛❜❝ ❤❛s t❤❛t s❛♠❡ ♣❡r✐♦❞✱ ❛❜❝✱ ✇✐t❤ ❢♦✉r ♦❝❝✉rr❡♥❝❡s✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ ♣❡r✐♦❞ ♦❢ ❛❜❝❛❜❝✱ ✇✐t❤ t✇♦ ♦❝❝✉rr❡♥❝❡s✳ r ✐s ❝❛❧❧❡❞ ❛ sq✉❛r❡ ✇❤❡♥ t❤❡ ❡①♣♦♥❡♥t h ❤❛s t❤❡ ✈❛❧✉❡ ♦❢ t✇♦✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❛ sq✉❛r❡ ✐s ♦❢ t❤❡ ❢♦r♠ uu ♦r u2 ✲ t❤❡ ♣❡r✐♦❞ ✐s r❡♣❡❛t❡❞ t✇✐❝❡✳ ❋♦r ❡①❛♠♣❧❡✱ ❛❜❝❛❜❝ ✐s ❛ sq✉❛r❡ ✭✇✐t❤ u ❛s ❛❜❝✮✱ ✇❤✐❧❡ ❛❜❝❛❜ ✐s ♥♦t✳ r ✐s ❝❛❧❧❡❞ ♣r✐♠✐t✐✈❡ ✐❢ ✐t ❝❛♥♥♦t ❜❡ ❡①♣r❡ss❡❞ ❛s uh✱ ❢♦r ❛♥② ✐♥t❡❣❡r h > 1✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❛ ♣r✐♠✐t✐✈❡ str✐♥❣ ✐s ❡✐t❤❡r ♦♥❡ t❤❛t ✐s ♥♦t ♣❡r✐♦❞✐❝ ✭s✉❝❤ ❛s ❛❜❝❞✮✱ ♦r✱ ✐❢ ✐t ✐s ♣❡r✐♦❞✐❝✱ ✐ts u′ ✐s ♥♦t ♥✉❧❧✱ ♠❡❛♥✐♥❣ t❤❛t r✬s ❧❛st ♣❡r✐♦❞ ✐s ♣❛rt✐❛❧✳ ❋♦r ❡①❛♠♣❧❡✱ ❛❜❝❞❛❜❝❞❛❜ ✐s ♣r✐♠✐t✐✈❡✱ s✐♥❝❡ ✐t ❤❛s ✷✳✺ ♦❝❝✉rr❡♥❝❡s ♦❢ p ❂ ❛❜❝❞ ✲ t❤❡r❡ ❛r❡ t✇♦ ♦❝❝✉rr❡♥❝❡s ♦❢ ❛❜❝❞✱ ❢♦❧❧♦✇❡❞ ❜② ❛ ♣❛rt✐❛❧ ♣❡r✐♦❞✳ ❚❤❡ ✐♥t❡r❡st✐♥❣ ♣❡r✐♦❞s ❛r❡ t❤♦s❡ ✇❤❡r❡ t❤❡ ♣❡r✐♦❞ ✐s ♣r✐♠✐t✐✈❡✳ ❋♦r ❛❜❝❛❜✲ ❝❛❜❝❛❜❝❛❜✱ ❢♦r ✐♥st❛♥❝❡✱ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ t❤❡ ♣❡r✐♦❞ ♦❢ s✐③❡ t❤r❡❡✱ ❛❜❝✱ ✇✐t❤ ✹ 2

3 ♦❝❝✉rr❡♥❝❡s✱ s✐♥❝❡ ❛❜❝ ✐s ♣r✐♠✐t✐✈❡✱ ❜✉t ♥♦t ✐♥ t❤❡ ♣❡r✐♦❞ ♦❢ s✐③❡ s✐①✱ ❛❜❝❛❜❝✱

✇✐t❤ ✷ 1

3 ♦❝❝✉rr❡♥❝❡s✱ ❛s ❛❜❝❛❜❝ ✐s ♥♦t ♣r✐♠✐t✐✈❡✳ ❙✐♠✐❧❛r❧②✱ an ❤❛s ❛ ♣❡r✐♦❞ ♦❢

s✐③❡ ♦♥❡ ✇✐t❤ n ♦❝❝✉rr❡♥❝❡s❀ ✐t ❛❞❞✐t✐♦♥❛❧❧② ❤❛s ❛ ♣❡r✐♦❞ ♦❢ s✐③❡ t✇♦ ✇✐t❤ n/2 ♦❝✲ ✼

❝✉rr❡♥❝❡s✳ a✱ t❤❡ ♣r✐♠✐t✐✈❡ ♣❡r✐♦❞✱ ✐s t❤❡ ♦♥❡ ♦❢ ✐♥t❡r❡st✳ ❚♦ ❜❡ ❝❧❡❛r✱ ✇❤❡♥❡✈❡r ✇❡ r❡❢❡r t♦ t❤❡ ♣❡r✐♦❞✱ ✇❡ ❛r❡ r❡❢❡rr✐♥❣ t♦ ♦♥❡ t❤❛t ✐s ♣r✐♠✐t✐✈❡✳ ❆ ♣❡r✐♦❞✐❝ str✐♥❣ ✐s ❛❧s♦ ❝❛❧❧❡❞ ❛ r❡♣❡t✐t✐♦♥ ♦r ❛♥ ❡①❛❝t t❛♥❞❡♠ r❡♣❡❛t ✭❊❚❘✮ ✲ ❛ r❡♣❡❛t✱ ❢♦r s❤♦rt✳ ❆ str✐♥❣ ♠❛② ❤❛✈❡ s✉❜str✐♥❣s ✇❤✐❝❤ ❛r❡ ♣❡r✐♦❞✐❝ ❛♥❞ s✉❜str✐♥❣s ✇❤✐❝❤ ❛r❡ ♥♦t✱ ❡✳❣✳ ❝❛❜❛❜❛❜❛❜❛❞ ✐s ♣❡r✐♦❞✐❝ ✐♥ ❛❧❧ ❝❤❛r❛❝t❡rs ❜✉t t❤❡ ✜rst ❛♥❞ ❧❛st✳ ❆ ♣r♦❜❧❡♠ ✐s t❤❛t ♦❢ s❡❛r❝❤✐♥❣ ❛ str✐♥❣ ❢♦r r❡♣❡❛ts ✲ ✐♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ❧♦♦❦ ❢♦r ♠❛①✐♠❛❧ r❡♣❡❛ts✳ ❆ r❡♣❡❛t ✐s ♠❛①✐♠❛❧ ✇❤❡♥ ✐t ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞ t♦ t❤❡ r✐❣❤t ♦r t♦ t❤❡ ❧❡❢t ❜② ❡✈❡♥ ♦♥❡ ❝❤❛r❛❝t❡r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❡♥ ❛ r❡♣❡❛t st❛rts ❛t ♣♦s✐t✐♦♥ i✱ ✐t ♠❡❛♥s t❤❛t ❡✐t❤❡r i ❂ ✶ ♦r t❤❛t r[i−1] = r[i+p−1]❀ ❢♦r ♦t❤❡r✇✐s❡✱ ✐t ✐s ♥♦t ♠❛①✐♠❛❧❧② ❡①t❡♥❞❡❞✳ ❙✐♠✐❧❛r❧②✱ ✐❢ ❛ r❡♣❡❛t ❡♥❞s ❛t ♣♦s✐t✐♦♥ j✱ t❤❡♥ ❡✐t❤❡r j ❂ n ♦r r[j + 1] = r[j + 1 − p]✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ ❝❛❛❛❛❞✱ t❤❡ r❡♣❡❛t ❛❛❛❛ ✐s ♠❛①✐♠❛❧✱ ✇❤✐❧❡ r❡♣❡❛ts ❛❛ ❛♥❞ ❛❛❛ ❛r❡ ♥♦t✳ ❚♦ ❜❡ ❝❧❡❛r✱ ✇❤❡♥ ✇❡ r❡❢❡r t♦ t❤❡ r❡♣❡❛t✱ ✇❡ ❛r❡ r❡❢❡rr✐♥❣ t♦ ♦♥❡ t❤❛t ✐s ♠❛①✐♠❛❧ ❛♥❞ ✇❤♦s❡ ♣❡r✐♦❞ ✐s ♣r✐♠✐t✐✈❡✳ ❚❤❡r❡ ❛r❡ s❡✈❡r❛❧ O(n log n) ❛❧❣♦r✐t❤♠s t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠ ♦❢ ✜♥❞✐♥❣ r❡✲ ♣❡❛ts ✐♥ ❛ str✐♥❣ ♦❢ ❧❡♥❣t❤ n✳ ❚❤✐s ✐♥❝❧✉❞❡s ❬✾❪✱ ✇❤♦ ✜♥❞s t❤❡♠ ✇✐t❤ ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ✐s ❜❛s❡❞ ♦♥ ❑▼P ❬✶✽❪✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ✉s❡s ❡q✉✐✈❛❧❡♥❝❡s✱ ✇❤✐❝❤ ❛r❡ ♣♦s✐t✐♦♥s ✐♥ t❤❡ str✐♥❣ t❤❛t ❛r❡ ❝♦♥s✐❞❡r❡❞ ❡q✉✐✈❛❧❡♥t✳ ❆♥♦t❤❡r ❛❧❣♦r✐t❤♠ ✇✐t❤ t❤✐s r✉♥✲ t✐♠❡ ✐s ❬✺❪✱ ✇❤✐❝❤ ✉s❡s ❛♥ ❙❚ t❡❝❤♥✐q✉❡ t♦ ✜♥❞ ❛❧❧ r✐❣❤t✲♠❛①✐♠❛❧ r❡♣❡❛ts ✭✇❤❡r❡ t❤❡ r❡♣❡❛ts ❛r❡ ♠❛①✐♠❛❧ ❥✉st ♦♥ t❤❡ r✐❣❤t s✐❞❡✮✳ ❋✐♥❛❧❧②✱ ▼❛✐♥ ❛♥❞ ▲♦r❡♥t③ ❬✷✹❪ ❤❛✈❡ ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ✜♥❞s r❡♣❡❛ts✳ ■t ✐s✱ ❧✐❦❡ t❤❡ ✜rst✱ ❜❛s❡❞ ♦♥ ❑▼P ❬✶✽❪✱ ❜✉t ✐t ❞✐✛❡rs ✐♥ t❤❛t ✐t ✉s❡s ❛ ❞✐✈✐❞❡ ❛♥❞ ❝♦♥q✉❡r ❛♣♣r♦❛❝❤ t♦ ✜♥❞ t❤❡ r❡♣❡❛ts t❤❛t ❝r♦ss t❤❡ ♠✐❞❞❧❡ ♣♦s✐t✐♦♥✱ ❢♦❧❧♦✇❡❞ ❜② t❤♦s❡ ♦♥ ❡✐t❤❡r s✐❞❡ ♦❢ t❤❡ ♠✐❞❞❧❡ ♣♦s✐t✐♦♥✳ ❬✷✵❪ ❤❛s t❤❡ s❛♠❡ ❣♦❛❧✱ ❜✉t ✇✐t❤ ❛ r✉♥t✐♠❡ t❤❛t ✐s s✉♣❡r✐♦r ✲ O(n)✳ ❚♦ ❞♦ s♦✱ t❤❡② ✜rst ♣r♦✈❡ t❤❛t t❤❡ s✉♠ ♦❢ t❤❡ ❡①♣♦♥❡♥ts ♦❢ ❛❧❧ t❤❡ r❡♣❡❛ts ♦❢ ❛ str✐♥❣ ♦❢ ❧❡♥❣t❤ n ✐s ❜♦✉♥❞❡❞ ❜② ❛ ❧✐♥❡❛r ❢✉♥❝t✐♦♥ ✐♥ n✳ ❚❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡r❡ ✐s ♦♥❧② ❛ ❧✐♥❡❛r ♥✉♠❜❡r ♦❢ r❡♣❡❛ts ✐♥ ❛ str✐♥❣✱ ❛❧❧♦✇✐♥❣ t❤❡♠ t♦ ❝♦♥str✉❝t ❛ ❧✐♥❡❛r✲t✐♠❡ ❛❧❣♦r✐t❤♠ ❢♦r ✜♥❞✐♥❣ t❤❡♠✳

✷✳✷ ❆♣♣r♦①✐♠❛t❡ ❚❛♥❞❡♠ ❘❡♣❡❛t

❙✐♥❝❡ ♠✉t❛t✐♦♥s✱ tr❛♥s❧♦❝❛t✐♦♥s✱ r❡✈❡rs❛❧s✱ ❛♥❞ t❤❡ ❧✐❦❡ ❛r❡ ❝♦♠♠♦♥ ✐♥ ❜✐♦❧♦❣②✱ ♠♦r❡ ✐♥t❡r❡st✐♥❣ t❤❛♥ t❤❡ ❛❜♦✈❡✲♠❡♥t✐♦♥❡❞ ❊❚❘ ✐s ❛♥ ❛♣♣r♦①✐♠❛t❡ t❛♥❞❡♠ r❡✲ ♣❡❛t✱ ♦r ❛♥ ❆❚❘✳ ❚❤✐s ✐s ✇❤❡♥ t❤❡ t❛♥❞❡♠ r❡♣❡❛ts ❛r❡ ❛♣♣r♦①✐♠❛t❡✱ ♠❡❛♥✐♥❣ t❤❛t t❤❡r❡ ❛r❡ ✈❛r✐❛t✐♦♥s ❜❡t✇❡❡♥ ♦❝❝✉rr❡♥❝❡s✳ ❚❤❡ ✈❛r✐❛t✐♦♥s ❛r❡ ♠❡❛s✉r❡❞ ✉s✲ ✐♥❣ ❞✐✛❡r❡♥t ♠❡tr✐❝s✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ ♠❡tr✐❝✱ ✇❤❡r❡ t❤❡ str✐♥❣s ❛r❡ ♦❢ t❤❡ s❛♠❡ ❧❡♥❣t❤ ❜✉t s✉❜st✐t✉t✐♦♥ ✐s ❛❧❧♦✇❡❞✱ ❛♥❞ t❤❡ ❡❞✐t ❞✐st❛♥❝❡ ✭♦t❤❡r✲ ✇✐s❡ ❦♥♦✇♥ ❛s t❤❡ ▲❡✈❡♥s❤t❡✐♥ ❞✐st❛♥❝❡✮ ♠❡tr✐❝✱ ✇❤❡r❡ s✉❜st✐t✉t✐♦♥ ❛♥❞ ✐♥❞❡❧s ✭✐♥s❡rt✐♦♥ ❛♥❞ ❞❡❧❡t✐♦♥✮ ❛r❡ ❛❧❧♦✇❡❞✳ ❲❡ ✉s❡ ♠✐s♠❛t❝❤❡s ❢♦r ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ ✈❛r✐❛t✐♦♥s ❛♥❞ ❡rr♦rs ❢♦r ❡❞✐t ❞✐st❛♥❝❡ ✈❛r✐❛t✐♦♥s✳ ❆♥ ❛♣♣r♦①✐♠❛t❡ sq✉❛r❡ ✐s ♦❢ t❤❡ ❢♦r♠ uv✱ ✇❤❡r❡ u ❛♥❞ v✱ ✇❤❡♥ ✉s✐♥❣ ❛ ❝❡rt❛✐♥ ♠❡tr✐❝✱ ❞✐✛❡r ❜② ♥♦ ♠♦r❡ t❤❛♥ ❛ ❝❡rt❛✐♥ t❤r❡s❤♦❧❞✳ ❚❤❡ t❤r❡s❤♦❧❞ ♠❛② ❜❡ s❡t ✐♥ r❡❧❛t✐♦♥ t♦ t❤❡ str✐♥❣✬s ❧❡♥❣t❤ ✭s✉❝❤ ❛s ❛ ♣❡r❝❡♥t❛❣❡ ♦❢ t❤❡ ❧❡♥❣t❤✮✱ ♦r ✐t ♠❛② ❜❡ ❛ s❡t ✈❛❧✉❡ t❤❛t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ str✐♥❣✬s ❧❡♥❣t❤✳ ❲❡ ✇✐❧❧ ✉s❡ ✽

t❤❡ s❡t ✈❛❧✉❡✱ ❛♥❞ ✇❡ ✇✐❧❧ ✐♥❞✐❝❛t❡ ✐t ❛s k✱ ♠❡❛♥✐♥❣ t❤❛t ✉♣ t♦ k ♠✐s♠❛t❝❤❡s ♦r ❡rr♦rs ❛r❡ ❛❧❧♦✇❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ ❛❜❝❛❜❞ ✐s ❛♥ ❛♣♣r♦①✐♠❛t❡ sq✉❛r❡✱ ✉♥❞❡r t❤❡ ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ ♠❡tr✐❝ ❛♥❞ ✇❤❡♥ k ≥ 1 ✭✇❤❡♥ ❛t ❧❡❛st ♦♥❡ ♠✐s♠❛t❝❤ ✐s ❛❧❧♦✇❡❞✮✱ ❛♥❞ ❛❜❝❛❜❞❞ ✐s ❛♥ ❛♣♣r♦①✐♠❛t❡ sq✉❛r❡✱ ✉♥❞❡r t❤❡ ❡❞✐t ❞✐st❛♥❝❡ ♠❡tr✐❝ ❛♥❞ ✇❤❡♥ k ≥ 2 ✭✇❤❡♥ ❛t ❧❡❛st t✇♦ ❡rr♦rs ❛r❡ ❛❧❧♦✇❡❞✮✳ ❆ ❝✉rr❡♥t ♣r♦❜❧❡♠ ✐s t❤❛t ♦❢ s❡❛r❝❤✐♥❣✱ ✐♥ ❛ str✐♥❣✱ ❢♦r s✉❜str✐♥❣s t❤❛t ❛r❡ ❛♣♣r♦①✐♠❛t❡ r❡♣❡❛ts✳ ❲❡ ✇✐❧❧ ♣r♦✈✐❞❡ ✜✈❡ ❞❡✜♥✐t✐♦♥s t❤❛t ❛r❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ♣r♦❜❧❡♠✳ ❙✐♥❝❡ ✇❡ ✜♥❞ t❤❛t t❤r❡❡ ♦❢ t❤❡♠ ❛r❡ ❝♦♥❝❡♣t✉❛❧❧② s✐♠✐❧❛r✱ ✇❤✐❧❡ t❤❡ ♦t❤❡r t✇♦ ❛r❡ ❝♦♥❝❡♣t✉❛❧❧② s✐♠✐❧❛r✱ ✇❡ ✇✐❧❧ ✜rst ❧✐st t❤❡ ❞❡✜♥✐t✐♦♥s✱ ✐♥ t❤❡✐r ❣r♦✉♣s✿

• r♦✉♣ ✶✿ ♥❡✐❣❤❜♦r✐♥❣ ✲

k✲r❡♣❡t✐t✐♦♥ k✲r✉♥ k✲❡❞✐t r❡♣❡❛t

• r♦✉♣ ✷✿ ❝♦♥s❡♥s✉s ✲

❝♦♥s❡♥s✉s ♠♦❞❡❧ k✲♠❛①✐♠❛❧ ❛♣♣r♦①✐♠❛t❡ r✉♥ ✭♦r k✲▼❆❘✮ ❲❡ ✇✐❧❧ ♥♦✇ ♣r♦❝❡❡❞ ✇✐t❤ t❤❡ ❞❡✜♥✐t✐♦♥s✱ ❛❧♦♥❣ ✇✐t❤ ❡①❛♠♣❧❡s ❛♥❞ ❜❛s✐❝ ✐♥❢♦r♠❛t✐♦♥✳

• r♦✉♣ ✶✿ ♥❡✐❣❤❜♦r✐♥❣

❋♦r t❤♦s❡ ✐♥ t❤✐s ❣r♦✉♣✱ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ✐s s✐♠✐❧❛r t♦ ❛❞❥❛❝❡♥t ✭♦r ♥❡✐❣❤❜♦r✲ ✐♥❣✮ ♦❝❝✉rr❡♥❝❡s✳ ❉❡✜♥✐t✐♦♥ ❬✷✶❪ ❆ str✐♥❣ r[1 . . . n] ✐s ❝❛❧❧❡❞ ❛ ❦✲r❡♣❡t✐t✐♦♥ ♦❢ ♣❡r✐♦❞ p✱ p ≤ n/2✱ ✐✛ t❤❡ ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ ♦❢ r[1 . . . n − p] ❛♥❞ r[p + 1 . . . n] ≤ k✳ ❋♦r ❡①❛♠♣❧❡✱ ❛t❛❛ ❛tt❛ ❝tt❛ ❝t ✐s ❛ t✇♦✲r❡♣❡t✐t✐♦♥ ♦❢ ♣❡r✐♦❞ ❢♦✉r✱ ❛s ✐t ❤❛s ❛ t♦t❛❧ ♦❢ t✇♦ ♠✐s♠❛t❝❤❡s✱ ✇❤❡♥ ❝♦♠♣❛r✐♥❣ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ t♦ t❤❡ ♥❡①t ♦❝❝✉r✲ r❡♥❝❡✳ ✭■t ✐s ♥♦t ❛ ♦♥❡✲r❡♣❡t✐t✐♦♥ ♦❢ ♣❡r✐♦❞ ❢♦✉r✱ ❡✈❡♥ t❤♦✉❣❤ t❤❡r❡ ✐s ❥✉st ♦♥❡ ♠✐s♠❛t❝❤ ❢r♦♠ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ t♦ t❤❡ ♥❡①t✱ s✐♥❝❡ k✲r❡♣❡t✐t✐♦♥s ❧♦♦❦ ❛t t❤❡ t♦t❛❧ ♥✉♠❜❡r✳✮ ❆❧❧ k✲r❡♣❡t✐t✐♦♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ O(nk log k + S) t✐♠❡✱ ✇❤❡r❡ n ✐s t❤❡ s✐③❡ ♦❢ t❤❡ ✐♥♣✉t ❛♥❞ S ✐s t❤❡ ♦✉t♣✉t s✐③❡✱ ❛♥❞ O(n) s♣❛❝❡ ❬✷✶❪✳ ❉❡✜♥✐t✐♦♥ ❬✷✶❪ ❆ str✐♥❣ r[1 . . . n] ✐s ❝❛❧❧❡❞ ❛ ❦✲r✉♥✱ ♦❢ ♣❡r✐♦❞ p✱ p ≤ n/2✱ ✐✛ ❢♦r ❡✈❡r② i ∈ [1 . . . n − 2p + 1]✱ t❤❡ ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ ♦❢ r[i . . . i + p − 1] ❛♥❞ r[i + p . . . i + 2p − 1] ≤ k✳ ❋♦r ❡①❛♠♣❧❡✱ ❛t❛❛ ❛tt❛ ❝tt❛ ❣❣t❛ ❣❣ ✐s ❛ ♦♥❡✲r✉♥✱ ♦❢ ♣❡r✐♦❞ ❢♦✉r✳ ❚❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♠✐s♠❛t❝❤❡s ✐s t❤r❡❡✳ ◆♦t❡ t❤❛t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♠✐s♠❛t❝❤❡s ✐♥ ❛ k✲r✉♥ ✐s ✉♥❜♦✉♥❞❡❞✳ k✲r✉♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ s❛♠❡ t✐♠❡ ❛♥❞ s♣❛❝❡ ❜♦✉♥❞s ❛s k✲r❡♣❡t✐t✐♦♥s✿ O(nk log k + S) t✐♠❡✱ ✇❤❡r❡ n ✐s t❤❡ s✐③❡ ♦❢ t❤❡ ✐♥♣✉t ❛♥❞ S ✐s t❤❡ ♦✉t♣✉t s✐③❡✱ ❛♥❞ O(n) s♣❛❝❡ ❬✷✶❪✳ ❬✶✾❪ ❞✐s❝✉ss❡s ♠r❡♣s✱ ✇❤✐❝❤ ✐s t❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✱ ❛❧♦♥❣ ✇✐t❤ s♦♠❡ ❤❡✉r✐st✐❝s t❤❛t ✐t ❛❞❞s t♦ t❤❡ ❛❧❣♦r✐t❤♠✳ ❖♥❡ ✾

s✉❝❤ ❤❡✉r✐st✐❝ r❡❧❛t❡s t♦ ✇❤❡♥ t❤❡ s❛♠❡ r❡❣✐♦♥ ✐s r❡♣r❡s❡♥t❡❞ ❛s s❡✈❡r❛❧ ❞✐st✐♥❝t k✲r✉♥s✱ ✇✐t❤ ❞✐✛❡r❡♥t ♣❡r✐♦❞ s✐③❡s ✲ ✐❢ s♦✱ t❤❡② ❛r❡ ♠❡r❣❡❞✳ ❋♦r ❡①❛♠♣❧❡✱ ❢♦r k = 1✱ ❛t❛t❛t❛t❛t❛❛❛ ✐s ❛ ♦♥❡✲r✉♥ ♦❢ ♣❡r✐♦❞ t✇♦ ✭❛t ❛t ❛t ❛t ❛t ❛❛ ❛✮✱ ♦❢ ♣❡r✐♦❞ ❢♦✉r ✭❛t❛t ❛t❛t ❛t❛❛ ❛✮✱ ❛♥❞ ♦❢ ♣❡r✐♦❞ s✐① ✭❛t❛t❛t ❛t❛t❛❛ ❛✮✳ ❚❤❡r❡❢♦r❡✱ ❢♦r t❤❡ t❤r❡❡ ♣❡r✐♦❞ s✐③❡s✱ t❤❡ ❡rr♦r❘❛t❡ ✐s ❝♦♠♣✉t❡❞✿ errorRate = errorNumber

length−p

✱ ✇❤❡r❡ ❡rr♦r◆✉♠❜❡r ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♠✐s♠❛t❝❤❡s✱ ❡①❝❡♣t t❤❛t t✇♦ ♠✐s♠❛t❝❤❡s ❝♦✉♥t ❛s ♦♥❡ ✐❢ t❤❡ ♠✐s♠❛t❝❤ ✐s t❤❡ s❛♠❡ ❝❤❛r❛❝t❡r ✐♥ t❤❡ s❛♠❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡✐r ♣❡✲ r✐♦❞s ✭s✉❝❤ ❛s y ✐♥ ♣♦s✐t✐♦♥ i✱ x ✐♥ ♣♦s✐t✐♦♥ i + p✱ ❛♥❞ y ✐♥ ♣♦s✐t✐♦♥ i + 2p✮✱ ❛♥❞ ❧❡♥❣t❤ ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ r❡♣❡❛t✳ ■♥ t❤✐s ❝❛s❡✱ ❡rr♦r❘❛t❡ ❢♦r t❤❡ ♣❡r✐♦❞ ♦❢ s✐③❡ t✇♦ ✐s

1 13−2 = 1 11 = 0.09❀ s✐③❡ ❢♦✉r ✐s 1 13−4 = 1 9 = 0.11❀ ❛♥❞ s✐③❡ s✐① ✐s 1 13−6 = 1 7 = 0.14✳ ❙✐♥❝❡ ❡rr♦r❘❛t❡ ❢♦r t❤❡ ♣❡r✐♦❞ ♦❢ s✐③❡ t✇♦ ✐s ❧♦✇❡st✱ ✐t ✐s t❤❡

✏❜❡st✑ r❡♣r❡s❡♥t❛t✐✈❡ ❛♥❞ t❤❡ ♦t❤❡r t✇♦ ❛r❡ ♠❡r❣❡❞ ✐♥t♦ ✐t✳ ❆♥♦t❤❡r ❤❡✉r✐st✐❝ ✐s ✜❧t❡r✐♥❣ ♦✉t st❛t✐st✐❝❛❧❧② ✐♥s✐❣♥✐✜❝❛♥t r❡♣❡❛ts✳ ❋♦r ✐♥st❛♥❝❡✱ ❡✈❡♥ t❤♦✉❣❤ ❛❛❛ ✐s ♠✐s♠❛t❝❤✲❢r❡❡✱ ✐t ✐s st❛t✐st✐❝❛❧❧② ✐♥s✐❣♥✐✜❝❛♥t ❛s ✐ts ❧❡♥❣t❤ ✐s s♦ s♠❛❧❧✳ ❯♥❞❡r t❤❡ ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ ♠❡tr✐❝✱ k✲r✉♥s ♣r♦✈✐❞❡ t❤❡ ✇❡❛❦❡st ✏r❡❛s♦♥✲ ❛❜❧❡✑ ❞❡✜♥✐t✐♦♥✱ s✐♥❝❡ ✐t r❡q✉✐r❡s t❤❛t ❡✈❡r② ♦❝❝✉rr❡♥❝❡ ✇✐t❤ ✐ts ♥❡✐❣❤❜♦r ❝❛♥♥♦t ❝♦♥t❛✐♥ ♠♦r❡ t❤❛♥ k ♠✐s♠❛t❝❤❡s✱ ✇❤✐❝❤ s❡❡♠s t♦ ❜❡ ❛ ♠✐♥✐♠❛❧ r❡❛s♦♥❛❜❧❡ r❡✲ q✉✐r❡♠❡♥t✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ k✲r❡♣❡t✐t✐♦♥s ✐s t❤❡ str♦♥❣❡st s✉❝❤ ♥♦t✐♦♥✱ ❛s ✐t ❧✐♠✐ts ❜② k t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♠✐s♠❛t❝❤❡s✳ ❚❤✐s ❥✉st✐✜❡s ✇❤② ♦❢t❡♥t✐♠❡s ❛ s❡❛r❝❤ ✇✐❧❧ ❧♦♦❦ ❛t ❜♦t❤ k✲r❡♣❡t✐t✐♦♥s ❛♥❞ k✲r✉♥s ✲ s✐♥❝❡ t❤❡② ❛r❡ ❡①tr❡♠❡s ❛t ♦♣♣♦s✐t❡ ❡♥❞s✱ ❜♦t❤ ♦❢ t❤❡♠ ❛r❡ ♥❡❡❞❡❞ ✐♥ ♦r❞❡r t♦ ✏❡♠❜r❛❝❡✑ ✇❤❛t ✐s ✐♥t❡r♠❡✲ ❞✐❛t❡✳ ❉❡✜♥✐t✐♦♥ ❬✸✹❪ ❬✸✺❪ ❆ str✐♥❣ r ✐s ❝❛❧❧❡❞ ❛ ❦✲❡❞✐t r❡♣❡❛t ✐❢ ✐t ❝❛♥ ❜❡ ♣❛rt✐t✐♦♥❡❞ ✐♥t♦ ❝♦♥s❡❝✉t✐✈❡ s✉❜str✐♥❣s✱ r = u′′u1u2...ulu′, l ≥ 2✱ s✉❝❤ t❤❛t t❤❡ s✉♠ ♦❢ ❡❞✐t ❞✐st❛♥❝❡ ❡rr♦rs ♦❢ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ✇✐t❤ ✐ts ♥❡✐❣❤❜♦r ❞♦❡s ♥♦t ❡①❝❡❡❞ k✳ ❋♦r ❡①❛♠♣❧❡✱ ❝❛❛❣❝t ❝❝❛❣❝t ❝❝❣❝ ✐s ❛ t✇♦✲❡❞✐t r❡♣❡❛t✱ ❛s t❤❡r❡ ❛ s✉❜st✐t✉t✐♦♥ ✭♦❢ a ✇✐t❤ c✮ ❛♥❞ ❛ ❞❡❧❡t✐♦♥ ✭♦❢ a✮❀ ccgc ✐s ❛ ♣❛rt✐❛❧ ♣❡r✐♦❞✱ s♦ ✐t ✐s ♥♦t ❝♦✉♥t❡❞ ❛s ❛ ❞❡❧❡t✐♦♥ ♦❢ t✳ ✭❲✐t❤ r❡♣❡❛ts t❤❛t ✐♥✈♦❧✈❡ ❡❞✐t ❞✐st❛♥❝❡✱ ❛ ♣❡r✐♦❞ s✐③❡ ✐s ♥♦t ✜①❡❞✳ ❆s s✉❝❤✱ ✐t ✐s ♥♦t ❣✐✈❡♥✳✮ ◆♦t❡ t❤❛t ❛ k✲❡❞✐t r❡♣❡❛t ✐s t❤❡ s❛♠❡ ❛s ❛ k✲r❡♣❡t✐t✐♦♥✱ ❡①❝❡♣t t❤❛t t❤❡ ❢♦r♠❡r ✉s❡s ❡❞✐t ❞✐st❛♥❝❡ ❛♥❞ t❤❡ ❧❛tt❡r ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡✳ k✲❡❞✐t r❡♣❡❛ts ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ O(n4) t✐♠❡ ❛♥❞ O(n2) s♣❛❝❡✱ ✇✐t❤ t❤❡ ♥❛✐✈❡ ❛❧❣♦r✐t❤♠✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ t❤❡ ♥❛✐✈❡ ❛❧❣♦r✐t❤♠ ✉s❡s ❛ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❡❞✐t ❞✐st❛♥❝❡ ♠❛tr✐①✱ ♦❢ O(n2) t✐♠❡ ❛♥❞ s♣❛❝❡✱ ❢♦r ❡❛❝❤ ♣❛✐r ♦❢ ♣♦s✐t✐♦♥s (i, j)✳ i ✐s t❤❡ st❛rt ♦❢ ❛ r❡♣❡❛t✱ ❛♥❞ ❛❧❧ n ♣♦s✐t✐♦♥s ❛r❡ ❝♦♥s✐❞❡r❡❞✳ j ✐s t❤❡ st❛rt ♦❢ t❤❡ ♣r♦♣❡r s✉✣① ♦❢ t❤❡ r❡♣❡❛t✱ ❛♥❞ ❝❡rt❛✐♥ ♣♦s✐t✐♦♥s t♦ t❤❡ r✐❣❤t ♦❢ i ❛r❡ ❝♦♥s✐❞❡r❡❞✳ ❚❤❡r❡ ❛r❡ O(n2) ♣❛✐rs✳ ❬✸✹❪ ❣✐✈❡s ❛♥ ❛❧❣♦r✐t❤♠ t♦ ✜♥❞ k✲❡❞✐t r❡♣❡❛ts ✐♥ O(nk log k log(n/k) + S)✱ ✇❤❡r❡ n ✐s t❤❡ s✐③❡ ♦❢ t❤❡ ✐♥♣✉t ❛♥❞ S ✐s t❤❡ ♦✉t♣✉t s✐③❡✱ ❛♥❞ O(n + k2) s♣❛❝❡✱ ❜❡❝❛✉s❡ ✐t ✐♥❝❧✉❞❡s t❤r❡❡ s♣❡❡❞✉♣s✳ ❚❤❡ ✜rst s♣❡❡❞✉♣ r❡❞✉❝❡s t❤❡ ♥✉♠❜❡r ♦❢ ✐t❡r❛t✐♦♥s ✐♥ t❤❡ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❡❞✐t ❞✐st❛♥❝❡ ♠❛tr✐①✱ ❜② ✉s✐♥❣ t❤❡ ✐❞❡❛s ♦❢ ▼❛✐♥ ❛♥❞ ▲♦r❡♥t③ ❬✷✹❪✿ t❤❡ ❛❧❣♦r✐t❤♠ ❝♦♥s✐❞❡rs ❛❧❧ ♣♦ss✐❜❧❡ st❛rts ❢r♦♠ t❤❡ ♠✐❞❞❧❡ ♣♦s✐t✐♦♥✱ ❛♥❞ t❤❡♥ r❡❝✉rs✐✈❡❧② ♦♥ t❤❡ ✜rst ❤❛❧❢ ❛♥❞ s❡❝♦♥❞ ❤❛❧❢ ♦❢ t❤❡ ✶✵

str✐♥❣✳ ❚❤❡♥✱ ❛s ❛ s❡❝♦♥❞ s♣❡❡❞✉♣✱ t❤❡ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❡❞✐t ❞✐st❛♥❝❡ ♠❛tr✐①✬s s✐③❡ ✐s r❡❞✉❝❡❞✱ ✉s✐♥❣ ✐❞❡❛s ❢r♦♠ ❬✷✸❪✳ ❚❤❡ t❤✐r❞ s♣❡❡❞✉♣ r❡❞✉❝❡s t❤❡ ❝♦♠♣✉t❛t✐♦♥ t✐♠❡ ♦❢ t❤❡ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ❡❞✐t ❞✐st❛♥❝❡ ♠❛tr✐① ❜② ✉s✐♥❣ ✐♥❢♦r♠❛t✐♦♥ ❢r♦♠ ❛ ❞✐✛❡r❡♥t ♣❡r✐♦❞ s✐③❡✳ ❬✸✺❪ r❡❞✉❝❡s t❤❡ r✉♥t✐♠❡ t♦ O(nk log2 k+S)✱ ❜② ✉s✐♥❣ t❤❡ ❢❛❝t♦rs ♦❢ ❛ ▲❡♠♣❡❧✲ ❩✐✈ ❢❛❝t♦r✐③❛t✐♦♥ ✭✇❤✐❝❤ ✐s ❛❧❣♦r✐t❤♠ ✉s❡❞ ❢♦r ❞❛t❛ ❝♦♠♣r❡ss✐♦♥✮✱ ♣❧✉s s♣❡❡❞✉♣s r❡❧❛t✐♥❣ t♦ t❤❡ ❡❞✐t ❞✐st❛♥❝❡✳

• r♦✉♣ ✷✿ ❝♦♥s❡♥s✉s

❚❤♦s❡ ✐♥ t❤✐s ❣r♦✉♣ ✉s❡ ❛ ❝♦♥s❡♥s✉s str✐♥❣✱ ✇❤✐❝❤ ✐s ❛ str✐♥❣ t♦ ✇❤✐❝❤ ❡❛❝❤ ♦❝❝✉rr❡♥❝❡ ✐s s✐♠✐❧❛r✳ ❉❡✜♥✐t✐♦♥ ❬✸✶❪ ❊❛❝❤ ♦❝❝✉rr❡♥❝❡ ✐s ❝❛❧❧❡❞ ❛ ✇❛❣♦♥✱ ❛♥❞ t❤❡ s❡q✉❡♥❝❡ ♦❢ ✇❛❣♦♥s ✐s ❛ tr❛✐♥✳ ❆ ♣r❡✜① ♠♦❞❡❧ ✐s ✇❤❡♥ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❡❞✐t ❞✐st❛♥❝❡ ❡rr♦rs✱ ❢r♦♠ ❡❛❝❤ ✇❛❣♦♥ t♦ t❤❡ ❝♦♥s❡♥s✉s str✐♥❣✱ ❞♦❡s ♥♦t ❡①❝❡❡❞ k✳ ❏✉♠♣s ♦❢ ❛ ❝❡rt❛✐♥ s✐③❡ ✭❛s ✇✐❧❧ ❜❡ ❡①♣❧❛✐♥❡❞ ❧❛t❡r✮ ❛r❡ ❛❧❧♦✇❡❞✳ ❆ ❝♦♥s❡♥s✉s ♠♦❞❡❧ ✐s ❛ ♣r❡✜① ♠♦❞❡❧✱ ✇✐t❤ ❛♥ ❛❞❞✐t✐♦♥❛❧ ❛❧❧♦✇❛♥❝❡ ✲ t❤❛t ❣❛♣s ♦❢ ❛ ❝❡rt❛✐♥ s✐③❡ ✭❡①♣❧❛✐♥❡❞ ❧❛t❡r✮ ❛r❡ ❛❧❧♦✇❡❞✳ ❚❤✐s t②♣❡ ♦❢ ❆❚❘ ✐s ✜rst ❞✐s❝✉ss❡❞ ❜② ❙❛❣♦t ❛♥❞ ▼②❡rs ❬✸✶❪✱ ✇❤❡r❡ t❤❡② ♣r❡s❡♥t ❛♥ ❛❧❣♦r✐t❤♠ t♦ ✜♥❞ tr❛✐♥s ✐♥ s❛t❡❧❧✐t❡ ❉◆❆ ✭t❤✐s ✐s ❛ t②♣❡ ♦❢ ❉◆❆ t❤❛t ✐s ❧♦❝❛t❡❞ ♥❡❛r t❤❡ r❡❣✐♦♥ ♦❢ t❤❡ ❝❡♥tr♦♠❡r❡ ❛♥❞ ✐s ✐♥✈♦❧✈❡❞ ✐♥ ❝❤r♦♠♦s♦♠❡ s❡❣r❡❣❛t✐♦♥ ❞✉r✐♥❣ ♠✐t♦s✐s ❛♥❞ ♠❡✐♦s✐s✮✳ ■◆P❯❚✿ ⋄ r✱ t❤❡ str✐♥❣ r❡♣r❡s❡♥t✐♥❣ t❤❡ s❛t❡❧❧✐t❡ ❉◆❆✱ ✇❤✐❝❤ ✐s ♦❢ ❧❡♥❣t❤ ♥ ⋄ ♠✐♥❴r❡♣❡❛t✱ ✇❤✐❝❤ ✐s t❤❡ ♠✐♥✐♠✉♠ ♥✉♠❜❡r ♦❢ ✇❛❣♦♥s ✐♥ ❛ tr❛✐♥ ⋄ ♠✐♥❴r❛♥❣❡ ❛♥❞ ♠❛①❴r❛♥❣❡✱ ✇❤✐❝❤ ❞❡✜♥❡ t❤❡ ♠✐♥✐♠✉♠ ❛♥❞ ♠❛①✐♠✉♠ ❧❡♥❣t❤s ♦❢ ❛ ✇❛❣♦♥ ⋄ ♠❛①❴❥✉♠♣ ✲ ❡①♣❧❛✐♥❡❞ ❜❡❧♦✇ ⋄ ❦✱ t❤❡ ❛❧❧♦✇❡❞ ♥✉♠❜❡r ♦❢ ❡❞✐t ❞✐st❛♥❝❡ ❡rr♦rs ❜❡t✇❡❡♥ ❛ ✇❛❣♦♥ ❛♥❞ t❤❡ ❝♦♥s❡♥s✉s str✐♥❣ ❖❯❚P❯❚✿ ❛♥② ♥✉♠❜❡r ♦❢ tr❛✐♥s✱ ✇❤❡r❡ ❛❧❧ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞ tr✉❡✿ ⋄ t❤❡r❡ ❛r❡ ❛t ❧❡❛st ♠✐♥❴r❡♣❡❛t ✇❛❣♦♥s ✐♥ ❛ tr❛✐♥✱ ❛♥❞ ❡❛❝❤ ✇❛❣♦♥✬s ❧❡♥❣t❤ ✐s ❜❡t✇❡❡♥ ♠✐♥❴r❛♥❣❡ ❛♥❞ ♠❛①❴r❛♥❣❡ ⋄ t❤❡r❡ ❛r❡ ❛t ♠♦st ✶✺✲✷✵✪ ❡❞✐t ❞✐st❛♥❝❡ ❡rr♦rs ❢r♦♠ ❡❛❝❤ ✇❛❣♦♥ t♦ t❤❡ ❝♦♥s❡♥s✉s str✐♥❣ ✭♠❡❛♥✐♥❣ t❤❛t ❡❛❝❤ ✇❛❣♦♥ ♠❛② ❞✐✛❡r ❢r♦♠ ❛♥② ♦t❤❡r ✇❛❣♦♥ ❜② ✸✵✲✹✵✪✮ ✲ ♥♦t❡ t❤❛t ✐t ✐s ♣❡r♠✐tt❡❞ ❢♦r t❤❡ ❝♦♥s❡♥s✉s str✐♥❣ t♦ ♥♦t ❜❡ ♣r❡s❡♥t ✐♥ t❤❡ str✐♥❣ ✭✐♥ ♦t❤❡r ✇♦r❞s✱ ❡✈❡r② ✇❛❣♦♥ ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ❝♦♥s❡♥s✉s str✐♥❣✮ ⋄ t❤❡ ❏❯▼P ♣r♦♣❡rt②✿ t❤❡ ❧❡♥❣t❤ ❢r♦♠ t❤❡ ❧❡❢t ♦❢ ♦♥❡ ✇❛❣♦♥ t♦ t❤❡ ❧❡❢t ♦❢ t❤❡ ♥❡①t ✇❛❣♦♥ ✐s [♠✐♥❴r❛♥❣❡✱ ♠❛①❴r❛♥❣❡ ✰ ♠❛①❴❥✉♠♣] ✶✶

❚❛❜❧❡ ✶✿ ●❆P ❛♥❞ ❏❯▼P ♣r♦♣❡rt✐❡s ❣ ❛ ❝ ❣ ❛ ❝ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ❣ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ❛ ✵ ✵ ✷ ✶ ✵ ✷ ✶ ❝ ✵ ✵ ✶ ✸ ✷ ✶ ✸ ❣ ✵ ✶ ✵ ✷ ✹ ✸ ✷ ❛ ✵ ✵ ✷ ✶ ✸ ✺ ✹ ❝ ✵ ✵ ✶ ✸ ✷ ✹ ✻ ❣ t ❝ ❛ ❣ t ✵ ✵ ✵ ✵ ✵ ✵ ✵ ❣ ✵ ✶ ✵ ✵ ✵ ✶ ✵ t ✵ ✵ ✷ ✵ ✵ ✵ ✷ ❝ ✵ ✵ ✵ ✸ ✶ ✵ ✵ ❛ ✵ ✵ ✵ ✶ ✹ ✷ ✵ ❣ ✵ ✶ ✵ ✵ ✷ ✺ ✸ t ✵ ✵ ✷ ✵ ✵ ✸ ✻ ❚❛❜❧❡ ✷✿ ❚❤❡ ❧♦❝❛❧ ❛❧✐❣♥♠❡♥ts ♦❢ str✐♥❣s ❣❛❝❣❛❝ ✭❊❚❘✮ ❛♥❞ ❣t❝❛❣t ✭♥♦t ❊❚❘✮ ⋄ t❤❡ ●❆P ♣r♦♣❡rt②✿ t❤❡ ❧❡♥❣t❤ ❢r♦♠ t❤❡ r✐❣❤t ♦❢ ♦♥❡ ✇❛❣♦♥ t♦ t❤❡ ❧❡❢t ♦❢ t❤❡ ♥❡①t ✇❛❣♦♥ ✐s [✵✱ ♠❛①❴❥✉♠♣] ✭♥♦t❡ t❤❛t ✇❤❡♥ ✐t ✐s ③❡r♦✱ t❤❡ ✇❛❣♦♥s ❛r❡ ❝♦♥s❡❝✉t✐✈❡✮ ❚♦ ✐❧❧✉str❛t❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❏❯▼P ❛♥❞ ●❆P ♣r♦♣❡rt✐❡s✱ s❡❡ t❛❜❧❡ ✶✱ ✇❤❡r❡ ♠✐♥❴r❛♥❣❡ ✐s s❡t t♦ ✷✱ ♠❛①❴r❛♥❣❡ t♦ ✹✱ ❛♥❞ ♠❛①❴❥✉♠♣ t♦ ✷✳ r ❝♦♥t❛✐♥s t✇♦ ❝♦♥s❡❝✉t✐✈❡ ✇❛❣♦♥s ✈ ❛♥❞ ✇✳ ✯ ✐♥❞✐❝❛t❡s ❛♥② ♥✉♠❜❡r ♦❢ ❝❤❛r❛❝t❡rs✳ ✰ ✐♥❞✐❝❛t❡s ❛♥② ♦♥❡ ❝❤❛r❛❝t❡r✱ ❛s ❧♦♥❣ ❛s ✐t ✐s ♥♦t ✈❀ s♦ s❤♦✇♥ t❤❡r❡ ❛r❡ ❡①❛❝t❧② t✇♦ ♦❢ ✈ ❛♥❞ t✇♦ ♦r ♠♦r❡ ♦❢ ✇✳ ❇♦t❤ ✈ ❛♥❞ ✇ ❛r❡ ♦❢ ✈❛❧✐❞ ❧❡♥❣t❤s✱ ❛s t❤❡✐r ❧❡♥❣t❤s ❛r❡ ❜❡t✇❡❡♥ ♠✐♥❴r❛♥❣❡ ❛♥❞ ♠❛①❴r❛♥❣❡✳ ❚❤❡ ❥✉♠♣ ✐s t♦ ❜❡ ✐♥ t❤❡ r❛♥❣❡ [♠✐♥❴r❛♥❣❡✱ ♠❛①❴r❛♥❣❡ ✰ ♠❛①❴❥✉♠♣] ❂ [✷✱ ✹ ✰ ✷] ❂ [✷✱ ✻]❀ ❛t ✺✱ ✐t ✐s✳ ❚❤❡ ❣❛♣ ✐s t♦ ❜❡ [✵✱ ♠❛①❴❥✉♠♣] ❂ [✵✱ ✷]❀ ❛t ✸✱ ✐t ✐s ♥♦t✳ ❙✐♥❝❡ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❤❛✈❡ ❥✉st ♦♥❡ ♦❢ t❤❡ t✇♦ ❢✉❧✜❧❧❡❞✱ ❏❯▼P ❛♥❞ ●❆P ❛r❡ ♥♦t r❡❞✉♥❞❛♥t✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ❝♦♥s✐sts ♦❢ t✇♦ st❛❣❡s✱ ✜❧t❡r✐♥❣ ❛♥❞ ❜r✉t❡ ❢♦r❝❡ s❡❛r❝❤✳ ❚❤❡ ✜rst st❛❣❡ ✐s ♦♣t✐♦♥❛❧✱ ❜✉t ✐t ✐s r❡❝♦♠♠❡♥❞❡❞ ❛s ✐t ✇✐❧❧ s❛✈❡ t✐♠❡ ✐♥ t❤❡ s❡❝♦♥❞ st❛❣❡✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡ ✜rst st❛❣❡ ✐♥✈♦❧✈❡s ❤❡✉r✐st✐❝s ❛♥❞ ✐s ♥♦t ❡①❛❝t✱ s♦ ✐t ✐s ♣♦ss✐❜❧❡ ✭t❤♦✉❣❤ ✉♥❧✐❦❡❧②✮ t❤❛t ❛ ❚❘ ❣♦t ✜❧t❡r❡❞ ♦✉t✳ ❚❤❡ ✉♥❞❡r❧②✐♥❣ ✐❞❡❛ ❜❡❤✐♥❞ t❤❡ ✜rst st❛❣❡ ✐s t❤❛t ❛ ❤✐❣❤ s✉♠ ♦❢ t❤❡ ✈❛❧✉❡s ✐♥ t❤❡ ✉♣♣❡r r✐❣❤t r❡❣✐♦♥ ♦❢ t❤❡ ❧♦❝❛❧ ❛❧✐❣♥♠❡♥t ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ ♠❛tr✐① ♦❢ r ✐s ✐♥❞✐❝❛t✐✈❡ ♦❢ ✇❤✐❝❤ s✉❜str✐♥❣s ❛r❡ ♠♦r❡ ❧✐❦❡❧② t♦ ❝♦♥t❛✐♥ r❡♣❡❛ts✳ ❙❡❡ t❛❜❧❡ ✷ ❢♦r t❤❡ ❡♥t✐r❡ ♠❛tr✐❝❡s ♦❢ str✐♥❣s ❣❛❝❣❛❝ ✭❊❚❘✮ ❛♥❞ ❣t❝❛❣t ✭♥♦t ❊❚❘✮✱ ❝❛❧❝✉❧❛t❡❞ ✉s✐♥❣ t❤❡ ❙♠✐t❤✲❲❛t❡r♠❛♥ ❧♦❝❛❧ ❛❧✐❣♥♠❡♥t ❛❧❣♦r✐t❤♠ ❬✸✸❪ ✇✐t❤ ♣❛r❛♠❡t❡rs ♦❢ ♠❛t❝❤ ❂ ✰✶ ❛♥❞ ♠✐s♠❛t❝❤ ❂ ❣❛♣ ❂ ✲✶✳ ❚❤❛t r❡❣✐♦♥ ✭t❤❡ ✉♥❞❡r❧✐♥❡❞ ♣♦s✐t✐♦♥s✮ s✉♠s t♦ ✷✵ ❛♥❞ ✾✱ r❡s♣❡❝t✐✈❡❧②✱ ✐♥ t❤❡ ❧❡❢t ❛♥❞ r✐❣❤t ♠❛tr✐❝❡s✳ ❚❤❡ ❞❡t❛✐❧s ♦❢ t❤✐s st❛❣❡ ❛r❡ ❛s ❢♦❧❧♦✇s✿ ❛❢t❡r s❝❛♥♥✐♥❣ ri✱ t❤❡ r❡❣✐♦♥ ♦❢ t❤❡ ✶✷

♠❛tr✐① t❤❛t ✐s ❝♦♠♣✉t❡❞ ✐s t❤❡ tr✐❛♥❣❧❡ ♦❢ (i − band, i − band) t♦ (i − band, i) t♦ (i, i)✱ ✇❤❡r❡ t❤❡ ♣❛r❛♠❡t❡r ❜❛♥❞ ✐s ❝❤♦s❡♥ s♦ t❤❛t ✐t ✐s ♦❢ s✐③❡ ❛t ❧❡❛st max(3.5 · max❴range, 50)✳ ✭❚❤❡ r❡❛s♦♥ ❢♦r t❤❡ s✐③❡ ✐s t❤❛t ❡✈❡♥ ✐❢ t❤❡ ♣❡r✐♦❞ ✐s ❛s ❜✐❣ ❛s ✐t ❝❛♥ ❜❡ ✲ ❛s ❜✐❣ ❛s ♠❛①❴r❛♥❣❡ ✲ t❤❡♥ t❤❡r❡ ✐s ❡♥♦✉❣❤ ❞❛t❛ t♦ ❜❡ ❛❜❧❡ t♦ ✐♥❞✐❝❛t❡ r❡♣❡t✐t✐✈❡ s✉❜str✐♥❣s❀ ✐❢ t❤❡ ♣❡r✐♦❞ ✐s s♠❛❧❧✱ t❤❡♥ ♠♦r❡ ❞❛t❛✱ ✇✐t❤♦✉t ❜❡✐♥❣ ❡①❝❡ss✐✈❡✱ ✐s ❧♦♦❦❡❞ ❛t✳✮ ❇❡❝❛✉s❡ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞ ❞✉r✐♥❣ ♦♥❡ s❝❛♥ ♦❢ r ✭❧❡♥❣t❤ n✮✱ ❛♥❞ ✐t ❝♦♠♣✉t❡s ❛♥ O(max❴range) ❜❛♥❞ ❛r♦✉♥❞ t❤❡ ❞✐❛❣♦♥❛❧✱ t❤✐s st❛❣❡ ❤❛s ❛ r✉♥t✐♠❡ ♦❢ O(n · max❴range)✳ ❚❤❡ s❡❝♦♥❞ st❛❣❡ ✐s ❛ ❜r✉t❡ ❢♦r❝❡ s❡❛r❝❤ ♦❢ t❤❡ s✉❜str✐♥❣s t❤❛t ✇❡r❡ ♥♦t ✜❧t❡r❡❞ ♦✉t✳ ■t ✉s❡s ❛ s❡t Lm✱ ✇❤✐❝❤ st♦r❡s t❤❡ ❧❡❢t✲❡♥❞ ♣♦s✐t✐♦♥s ♦❢ ✇❛❣♦♥s t❤❛t ❛r❡ ❦♥♦✇♥ t♦ ❜❡ ✈❛❧✐❞ ❢♦r ❝♦♥s❡♥s✉s str✐♥❣ m✱ ❛♥❞ ❛❧s♦ ❛ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣ t❛❜❧❡ Dm✱ ✇❤✐❝❤ st♦r❡s ❡❞✐t ❞✐st❛♥❝❡ ✈❛❧✉❡s ❜❡t✇❡❡♥ ✇❛❣♦♥s ✇✐t❤ t❤❡✐r ❝♦♥s❡♥s✉s str✐♥❣ m✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ♣❡r❢♦r♠❡❞✿ ✐♥ O(nk)✱ t❤❡ ❡❞✐t ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❛ ✇❛❣♦♥ ❛♥❞ ❛ ❝♦♥s❡♥s✉s str✐♥❣ ✐s ❝♦♠♣✉t❡❞✱ ✐♥ ♦r❞❡r t♦ s❡❡ ✐❢ t❤❡ ✇❛❣♦♥ ✐s ✇✐t❤✐♥ k ❡rr♦rs ♦❢ t❤❡ ❝♦♥s❡♥s✉s str✐♥❣❀ ✐❢ s♦✱ t❤❡ ✈❛❧✉❡ ✐s st♦r❡❞✳ ❚❤❡♥ ❝❤❡❝❦ ✐❢ t❤❡r❡ ❛r❡ ❛t ❧❡❛st ♠✐♥❴r❡♣❡❛t ♦r ♠♦r❡ ✇❛❣♦♥s✱ ✉s✐♥❣ ♠❛①❴❥✉♠♣ ✐♥ ♦r❞❡r t♦ ❦♥♦✇ ✇❤❛t ✇♦✉❧❞ ❜❡ t❤❡ ❧❡❢t✲❡♥❞ ♣♦s✐t✐♦♥ ♦❢ t❤❡ ♥❡①t ✇❛❣♦♥✳ ❚❤✐s t❛❦❡s O(n · max❴jump) t✐♠❡✳ ■❢ ❛❧❧ ✐s ✇❡❧❧✱ ✐t ✐s ❛ ♣r❡✜① ♠♦❞❡❧✱ ❛♥❞ s♦ ❢❛r ✐t✬s ❜❡❡♥ O(n·(k+max❴jump))✳ ◆♦✇ t❤❡ ❛❧❣♦r✐t❤♠ ❛tt❡♠♣ts t♦ ✜♥❞ ❛♥♦t❤❡r ✈❛❧✐❞ ❝♦♥s❡♥s✉s str✐♥❣ ❜② t❛❦✐♥❣ t❤✐s ♣r❡✈✐♦✉s❧② ❣♦♦❞ ❝♦♥s❡♥s✉s str✐♥❣ m ❛♥❞ ❡✐t❤❡r ✐♥s❡rt✐♥❣ t♦ t❤❡ ❧❡❢t✱ s✉❜✲ st✐t✉t✐♥❣ ♦♥ t❤❡ ❧❡❢t✱ ♦r ❞❡❧❡t✐♥❣ ♦♥ t❤❡ ❧❡❢t✳ ❚❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡s ✐t ❝❛♥ ❞♦ s♦ ✐s ❜♦✉♥❞❡❞ ❜② k ✭✐t ♠❛② ♥♦t ❡①❝❡❡❞ k ❡rr♦rs✮ ❛♥❞ ❜② ♠❛①❴r❛♥❣❡ ✭✐❢ ✐t ✐♥s❡rts ♦♥ t❤❡ ❧❡❢t✱ ✐t ♠❛② ♥♦t ✐♥s❡rt s✉❝❤ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝♦♥s❡♥s✉s str✐♥❣ ❡①❝❡❡❞s ♠❛①❴r❛♥❣❡✮✳ ❚❤✉s✱ ❜② ❞❡✜♥✐t✐♦♥✱ t❤❡ ♥✉♠❜❡r ♦❢ ♣r❡✜① ♠♦❞❡❧s t❤❛t ❝♦✉❧❞ ♠❛t❝❤ ❛ ❣✐✈❡♥ ♣♦s✐t✐♦♥ ✇✐t❤ k ♦r ❢❡✇❡r ❡rr♦rs ✐s max❴range

m=1

N(k, m)✱ ✇❤❡r❡ N(k, m) ✐s t❤❡ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ str✐♥❣s t❤❛t ❛r❡ ♥♦t ♠♦r❡ t❤❛♥ k ❡rr♦rs ❢r♦♠ ❛♥♦t❤❡r str✐♥❣ ♦❢ ❧❡♥❣t❤ m✳ ❯s✐♥❣ ❬✸✷❪✱ ✐t ✐s s❤♦✇♥ t❤❛t N(k, m) ✐s ✉♣♣❡r✲❜♦✉♥❞❡❞ ❜② (m·⑤⑤)k✳ ■♥ t❤✐s ❝❛s❡✱ N(k, m) ✐s N(k, max❴range)✱ ❛♥❞ ✐t✬s ❞♦♥❡ ♠❛①❴r❛♥❣❡ t✐♠❡s✳ ❚❤❛t ❜r✐♥❣s t❤❡ r✉♥t✐♠❡ t♦ O(n · max❴range · N(k, max❴range))✳ ❋✐♥❛❧❧②✱ t❤❡ ❛❧❣♦r✐t❤♠ ✈❡r✐✜❡s ✐❢ ❡❛❝❤ ♣r❡✜① ♠♦❞❡❧ ✐s ❛❧s♦ ❛ ❝♦♥s❡♥s✉s ♠♦❞❡❧✱ ❜② ❝❤❡❝❦✐♥❣ ✐❢ t❤❡ ●❆P ♣r♦♣❡rt② ✐s s❛t✐s✜❡❞✳ ❙✐♥❝❡ t❤❡ ❣❛♣ s✐③❡ ✐s s✉r❡❧② s♠❛❧❧❡r t❤❛♥ ♠❛①❴❥✉♠♣✱ t❤❡♥ t❤❡ r✉♥t✐♠❡ ✇✐❧❧ ❜❡ ♣✉t ✐♥ t❡r♠s ♦❢ ♠❛①❴❥✉♠♣✱ ❛♥❞ t❤❡ r✉♥t✐♠❡ ♦❢ t❤✐s ♣❛rt ✐s O(n · (k + max❴jump))✳ ❚❤✉s✱ t❤❡ t♦t❛❧ t✐♠❡ t❛❦❡♥ ❜② t❤❡ s❡❝♦♥❞ st❛❣❡ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✱ ❛♥❞ ✉❧t✐✲ ♠❛t❡❧② ✐s t❤❡ r✉♥t✐♠❡ ♦❢ ❜♦t❤ st❛❣❡s✱ ✐s ❜♦✉♥❞❡❞ ❜② O(n · (k + max❴jump + k + max❴jump) · max❴range · N(k, max❴range)) ❂ ✭❜② ❝♦♥s♦❧✐❞❛t✐♥❣ ❧✐❦❡ t❡r♠s✮ O(n · (k + max❴jump) · max❴range · N(k, max❴range))✳ ❇❡❝❛✉s❡ k < ♠❛①❴r❛♥❣❡✱ t❤❛t✬s O(n · max❴range2 · max❴jump · N(k, max❴range))✳ ❚❤❡ s♣❛❝❡ r❡q✉✐r❡♠❡♥t ✐s t❤❛t ♦❢ ❦❡❡♣✐♥❣ ❛❧❧ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ❝♦♥❝❡r♥✐♥❣ ❛t ♠♦st ♠❛①❴r❛♥❣❡ ❝♦♥s❡♥s✉s str✐♥❣s ❛t ❛ t✐♠❡ ✭m ❛♥❞ ❛❧❧ ✐ts ♣r❡✜①❡s✮✳ ■t ✐s t❤❡r❡❢♦r❡ ♦♥ t❤❡ ♦r❞❡r ♦❢ ❛t ♠♦st O(n · max❴range · k) ❛s ♦♥❧② O(nk) st♦r❛❣❡ ✐s r❡q✉✐r❡❞ t♦ r❡❝♦r❞ t❤❡ ❧❡❢t✲❡♥❞ ♣♦s✐t✐♦♥s ❛♥❞ ❡❞✐t ❞✐st❛♥❝❡ ✈❛❧✉❡s ❛t ❡❛❝❤ ♣♦ss✐❜❧❡ r✐❣❤t ❡♥❞✳ ❉❡✜♥✐t✐♦♥ ❬✹❪ ❆ ❦✲♠❛①✐♠❛❧ ❛♣♣r♦①✐♠❛t❡ r✉♥ ✭♦r ❦✲▼❆❘✮ ✐s ❛ ♥♦♥✲❡♠♣t② str✐♥❣ r✱ s✉❝❤ t❤❛t t❤❡ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ ❛t ♠♦st k ❝❤❛r❛❝t❡rs ✐♥ r ❣❡♥❡r❛t❡s ❛ ♠❛①✐♠❛❧ ✶✸

✭❡①❛❝t✮ r✉♥✳ ❆ ♠♦❞✐✜❝❛t✐♦♥ ✐s ❛ s✉❜st✐t✉t✐♦♥ ♦❢ ❛ s✐♥❣❧❡ ❝❤❛r❛❝t❡r ✇✐t❤ ❛ ❞✐✛❡r❡♥t s✐♥❣❧❡ ❝❤❛r❛❝t❡r✳ ❋♦r ❡①❛♠♣❧❡✱ ❛❜❛❛❜❝❛❜❛ ✐s ❛ ♦♥❡✲♠❛①✐♠❛❧ ❛♣♣r♦①✐♠❛t❡ r✉♥ ✇✐t❤ ❛ ♣❡r✐♦❞ ♦❢ ❧❡♥❣t❤ t❤r❡❡ ❛♥❞ t❤r❡❡ ♦❝❝✉rr❡♥❝❡s✳ ◆♦t❡ t❤❡ ❞✐st✐♥❝t✐♦♥ ❜❡t✇❡❡♥ ❛ ♠✐s♠❛t❝❤ ❛♥❞ ❛ ♠♦❞✐✜❡❞ ❝❤❛r❛❝t❡r✳ ■❢ r[i] = r[i + p]✱ t❤❡♥ t❤❡r❡ ✐s ❛ ♠✐s♠❛t❝❤ ❜❡t✇❡❡♥ ♣♦s✐t✐♦♥s i ❛♥❞ i + p ✐♥ r✳ ❆ ♠♦❞✐✜❡❞ ❝❤❛r❛❝t❡r ✐s ✇❤❡♥ ❛ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ ❛ ❝❤❛r❛❝t❡r ❝♦♥✈❡rts t❤❡ ❛♣♣r♦①✐♠❛t❡ r✉♥ ✐♥t♦ ❛ ♠❛①✐♠❛❧ r✉♥✳ ❋♦r ✐♥st❛♥❝❡✱ ❢♦r t❤r❡❡ ♦❝❝✉rr❡♥❝❡s ❛♥❞ ♣❡r✐♦❞ ♦❢ s✐③❡ t❤r❡❡✱ ❛❜❝❛❜❝❛❜❞ ❤❛s ♦♥❡ ♠✐s♠❛t❝❤ ✭❜❡t✇❡❡♥ t❤❡ t✇♦ ✉♥❞❡r❧✐♥❡❞ ❝❤❛r❛❝t❡rs✮ ❛♥❞ t❤❡r❡ ✐s ♦♥❡ ♠♦❞✐✜❡❞ ❝❤❛r❛❝t❡r ✭t❤❡ s❡❝♦♥❞ ✉♥❞❡r❧✐♥❡❞ ❝❤❛r❛❝t❡r✮✳ ❆♥♦t❤❡r ✐♥st❛♥❝❡✿ ✐♥ ❛❜❛❛❜❝❛❜❛✱ t❤❡r❡ ❛r❡ t❤r❡❡ ♦❝❝✉rr❡♥❝❡s ❛♥❞ ❛ ♣❡r✐♦❞ ♦❢ s✐③❡ t❤r❡❡✱ ✇✐t❤ t✇♦ ♠✐s♠❛t❝❤❡s ✭❜❡t✇❡❡♥ t❤❡ ✜rst t✇♦ ✉♥❞❡r❧✐♥❡❞ ❝❤❛r❛❝t❡rs ❛♥❞ ❜❡t✇❡❡♥ t❤❡ ❧❛st t✇♦ ✉♥❞❡r❧✐♥❡❞ ❝❤❛r❛❝t❡rs✮❀ ❜✉t ♦♥❧② ♦♥❡ ♠♦❞✐✜❡❞ ❝❤❛r❛❝t❡r ✭t❤❡ s❡❝♦♥❞ ✉♥❞❡r✲ ❧✐♥❡❞ ❝❤❛r❛❝t❡r✮ ✇✐❧❧ ❝♦♥✈❡rt r ✐♥t♦ ❛ ♠❛①✐♠❛❧ r✉♥✳ ■♥ ❛❜❝❛❜❝❛❜❞❛❜❞❛❜❞✱ t❤❡r❡ ❛r❡ ✜✈❡ ♦❝❝✉rr❡♥❝❡s ❛♥❞ ❛ ♣❡r✐♦❞ ♦❢ s✐③❡ t❤r❡❡✳ ❚❤❡r❡ ✐s ♦♥❡ ♠✐s♠❛t❝❤ ✭❜❡t✇❡❡♥ t❤❡ ❧❛st t✇♦ ✉♥❞❡r❧✐♥❡❞ ❝❤❛r❛❝t❡rs✮✱ ✇✐t❤ t✇♦ ♠♦❞✐✜❡❞ ❝❤❛r❛❝t❡rs ✭t❤❡ ✜rst t✇♦ ✉♥❞❡r❧✐♥❡❞ ❝❤❛r❛❝t❡rs✮✳ ❬✹❪ ♣r♦✈✐❞❡ t❤r❡❡ ❛❧❣♦r✐t❤♠s t♦ ✜♥❞ ❛❧❧ k✲▼❆❘ s✉❜str✐♥❣s ✐♥ ❛ str✐♥❣✳ ■◆P❯❚✿ r✱ ❛ str✐♥❣ ♦❢ ❧❡♥❣t❤ ♥❀ ❛♥❞ k✱ t❤❡ ❛❧❧♦✇❡❞ ♥✉♠❜❡r ♦❢ ♠♦❞✐✜❝❛t✐♦♥s✳ ❖❯❚P❯❚✿ ❛❧❧ s✉❜str✐♥❣s ♦❢ r s✉❝❤ t❤❛t t❤❡ ♠♦❞✐✜❝❛t✐♦♥ ♦❢ ❛t ♠♦st k ❝❤❛r❛❝t❡rs ❣❡♥❡r❛t❡s ❛ ♠❛①✐♠❛❧ r✉♥✳ ❚❤❡ ✜rst ❛❧❣♦r✐t❤♠ ❤❛s ❛ r✉♥t✐♠❡ ♦❢ O(n2)✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ✐t r✉♥s ♦♥❡ ✐t❡r❛t✐♦♥ ❢♦r ❡❛❝❤ ♦❢ t❤❡ p ≤ n

2 ♣❡r✐♦❞ s✐③❡s✱ ❛♥❞ ❡❛❝❤ ✐t❡r❛t✐♦♥ t❛❦❡s O(n)✳

❋♦r ❡❛❝❤ ✐t❡r❛t✐♦♥✿ t❤❡r❡ ❛r❡ ♣♦✐♥t❡rs t♦ t❤❡ ❧❡❢t♠♦st ❛♥❞ r✐❣❤t♠♦st ❝❤❛r✲ ❛❝t❡rs ♦❢ t❤❡ k✲▼❆❘✱ ❦♥♦✇♥ ❛s pl ❛♥❞ pr✱ ❛♥❞ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❡❛❝❤ ✐t❡r❛✲ t✐♦♥ t❤❡② ❜♦t❤ ♣♦✐♥t t♦ t❤❡ ✜rst ❝❤❛r❛❝t❡r ♦❢ r✳ ❆❧s♦✱ ❛ ❞❛t❛ str✉❝t✉r❡ ❝❛❧❧❡❞ t❤❡ P❛r✐❦❤ ♠❛tr✐① ✐s ❝r❡❛t❡❞ ❛t t❤❡ st❛rt ♦❢ ❡❛❝❤ ✐t❡r❛t✐♦♥ ✭❛♥❞ ✐t ✇✐❧❧ ❜❡ ❞❡✲ str♦②❡❞ ❛t ✐ts ❡♥❞✮✳ ❙❛♠♣❧❡ P❛r✐❦❤ ♠❛tr✐❝❡s✱ ❢♦r ♣❡r✐♦❞ ❧❡♥❣t❤s ♦❢ ✸ ❛♥❞ ✺✱ ✇❤❡♥ r = abcaccaccaab✱ ❛r❡ s❤♦✇♥ ✐♥ ✜❣✉r❡ ✺✳ ❚❤❡ ❞❛t❛ str✉❝t✉r❡ ✐s ♦❢ s✐③❡ p ✇✐❞❡ ❛♥❞ ⑤Σ⑤ ❤✐❣❤ ✭✇❤❡r❡ ⑤Σ⑤ ✐s t❤❡ s✐③❡ ♦❢ t❤❡ ❛❧♣❤❛❜❡t✮✱ ❛♥❞ ✐t ✐s s✐♠♣❧② ❛ ❝♦✉♥t ♦❢ t❤❡ ❝❤❛r❛❝t❡rs ✐♥ ❛ ❝♦❧✉♠♥✳ ✭❆ ❝♦❧✉♠♥ c ✐s ❛❧❧ t❤❡ ❝❤❛r❛❝t❡rs ❧♦❝❛t❡❞ ❛t ♣♦s✐t✐♦♥s i✱ ✇❤❡r❡ i mod p = c✳ ❋♦r ✐♥st❛♥❝❡✱ ✇❤❡♥ p ❂ ✸✱ ❝♦❧✉♠♥ ✵ ❤❛s a ❢♦✉r t✐♠❡s ✲ abcaccaccaab❀ ❛♥❞ ✇❤❡♥ p ❂ ✺✱ ❝♦❧✉♠♥ ✵ ❤❛s ✷ a✬s ❛♥❞ ✶ c ✲ abcaccaccaab✳✮ ❋♦r t❤✐s ❛❧❣♦r✐t❤♠✱ t❤❡ P❛r✐❦❤ ♠❛tr✐①✬s ❝♦✉♥ts ❛r❡ ✉s❡❞ ✐♥ ♦r❞❡r t♦ ❦♥♦✇ t❤❡ ♥✉♠❜❡r ♦❢ ♠♦❞✐✜❡❞ ❝❤❛r❛❝t❡rs t❤❛t ❛ str✐♥❣ ✭♦r s✉❜str✐♥❣✮ ♥❡❡❞s ✐♥ ♦r❞❡r ❢♦r t❤❡ str✐♥❣ ✭♦r s✉❜str✐♥❣✮ t♦ ❜❡ ❛♥ ❡①❛❝t r✉♥✳ ❋♦❧❧♦✇✐♥❣ t❤❡ ❝r❡❛t✐♦♥ ♦❢ t❤❡ P❛r✐❦❤ ♠❛tr✐① ❛♥❞ t❤❡ s❡t✉♣ ♦❢ t❤❡ t✇♦ ♣♦✐♥t❡rs✱ ❛♥ ✐t❡r❛t✐♦♥ r✉♥s ❛s ❢♦❧❧♦✇s✿ pr ♠♦✈❡s t♦ t❤❡ r✐❣❤t ♦♥❡ ❝❤❛r❛❝t❡r ❛t ❛ t✐♠❡✱ ❛♥❞ t❤❡ P❛r✐❦❤ ♠❛tr✐① ✐♥❝r❡♠❡♥ts t❤❡ ❝♦✉♥t ♦❢ t❤❡ ❝❤❛r❛❝t❡r ♣♦✐♥t❡❞ t♦ ❜② pr✳ ❯s✐♥❣ t❤❡ ❝♦✉♥ts✱ t❤❡ P❛r✐❦❤ ♠❛tr✐① ❦♥♦✇s ✇❤✐❝❤ ❝❤❛r❛❝t❡rs ♥❡❡❞ t♦ ❜❡ ♠♦❞✐✜❡❞ ✐♥ ♦r❞❡r t♦ ❜❡ ♣❛rt ♦❢ ❛♥ ❡①❛❝t r✉♥❀ ❛♥❞ ♦♥❝❡ k ♦❢ t❤♦s❡ ❛r❡ ❢♦✉♥❞✱ t❤❡ ❛❧❣♦r✐t❤♠ ♦✉t♣✉ts t❤❡ s✉❜str✐♥❣ t❤❛t ✐s ❛ k✲▼❆❘✳ ❚♦ ✜♥❞ ❛ s✉❜s❡q✉❡♥t k✲▼❆❘✱ pl ♠♦✈❡s t♦ t❤❡ r✐❣❤t ✉♥t✐❧ t❤❡ s✉❜str✐♥❣ ❜❡t✇❡❡♥ pl ❛♥❞ pr ✐s ❛ (k✲1)✲▼❆❘❀ t❤❡♥ pr ♠♦✈❡s t♦ t❤❡ r✐❣❤t ♦♥❡ ❝❤❛r❛❝t❡r ❛t ❛ t✐♠❡ ✉♥t✐❧ ❛♥♦t❤❡r k✲▼❆❘ ✐s ❢♦✉♥❞✳ ✶✹

❋✐❣✉r❡ ✺✿ P❛r✐❦❤ ♠❛tr✐❝❡s ❢♦r ♣❡r✐♦❞ ❧❡♥❣t❤s ♦❢ ✸ ❛♥❞ ✺✱ ✇❤❡♥ r = abcaccaccaab✳ ❚❤❡ s❡❝♦♥❞ ❛❧❣♦r✐t❤♠ ❤❛s ❛ r✉♥t✐♠❡ ♦❢ O(nk3 log n)❀ t❤❡ t❤✐r❞ ✐s ♦❢ O(nk2 log k log n)✲ t✐♠❡✳✶ ❚❤❡s❡ t✇♦ ❛❧❣♦r✐t❤♠s ✉s❡ ▼❛✐♥ ❛♥❞ ▲♦r❡♥t③ ❬✷✹❪✬s ❞✐✈✐❞❡✲❛♥❞✲❝♦♥q✉❡r r❡❝✉r✲ s✐✈❡ s❝❤❡♠❡✿ ✜rst ❧♦❝❛t❡ ❛❧❧ k✲▼❆❘s✱ ❢♦r ❛❧❧ ♣❡r✐♦❞ s✐③❡s✱ t❤❛t ❝♦♥t❛✐♥ t❤❡ ♠✐❞❞❧❡ ♣♦s✐t✐♦♥ ✭♣♦s✐t✐♦♥ n

2 ✮ ♦❢ r✱ ❛♥❞ t❤❡♥ ♣❡r❢♦r♠ t❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡ r❡❝✉rs✐✈❡❧②✱ ♦♥

t❤❡ t✇♦ ❤❛❧✈❡s✳ ❚❤❡r❡ ❛r❡ O(log n) ✐t❡r❛t✐♦♥s✱ ❡❛❝❤ ❧♦♦❦✐♥❣ ❢♦r n

2 ♣❡r✐♦❞ s✐③❡s✳

❚❤✐s ❥✉st✐✜❡s t❤❡ O(n log n) ♣❛rt ♦❢ t❤❡✐r r✉♥t✐♠❡s✳ ❚❤❡ s❡❝♦♥❞ ❛❧❣♦r✐t❤♠✬s O(k3) ♣❛rt ✐s ❜❛s❡❞ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ t❤r❡❡ ✭❜♦❧❞❡❞✮ ♦❜s❡r✈❛t✐♦♥s✿ ✶✳ ❚❤❡ r❛t✐♦ ♦❢ ♠✐s♠❛t❝❤❡s t♦ ♠♦❞✐✜❡❞ ❝❤❛r❛❝t❡rs ✐s ♦♥ t❤❡ ♦r❞❡r ♦❢ t✇♦ t♦ ♦♥❡✳❚❤❡r❡❢♦r❡✱ ❛ k✲▼❆❘✱ ✇❤✐❝❤✱ ❜② ❞❡✜♥✐t✐♦♥✱ ✐s ❛❧❧♦✇❡❞ ✉♣ t♦ k ♠♦❞✐✜❡❞ ❝❤❛r❛❝t❡rs✱ ❤❛s ✉♣ t♦ ✷k ♠✐s♠❛t❝❤❡s✳ ◆♦t❡ t❤❛t t❤❡s❡ O(k) ♠✐s♠❛t❝❤❡s ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✐♥ O(k)✲t✐♠❡✱ ❜② ♣❡r❢♦r♠✐♥❣ ▲❈❆ q✉❡r✐❡s ♦♥ ❛♥ ❙❚ ♦❢ r ❛♥❞ rR ✭r ✐♥ r❡✈❡rs❡✮✳ ✷✳ ❚❤❡ ♠✐s♠❛t❝❤❡s ❛r❡ ✉s❡❞ ✐♥ ♦r❞❡r t♦ ❦♥♦✇✱ ❢♦r ❡❛❝❤ ❝♦❧✉♠♥✱ ✐❢ ✐t ✐s ❛ ♣r♦❜✲ ❧❡♠❛t✐❝ ❝♦❧✉♠♥✱ ✐✳❡✳ ❛ ❝♦❧✉♠♥ ✇❤✐❝❤ ♠❛② ❤❛✈❡ ❛ ♠♦❞✐✜❡❞ ❝❤❛r❛❝t❡r✳ ✭❆ ❝♦❧✉♠♥ t❤❛t ✐s ♥♦t ♣r♦❜❧❡♠❛t✐❝ ❝♦♥t❛✐♥s t❤❡ s❛♠❡ ❝❤❛r❛❝t❡r ✐♥ t❤❛t ❝♦❧✲ ✉♠♥✱ ✐♥ ❡✈❡r② ♣❡r✐♦❞ ✲ ✐✳❡✳✱ ❝♦❧✉♠♥ ✵✱ ✇❤❡♥ ♣ ❂ ✸✱ ✇✐t❤ r = abcaccaccaab✳ ❙✉❝❤ ❛ ❝♦❧✉♠♥ ♥❡❡❞ ♥♦t ❜❡ ✈✐s✐t❡❞✱ s✐♥❝❡ ✐t ❝❡rt❛✐♥❧② ✇✐❧❧ ♥❡✈❡r r❡q✉✐r❡ ❛ ♠♦❞✐✜❡❞ ❝❤❛r❛❝t❡r✳✮ ❆ k✲▼❆❘ ❤❛s ❛t ♠♦st O(k) ♣r♦❜❧❡♠❛t✐❝ ❝♦❧✉♠♥s ❢♦r ❡❛❝❤ ♣❡r✐♦❞ s✐③❡✳ ✸✳ ❚❤❡ ♠✐s♠❛t❝❤❡s ❛r❡ ❛❧s♦ ✉s❡❞ t♦ ❞❡♠❛r❝❛t❡ ③♦♥❡s✳ ❆ ③♦♥❡ ✐s ❛♥ ❡①❛❝t r✉♥ ✭♦r ❛ ♣r❡✜① ♦❢ ❛♥ ❡①❛❝t r✉♥✱ ✐❢ t❤❡ ③♦♥❡ ✐s s❤♦rt❡r t❤❛♥ t❤❡ ♣❡r✐♦❞ s✐③❡✮❀ ❛♥❞ ❡❛❝❤ ♠✐s♠❛t❝❤ st❛rts ❛ ♥❡✇ ③♦♥❡✳ ❚❤❡r❡ ❛r❡ O(k) ③♦♥❡s✳ ❆ k✲▼❆❘ ♠✉st st❛rt ♦♥ t❤❡ ❧❡❢t♠♦st ♣♦s✐t✐♦♥ ♦❢ ❛ ③♦♥❡ ✭t❤❡ ♠✐s♠❛t❝❤✮ ♦r ♦♥ t❤❡ r✐❣❤t♠♦st k+1 ♣❡r✐♦❞s ♦❢ ❛ ③♦♥❡ ✭✐❢ ✐t st❛rts t♦ t❤❡ ❧❡❢t t❤❛♥ t❤❛t✱ t❤❡♥ k ✇✐❧❧ ❜❡ ❡①❝❡❡❞❡❞ ✇❤❡♥ tr②✐♥❣ t♦ ❥♦✐♥ t❤❡ r✐❣❤t♠♦st k + 1 ♣❡r✐♦❞s ♦❢ t❤✐s ③♦♥❡ ✇✐t❤ t❤❡ ♥❡①t ③♦♥❡ ❛s ❛ k✲▼❆❘✮✱ ❛♥❞ s♦ ❡✈❡r② ③♦♥❡ ❤❛s ✉♣ t♦ O(k2) ♣♦s✐t✐♦♥s t♦ ✈✐s✐t✳ ❚❤❡r❡❢♦r❡ ❡❛❝❤ ✐t❡r❛t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ❛❧❣♦r✐t❤♠ r✉♥s ✐♥ t✇♦ st❡♣s✿ ✭✶✮ ✜♥❞ t❤❡ O(k) ♠✐s♠❛t❝❤❡s ✐♥ O(k) t✐♠❡✳ ✭✷✮ ❯s❡ t❤❡ ♠✐s♠❛t❝❤❡s ✐♥ ♦r❞❡r t♦ ❞❡♠❛r✲

✶❚❤❡ r✉♥t✐♠❡s ❛r❡ r❡❛❧❧② O(nk3 log n k ) ❛♥❞ O(nk2 log k log n k )✱ r❡s♣❡❝t✐✈❡❧②✱ ❜✉t ❢♦r s✐♠✲

♣❧✐❝✐t② t❤❡ ❛✉t❤♦r ❢♦❝✉s❡s ♦♥ ❞❡s❝r✐❜✐♥❣ t❤❡ ❛❧❣♦r✐t❤♠s ❛s log n ❛♥❞ ♥♦t log n

k ✳

✶✺

❘✉♥t✐♠❡ ❙♣❛❝❡ k✲r❡♣❡t✐t✐♦♥ O(nk log k + S) O(n) k✲r✉♥ O(nk log k + S) O(n) k✲❡❞✐t r❡♣❡❛t O(nk log2 k + S) O(n + k2) ❝♦♥s❡♥s✉s ♠♦❞❡❧ O(n · max❴range2 · max❴jump · N(k, max❴range)) O(n · max❴range · k) k✲▼❆❘ O(nk2 log k log n

k )

O(⑤Σ⑤n + k) ❚❛❜❧❡ ✸✿ ❘✉♥t✐♠❡ ❛♥❞ s♣❛❝❡ ❢♦r ❞✐✛❡r❡♥t ❆❚❘ ❞❡✜♥✐t✐♦♥s ❝❛t❡ t❤❡ O(k) ③♦♥❡s❀ ❛♥❞ ❢♦r ❡❛❝❤ ③♦♥❡ t❛❦❡ O(k3) t✐♠❡ t♦ ✈✐s✐t O(k2) ♣♦s✐t✐♦♥s ✭t❤❡ ♣♦s✐t✐♦♥s t♦ ✈✐s✐t ❛r❡ t❤❡ ♦♥❡s ♦♥ t❤❡ ❧❡❢t♠♦st ♣♦s✐t✐♦♥ ♦❢ ❛ ③♦♥❡ ♦r ♦♥ t❤❡ r✐❣❤t♠♦st k + 1 ♣❡r✐♦❞s ♦❢ ❛ ③♦♥❡✮✳ ❆s ❝♦♠♣❛r❡❞ t♦ t❤❡ ✜rst ❛❧❣♦r✐t❤♠ ✲ r❛t❤❡r t❤❛♥ ✐♥st❡❛❞ ♦❢ pl ❛♥❞ pr ♠♦✈✐♥❣ ♦♥❡ ❝❤❛r❛❝t❡r ❛t ❛ t✐♠❡✱ ✐♥st❡❛❞ t❤❡② ♠♦✈❡ t♦ ♣♦✐♥t ❛t t❤❡ ❝❤❛r❛❝t❡r ✐♥ t❤❡ ♥❡①t ♣r♦❜❧❡♠❛t✐❝ ❝♦❧✉♠♥✳ ❚♦ ❣♦ ❢r♦♠ t❤❡ s❡❝♦♥❞ t♦ t❤❡ t❤✐r❞ ❛❧❣♦r✐t❤♠✱ t❤❡ ❛✉t❤♦rs ✐♥tr♦❞✉❝❡ t❤❡ ❝♦♥❝❡♣t ♦❢ ❛ ♣r♦❜❧❡♠❛t✐❝ ♣♦s✐t✐♦♥✳ ❆ ♣r♦❜❧❡♠❛t✐❝ ❝♦❧✉♠♥ ♠❛② ❛❝t ❛s ❛ ♥♦♥✲ ♣r♦❜❧❡♠❛t✐❝ ❝♦❧✉♠♥ ✐♥ s♦♠❡ ♣❛rts ♦❢ t❤❡ str✐♥❣✱ ❛♥❞ s♦ ✐♥ t❤♦s❡ ♣❛rts ♦❢ t❤❡ str✐♥❣ t❤❡ ❝♦❧✉♠♥ ✐s tr❡❛t❡❞ ❛s s✉❝❤❀ ❜✉t ✐♥ ♣❛rts ♦❢ t❤❡ str✐♥❣ t❤❛t t❤❡ ♣r♦❜✲ ❧❡♠❛t✐❝ ❝♦❧✉♠♥ ✐s ❛❝t✐♥❣ ♣r♦❜❧❡♠❛t✐❝✱ t❤❡♥ ♣♦s✐t✐♦♥s ✐♥ t❤❡ ♣r♦❜❧❡♠❛t✐❝ ❝♦❧✉♠♥ ❛r❡ ❝❛❧❧❡❞ ♣r♦❜❧❡♠❛t✐❝ ♣♦s✐t✐♦♥s✳ ❆ ❧✐st ♦❢ t❤❡ O(k) ♣r♦❜❧❡♠❛t✐❝ ♣♦s✐t✐♦♥s ✐s ♠❛✐♥t❛✐♥❡❞✱ ❛♥❞ t❤❡ P❛r✐❦❤ ♠❛tr✐① ❧♦♦❦s ❛t ❥✉st t❤❡ ♣r♦❜❧❡♠❛t✐❝ ♣♦s✐t✐♦♥s ✐♥ t❤✐s ♣❛rt ♦❢ t❤❡ str✐♥❣✳ ❚❤❡r❡ ✐s ❛ ❞❡❝r❡❛s❡ ✐♥ ♣♦s✐t✐♦♥s t❤❛t pl ❛♥❞ pr ♣♦✐♥t t♦ ✲ ❢r♦♠ O(k2) t♦ O(k) ✲ ❜✉t t❤❡ ❧✐st ✐s s♦rt❡❞✱ ✇❤✐❝❤ ❝♦♥tr✐❜✉t❡s O(log k) t♦ t❤❡ r✉♥t✐♠❡✳ ❚❤❡ s♣❛❝❡ r❡q✉✐r❡♠❡♥t ❢♦r t❤❡ t❤✐r❞ ❛❧❣♦r✐t❤♠ ✐s ❛s ❢♦❧❧♦✇s✿ ❢♦r ❡❛❝❤ ♦❢ t❤❡ p ✐t❡r❛t✐♦♥s✱ ❛ P❛r✐❦❤ ♠❛tr✐① ✐s st♦r❡❞ ❢♦r ❥✉st t❤❡ ❞✉r❛t✐♦♥ ♦❢ t❤❡ ✐t❡r❛t✐♦♥✱ ❛♥❞ ✐t ✐s ♦❢ s✐③❡ p ≤ n/2 ✇✐❞❡ ❛♥❞ ♦❢ s✐③❡ ⑤Σ⑤ ❤✐❣❤✳ ❚❤❡r❡❢♦r❡✱ ❛❧❧ t❤❡ t❡♠♣♦r❛r② P❛r✐❦❤ ♠❛tr✐❝❡s s❤❛r❡ ❛ s♣❛❝❡ ♦❢ O(⑤Σ⑤n)✳ ❆❞❞✐t✐♦♥❛❧ ❞❛t❛ ✐s st♦r❡❞ ✐♥ ❧✐sts ♦❢ s✐③❡ O(k)✳ ❚❤✉s✱ t❤❡ t♦t❛❧ s♣❛❝❡ r❡q✉✐r❡♠❡♥t ✐s O(⑤Σ⑤n + k)✳ ❲❡ ❣r♦✉♣❡❞ k✲▼❆❘ ✐♥t♦ t❤❡ ❝♦♥s❡♥s✉s ❝❛t❡❣♦r②✱ s✐♥❝❡ t❤❡ str✐♥❣ ✇✐❧❧ ❜❡ ♠♦❞✐✜❡❞ ✐♥ ♦r❞❡r t♦ ♠❛t❝❤ ❛ ❝♦♥s❡♥s✉s str✐♥❣✳ ❚❛❜❧❡ ✸ s✉♠s t❤❡ r✉♥t✐♠❡s ❛♥❞ s♣❛❝❡ r❡q✉✐r❡♠❡♥ts ♦❢ t❤❡ ❛❧❣♦r✐t❤♠s r❡❧❛t✐♥❣ t♦ t❤❡ ❛❜♦✈❡ ❞❡✜♥✐t✐♦♥s✳ ❙❤♦✇♥ ✐♥ ❚❛❜❧❡ ✹ ❛r❡ ❛❧t❡r♥❛t✐✈❡ ✇❛②s ♦❢ ❣r♦✉♣✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥s✳ ❚❤❡ ✜rst ❛❧t❡r♥❛t✐✈❡ ❣r♦✉♣✐♥❣ ✐s ✇❤❡t❤❡r t❤❡ ❞❡✜♥✐t✐♦♥ ✇♦r❦s ✇✐t❤ ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ ♠✐s♠❛t❝❤❡s ♦r ✇✐t❤ ❡❞✐t ❞✐st❛♥❝❡ ❡rr♦rs✳ ❚❤❡ s❡❝♦♥❞ ✐s ✇❤❡t❤❡r✱ ♦✈❡r❛❧❧✱ ❛❧❧ t❤❡ r❡♣❡❛t ♦❝❝✉rr❡♥❝❡s ♠✉st ❜❡❛r s✐♠✐❧❛r✐t② t♦ ❛❧❧ ♦t❤❡rs ♦r ✇❤❡t❤❡r t❤❡ str✐♥❣ ✐s ❡✈♦❧✉t✐✈❡✱ ♠❡❛♥✐♥❣ t❤❛t ✐t✬s ♦♥❧② ❛❞❥❛❝❡♥t ♦❝❝✉rr❡♥❝❡s t❤❛t ♠✉st ❜❡ s✐♠✐❧❛r ✭✐✳❡✳✱ ✐t ✐s ♣♦ss✐❜❧❡ t❤❛t t❤❡ r❡♣❡❛ts ❡✈♦❧✈❡ s♦ ♠✉❝❤ t❤❛t r❡♣❡❛ts t❤❛t ❛r❡ ❢❛r ♠❛② ❜❡❛r ❧✐tt❧❡ r❡s❡♠❜❧❛♥❝❡ t♦ ❡❛❝❤ ♦t❤❡r✮✳ ❚❤❡ ♦❝❝✉rr❡♥❝❡s ♦❢ k✲r❡♣❡t✐t✐♦♥✱ k✲r✉♥✱ ❛♥❞ k✲❡❞✐t r❡♣❡❛t ❤❛✈❡ ♥♦ s✐♠✐❧❛r✐t② r❡q✉✐r❡♠❡♥t✳ ■♥ t❤❡ ❝♦♥s❡♥s✉s ♠♦❞❡❧✱ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡ ♦❝❝✉rr❡♥❝❡s ♠✉st ❜❡ s♦♠❡✇❤❛t s✐♠✐❧❛r✱ ❛s ❛❧❧ ♦❝❝✉rr❡♥❝❡s ♠✉st ✶✻

♥❡✐❣❤❜♦r✐♥❣ ❝♦♥s❡♥s✉s k✲r❡♣❡t✐t✐♦♥ k✲r✉♥ k✲❡❞✐t r❡♣❡❛t ❝♦♥s❡♥s✉s ♠♦❞❡❧ k✲▼❆❘ ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ ① ① ① ❊❞✐t ❞✐st❛♥❝❡ ① ① ❆❧❧ ♦❝❝✉rr❡♥❝❡s s✐♠✐❧❛r r❡q✉✐r❡❞ ① ❆❧❧ ♦❝❝✉rr❡♥❝❡s s✐♠✐❧❛r ♥♦t r❡q✉✐r❡❞ ① ① ① ① ❚♦t❛❧ ❝♦✉♥t ❢♦r ♥✉♠❜❡r ♦❢ ✈❛r✐❛t✐♦♥s ① ① ① ■♥❞✐✈✐❞✉❛❧ ❝♦✉♥t ❢♦r ♥✉♠❜❡r ♦❢ ✈❛r✐❛t✐♦♥s ① ① ❚♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❛r✐❛✲ t✐♦♥s ❜♦✉♥❞❡❞ ① ① ① ① ❚♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❛r✐❛✲ t✐♦♥s ♥♦t ❜♦✉♥❞❡❞ ① ❚❛❜❧❡ ✹✿ ❆❧t❡r♥❛t✐✈❡ ●r♦✉♣✐♥❣ ♦❢ ❆❚❘s ❜❡❛r s✐♠✐❧❛r✐t② t♦ t❤❡ ❝♦♥s❡♥s✉s str✐♥❣✳ k✲▼❆❘s ❛r❡ ✉♥✉s✉❛❧ ✲ ❡✈❡♥ t❤♦✉❣❤ t❤❡② ❛r❡ ❣r♦✉♣❡❞ ❛s ❝♦♥s❡♥s✉s ✐♥ t❤❡ ♦r✐❣✐♥❛❧ ❣r♦✉♣✐♥❣ ✭♦❢ ♥❡✐❣❤❜♦r✐♥❣ ❛♥❞ ❝♦♥s❡♥s✉s✮✱ t❤❡② ❞♦ ♥♦t ❤❛✈❡ t❤❡ ♣r♦♣❡rt② t❤❛t ❛❧❧ ♦❝❝✉rr❡♥❝❡s ❛r❡ r❡q✉✐r❡❞ t♦ ❜❡ s✐♠✐❧❛r✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ✐❢ ❛❧❧ k ♠✐s♠❛t❝❤❡s ❛r❡ ✐♥ ❛ ❢❡✇ ♦❝❝✉rr❡♥❝❡s✱ t❤❛t ❝♦✉❧❞ r❡♥❞❡r t❤❡s❡ ❢❡✇ ♦❝❝✉rr❡♥❝❡s ❛s ♥♦♥✲s✐♠✐❧❛r t♦ t❤❡ ♦t❤❡r ♦❝❝✉rr❡♥❝❡s✳ ❆♥♦t❤❡r ❣r♦✉♣✐♥❣ ✐s ✇❤❡t❤❡r t❤❡ ❝♦✉♥t✐♥❣ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❛t✐♦♥s ✐s ❛ t♦t❛❧ ❝♦✉♥t ✭❝♦✉♥t ♦✈❡r ❛❧❧ ♦❝❝✉rr❡♥❝❡s✮ ♦r ❛♥ ✐♥❞✐✈✐❞✉❛❧ ❝♦✉♥t ✭t❤❛t t❤❡ ❝♦✉♥t❡r ✐s r❡s❡t ❢♦r ❡❛❝❤ ♦❝❝✉rr❡♥❝❡✮❀ ❛♥❞ t❤❡ ❧❛st ❣r♦✉♣✐♥❣ ❧♦♦❦s ❛t ✇❤❡t❤❡r t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❛r✐❛t✐♦♥s ✐s ❜♦✉♥❞❡❞ ♦r ♥♦t ✭❜♦✉♥❞❡❞ ✐s ✇❤❡♥ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❛r✐❛t✐♦♥s ❞♦❡s ♥♦t ❡①❝❡❡❞ k✱ ❛♥❞ k ✐s ❛ ✜①❡❞ ♥✉♠❜❡r ♦r ♣❡r❝❡♥t❛❣❡ t❤❛t ✐s ❦♥♦✇♥ ✐♥ ❛❞✈❛♥❝❡✱ ✇❤❡r❡❛s ✉♥❜♦✉♥❞❡❞ ♠❡❛♥s t❤❛t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❛r✐❛t✐♦♥s ❞❡♣❡♥❞s ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ♦❝❝✉rr❡♥❝❡s✱ s♦ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❛r✐❛t✐♦♥s ❤❛s ♥♦ ❜♦✉♥❞ t❤❛t ✐s ❦♥♦✇♥ ✐♥ ❛❞✈❛♥❝❡✮✳ ❚❤❡s❡ t✇♦ s❡❡♠ s✐♠✐❧❛r✱ ❜✉t t❤❡② ❛r❡✱ ✐♥ ❢❛❝t✱ ❞✐✛❡r❡♥t✳ k✲r❡♣❡t✐t✐♦♥s ❛♥❞ k✲❡❞✐t r❡♣❡❛ts ✇♦r❦ ✇✐t❤ ❛ t♦t❛❧ ❝♦✉♥t✱ ❛♥❞ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ✈❛r✐❛t✐♦♥s ✐s ❜♦✉♥❞❡❞ ❜② k✳ ❚❤❡ s❛♠❡ ❣♦❡s ❢♦r k✲▼❆❘s✳ k✲r✉♥s ❤❛✈❡ ❛♥ ✐♥❞✐✈✐❞✉❛❧ ❝♦✉♥t ❛♥❞ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♠✐s♠❛t❝❤❡s ✐s ✉♥❜♦✉♥❞❡❞ ✲ s✐♥❝❡ k ♠✐s♠❛t❝❤❡s ❛r❡ ❛❧❧♦✇❡❞ ❢♦r ❡❛❝❤ ♦❝❝✉rr❡♥❝❡✱ t❤❡♥ ♦✈❡r❛❧❧ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♠✐s♠❛t❝❤❡s ✐s ✉♥❜♦✉♥❞❡❞✳ ❚❤❡ ❝♦♥s❡♥s✉s ♠♦❞❡❧ ✐s t❤❡ ✉♥✉s✉❛❧ ♦♥❡✱ ✐♥ t❤❛t t❤❡ ❝♦✉♥t ✐s ✐♥❞✐✈✐❞✉❛❧✱ s✐♥❝❡ t❤❡ ❝♦✉♥t❡r ✐s r❡s❡t✱ ②❡t t❤❡ ♥✉♠❜❡r ♦❢ ✈❛r✐❛t✐♦♥s ✐s ❜♦✉♥❞❡❞ ✲ ❛s ♣r❡✈✐♦✉s❧② ♠❡♥t✐♦♥❡❞✱ t❤❡ ❛❧❣♦r✐t❤♠ ✇♦r❦s ❢♦r ✇❤❡♥ t❤❡r❡ ❛r❡ ❛t ♠♦st ✶✺✲✷✵✪ ❡❞✐t ❞✐st❛♥❝❡ ❡rr♦rs ❜❡t✇❡❡♥ t❤❡ ✐♥♣✉t str✐♥❣ ❛♥❞ t❤❡ ❝♦♥s❡♥s✉s str✐♥❣✳ ✶✼

✷✳✸ ❘❡❧❛t❡❞ ❘❡s❡❛r❝❤

✷✳✸✳✶ ◆❡st❡❞ ❚❛♥❞❡♠ ❘❡♣❡❛t ❆ r❡❧❛t❡❞ t♦♣✐❝ ✐s t❤❛t ♦❢ ♥❡st❡❞ t❛♥❞❡♠ r❡♣❡❛ts ✭◆❚❘s✮✱ ✜rst ❞❡✜♥❡❞ ❜② ❬✶✻❪✳ ❖♥❡ t②♣❡ ♦❢ ◆❚❘ ✐s ✇❤❡♥ t❤❡r❡ ❛r❡ t✇♦ ❞✐st✐♥❝t ❚❘s✱ ✇❤❡r❡ ♦♥❡ ✐s ♥❡st❡❞ ✇✐t❤✐♥ t❤❡ ♦t❤❡r✱ ♦r t❤❡② ❛r❡✱ ✐♥ s♦♠❡ ✇❛②✱ ✐♥t❡rs♣❡rs❡❞✳ ❆♥ ❡①❛♠♣❧❡ ◆❚❘ ✐s ♦❢ t❤❡ ❢♦r♠ ❳①①①❳①①①①❳①①❳①①❳①❳①①①✱ s✉❝❤ ❛s✿ ❈❚❚●❆●❆❚❚ ❆❈❆❚ ❆❈❆❚ ❆❈❆❚ ❈❚❚●❆●❆❚❚ ❆❈❆❚ ❆❈❆❚ ❆❈❆❚ ❆❈❆❚ ❈❚❚●❆●❆❚❚ ❆❈❆❚ ❆❈❆❚ ❈❚❚●❆●❆❚❚ ❆❈❆❚ ❆❈❆❚ ❈❚❚●❆●❆❚❚ ❆❈❆❚ ❈❚❚●❆●❆❚❚ ❆❈❆❚ ❆❈❆❚ ❆❈❆❚ ❆♣♣r♦①✐♠❛t❡ ♥❡st❡❞ t❛♥❞❡♠ r❡♣❡❛ts ❛r❡ ✇❤❡♥ t❤❡ ◆❚❘s ❤❛✈❡ ✈❛r✐❛t✐♦♥s✳ ❘❡s❡❛r❝❤ ♦♥ t❤❡ ♣r♦❜❧❡♠ ♦❢ ✜♥❞✐♥❣ ❛♣♣r♦①✐♠❛t❡ ◆❚❘s✱ ✇❤❡r❡ t❤❡r❡ ❛r❡ ❍❛♠✲ ♠✐♥❣ ❞✐st❛♥❝❡ ♠✐s♠❛t❝❤❡s✱ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✶✻❪✳ ❆♣♣r♦①✐♠❛t❡ ◆❚❘s ❛♥❞ t❤❡ ❛❧✐❣♥♠❡♥t ♣r♦❜❧❡♠ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ ❬✷✻❪✳ ❬✷✼❪✬s ❛❧❣♦r✐t❤♠✱ ✐♠♣❧❡♠❡♥t❡❞ ✐♥ t❤❡ s♦❢t✇❛r❡ ❝❛❧❧❡❞ ◆❚❘❋✐♥❞❡r✱ ✇♦r❦s ✇✐t❤ ✜♥❞✐♥❣ ❛♣♣r♦①✐♠❛t❡ ◆❚❘s ✲ ❛♣♣r♦①✐✲ ♠❛t❡ ✉♥❞❡r t❤❡ ❡❞✐t ❞✐st❛♥❝❡ ♠❡tr✐❝ ✭s✉❜st✐t✉t✐♦♥s ❛♥❞ ✐♥❞❡❧s✮✱ ✐♥ ❛❞❞✐t✐♦♥ t♦ ❞✉♣❧✐❝❛t✐♦♥ ✭✇❤❡♥ ❛ s✉❜str✐♥❣ ✐s r❡♣❡❛t❡❞✮✳ ✷✳✸✳✷ ❆♣♣r♦①✐♠❛t❡ ❙❡❡❞ ❆♥♦t❤❡r r❡❧❛t❡❞ t♦♣✐❝ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❬✽❪✳ ❚❤✐s ✐s ❛❜♦✉t ❛♣♣r♦①✐♠❛t❡ s❡❡❞s ♦❢ str✐♥❣s✱ ✇❤✐❝❤ ❛r❡ s✉❜str✐♥❣s ♦❢ ❛ ❣✐✈❡♥ str✐♥❣ t❤❛t ❝♦✈❡r ✲ ❜② ❝♦♥❝❛t❡♥❛t✐♦♥ ♦r ♦✈❡r❧❛♣ ✲ ❛ s✉♣❡rstr✐♥❣ ♦❢ t❤❛t s❛♠❡ str✐♥❣✳ ■t ✐s ❛♣♣r♦①✐♠❛t❡ ✐♥ t❤❛t ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ ♠✐s♠❛t❝❤❡s✱ ❡❞✐t ❞✐st❛♥❝❡ ❡rr♦rs✱ ❛♥❞ ❛❧s♦ ✇❡✐❣❤t❡❞ ❡❞✐t ❞✐st❛♥❝❡ ❡rr♦rs ❛r❡ ❛❧❧♦✇❡❞✳ ❚❤✐s ❞✐✛❡rs ❢r♦♠ ♦✉r ❢♦❝✉s✱ ✐♥ t❤❛t ✇❡ ✇♦r❦ ✇✐t❤ ❝♦♥❝❛t❡♥❛t✐♦♥ ❜✉t ♥♦t ♦✈❡r❧❛♣✳ ✷✳✸✳✸ P❡r✐♦❞ ❘❡❝♦✈❡r② Pr♦❜❧❡♠ ❆❧s♦ ✐♥t❡r❡st✐♥❣ ✐s t❤❡ ♣❡r✐♦❞ r❡❝♦✈❡r② ♣r♦❜❧❡♠✱ ✇❤✐❝❤ ✐s ❞❡s❝r✐❜❡❞ ✐♥ ❬✸❪ ❛♥❞ ❬✶✵❪✳ ■t ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✿ ❧❡t r ❜❡ ❛♥ n✲❧♦♥❣ str✐♥❣ ✇✐t❤ ♣❡r✐♦❞ p✳ ●✐✈❡♥ r′✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ ❛s r ❜✉t ♣♦ss✐❜❧② ❝♦rr✉♣t❡❞ ❜② ❛t ♠♦st ❛ s♣❡❝✐✜❡❞ ♣❡r❝❡♥t❛❣❡ ♦❢ ❡❞✐t ❞✐st❛♥❝❡ ❡rr♦rs✱ r❡t✉r♥ r✳ ■t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ❞✐st✐♥❣✉✐s❤ t❤❡ ♦r✐❣✐♥❛❧ ♣❡r✐♦❞ ❢r♦♠ ♦t❤❡r ❢❛❧s❡ ❝❛♥❞✐❞❛t❡s✱ s♦✱ ✐♥ ❛❝t✉❛❧✐t②✱ t❤❡ ♣r♦❜❧❡♠ ✐s t♦ ✐❞❡♥t✐❢② ❛ s❡t ♦❢ ❝❛♥❞✐❞❛t❡s t❤❛t ✐s ❣✉❛r❛♥t❡❡❞ t♦ ✐♥❝❧✉❞❡ t❤❡ r❡❛❧ ❛♥s✇❡r✳ ❚❤❡r❡ ✐s ❛ r❡q✉✐r❡♠❡♥t t❤❛t p ✐s ❛ s✉❜str✐♥❣ ♦❢ r′✱ ♠❡❛♥✐♥❣ t❤❛t p ♠✉st ❛♣♣❡❛r ❛t ❧❡❛st ♦♥❝❡✱ ❡rr♦r✲❢r❡❡✱ ✐♥ r′✳ ❇② ❤❛✈✐♥❣ s✉❝❤ ❛ r❡q✉✐r❡♠❡♥t✱ t❤✐s ❧❡❛❞s t♦ t❤❡ ♥❛✐✈❡ ❛❧❣♦r✐t❤♠ t♦ s♦❧✈❡ t❤❡ ♣❡r✐♦❞ r❡❝♦✈❡r② ♣r♦❜❧❡♠✿ s✐♥❝❡ p ✐s ❛ s✉❜str✐♥❣ ♦❢ r′✱ t❤❡ ♥❛✐✈❡ ❛❧❣♦r✐t❤♠ s✐♠♣❧② ❧♦♦❦s ❛t t❤❡ s❡t ♦❢ t❤❡ O(n2) s✉❜str✐♥❣s ♦❢ r′ ❛s ❝❛♥❞✐❞❛t❡s✳ ❋♦r ♠♦r❡ ❡✣❝✐❡♥t ❛❧❣♦r✐t❤♠s✱ t❤❡ r❡❛❞❡r ✐s r❡❢❡rr❡❞ t♦ ❬✸❪ ❛♥❞ ❬✶✵❪✳ ✶✽

✸ P❛❧✐♥❞r♦♠❡s

P❛❧✐♥❞r♦♠❡s ❛r❡ str✐♥❣s t❤❛t r❡❛❞ t❤❡ s❛♠❡ ❢♦r✇❛r❞s ❛♥❞ ❜❛❝❦✇❛r❞s✳ ❚♦ ❜❡ ♣r❡❝✐s❡✱ ❛ str✐♥❣ r ✐s ❛ ♣❛❧✐♥❞r♦♠❡ ✐❢ ✐t ✐s ♦❢ t❤❡ ❢♦r♠ uauR✱ ✇❤❡r❡ u ✐s ❛ str✐♥❣ ❛♥❞ uR ✐s ✐ts r❡✈❡rs❡❀ a ✐s t❤❡ ❡♠♣t② str✐♥❣ ♦r ❛ s✐♥❣❧❡ ❝❤❛r❛❝t❡r✳ a ✐s ❝❛❧❧❡❞ t❤❡ ❣❛♣✱ ✇❤✐❧❡ u ❛♥❞ uR ❛r❡ ❝❛❧❧❡❞ r❡s♣❡❝t✐✈❡❧② t❤❡ ❧❡❢t ❛r♠ ❛♥❞ r✐❣❤t ❛r♠ ♦❢ t❤❡ ♣❛❧✐♥❞r♦♠❡✳ ❙♦♠❡ ✉s❡ t❤❡ t❡r♠✐♥♦❧♦❣② ♦❢ r❛❞✐✉s ❛♥❞ ❞✐❛♠❡t❡r ♦❢ ❛ ♣❛❧✐♥❞r♦♠❡✱ r❡❢❡rr✐♥❣ t♦ t❤❡ ❧❡♥❣t❤ ♦❢ ♦♥❡ ♦❢ t❤❡ ❛r♠s ❛♥❞ t♦ t❤❡ s✉♠ ♦❢ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ❛r♠s✱ r❡s♣❡❝t✐✈❡❧②✳ ❆ ♣r♦❜❧❡♠ t❤❛t ✐s r❡s❡❛r❝❤❡❞ ✐s t❤❛t ♦❢ s❡❛r❝❤✐♥❣✱ ✐♥ ❛ str✐♥❣✱ ❢♦r s✉❜str✐♥❣s t❤❛t ❛r❡ ♣❛❧✐♥❞r♦♠❡s✳ ❚♦ s♦❧✈❡ ✐t✱ ♦♥❡ ❝♦♠♠♦♥ ❛♣♣r♦❛❝❤ ✐s t♦ ✜♥❞ t❤❡ ❧♦♥❣❡st ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t ❡❛❝❤ ♣♦s✐t✐♦♥✳ ❆s ✇✐t❤ r❡♣❡❛ts✱ t❤❡ ♣❛❧✐♥❞r♦♠❡s t❤❛t ❛r❡ ♦❢ ✐♥t❡r❡st ❛r❡ t❤♦s❡ t❤❛t ❛r❡ ♠❛①✐♠❛❧✳ ❚❤✐s ✐s ✇❤❡♥ ❛ ♣❛❧✐♥❞r♦♠❡ ❝❛♥♥♦t ❜❡ ❡①t❡♥❞❡❞ ❛♥② ❢✉rt❤❡r✱ ❡✐t❤❡r t♦ t❤❡ ❧❡❢t ♦r t♦ t❤❡ r✐❣❤t✳ ❋♦r ✐♥st❛♥❝❡✱ t❤❡ ❧❡❢t ❛r♠ ♦❢ t❤❡ ♠❛①✐♠❛❧ ♣❛❧✐♥❞r♦♠❡ ♦❢ ❛❜❛❜❜❛❜❛ ✐s ❛❜❛❜✱ st❛rt✐♥❣ ❛t t❤❡ ✜rst ♣♦s✐t✐♦♥❀ ❛❜ ✭t❤❛t st❛rts ❛t t❤❡ t❤✐r❞ ♣♦s✐t✐♦♥✮ ✐s ♥♦t ♠❛①✐♠❛❧✳ ❚♦ ❜❡ ❝❧❡❛r✱ ✇❤❡♥ ✇❡ r❡❢❡r t♦ t❤❡ ♣❛❧✐♥❞r♦♠❡✱ ✇❡ ❛r❡ r❡❢❡rr✐♥❣ t♦ ♦♥❡ t❤❛t ✐s ♠❛①✐♠❛❧✳

✸✳✶ ▲✐♥❡❛r✲t✐♠❡ ❆❧❣♦r✐t❤♠ ★✶ ❢♦r ✭❊①❛❝t✮ P❛❧✐♥❞r♦♠❡s

❚❤❡ ✜rst ❧✐♥❡❛r✲t✐♠❡ ❛❧❣♦r✐t❤♠ t♦ ✜♥❞ ✭❡①❛❝t✮ ♣❛❧✐♥❞r♦♠❡s t❤❛t ❛r❡ s✉❜str✐♥❣s ♦❢ ❛ str✐♥❣✱ ❜② ✜♥❞✐♥❣ t❤❡ ❧♦♥❣❡st ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t ❡❛❝❤ ♣♦s✐t✐♦♥✱ ✐s ❜② ▼❛♥❛❝❤❡r ❬✷✺❪✷✳ ❚❤❡ ✐♥t✉✐t✐♦♥ ❜❡❤✐♥❞ t❤❡ ❛❧❣♦r✐t❤♠ ✐s t❤❛t ✐❢ r = uauR✱ t❤❡♥ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ♣❛❧✐♥❞r♦♠❡s ❝❡♥t❡r❡❞ ❛t t❤❡ ♣♦s✐t✐♦♥s ♦❢ t❤❡ ❧❡❢t ❛r♠✱ u✱ ✇✐❧❧ ❜❡ ❛ ♠✐rr♦r ✐♠❛❣❡ ♦❢ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ♣❛❧✐♥❞r♦♠❡s ❝❡♥t❡r❡❞ ❛t t❤❡ ♣♦s✐t✐♦♥s ♦❢ t❤❡ r✐❣❤t ❛r♠✱ uR✳ ❲❡ ✇✐❧❧ ♥♦✇ ❡①♣❧❛✐♥ t❤❡ ❛❧❣♦r✐t❤♠ ✐♥ ❞❡t❛✐❧✳ ■t ✉s❡s str✐♥❣ r✱ ✇❤✐❝❤ ✐s ♦❢ ❧❡♥❣t❤ n✳ ❙✐♥❝❡ ✇❡ ❧♦♦❦ ❢♦r ❜♦t❤ ♣❛❧✐♥❞r♦♠❡s ♦❢ ♦❞❞ ❧❡♥❣t❤ ❛♥❞ ♦❢ ❡✈❡♥ ❧❡♥❣t❤✱ ✇❡ ❧♦♦❦ ❛t t❤❡ ❝❤❛r❛❝t❡r ♣♦s✐t✐♦♥s ✭✇❤❡r❡ ✇❡ ✜♥❞ ♣❛❧✐♥❞r♦♠❡s ♦❢ ♦❞❞ ❧❡♥❣t❤s✮ ❛♥❞ ❛t t❤❡ ✐♥✲❜❡t✇❡❡♥ ♣♦s✐t✐♦♥s ✭✇❤❡r❡ ✇❡ ✜♥❞ ♣❛❧✐♥❞r♦♠❡s ♦❢ ❡✈❡♥ ❧❡♥❣t❤s✮❀ ❛♥❞ s♦ ❛ str✐♥❣ ♦❢ ❧❡♥❣t❤ n r❡❛❧❧② ❤❛s 2n − 1 ♣♦s✐t✐♦♥s✳ ❍❡r❡❛❢t❡r✱ ✇❡ r❡❢❡r t♦ t❤❡ 2n − 1 ♣♦s✐t✐♦♥s ❛s n✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ✉s❡s t✇♦ ♣♦✐♥t❡rs✳ ❚❤❡ ✜rst✱ ❝❛❧❧❡❞ ✉♣❚♦✱ ❦❡❡♣s tr❛❝❦ ♦❢ t❤❡ ❧❛st ♣♦s✐t✐♦♥ t❤❛t ✇❛s ❛❧r❡❛❞② ❢♦✉♥❞ t♦ ❜❡ ♣❛rt ♦❢ ❛ ♣❛❧✐♥❞r♦♠❡✳ ❆s ✇✐❧❧ ❜❡ ❡①♣❧❛✐♥❡❞ s❤♦rt❧②✱ ✉♣❚♦ ✐s ✉s❡❞ t♦ ❤❡❧♣ ❝❛❧❝✉❧❛t❡ ✈❛❧✉❡s ♦❢ ♣❛❧✐♥❞r♦♠❡s ❧♦❝❛t❡❞ ❛t ♦t❤❡r ♣♦s✐t✐♦♥s✳ ❚❤❡ s❡❝♦♥❞ ✐s ❝❛❧❧❡❞ t❤❡ ❝✉rs♦r✳ ❚❤✐s ✐s t❤❡ ♣♦s✐t✐♦♥ ❢♦r ✇❤✐❝❤ t❤❡ ❛❧❣♦r✐t❤♠ ✐s ❝✉rr❡♥t❧② ❛tt❡♠♣t✐♥❣ t♦ ✜♥❞ t❤❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t✳ ■t st❛rts ❛t t❤❡ ✜rst ♣♦s✐t✐♦♥ ❛♥❞ ♠♦✈❡s t♦ t❤❡ r✐❣❤t✱ ♦♥❡ ♣♦s✐t✐♦♥ ❛t ❛ t✐♠❡✳ ❚❤❡ ✇❛② t❤❡ ❛❧❣♦r✐t❤♠ ✇♦r❦s ✐s t❤❛t✱ ❢♦r ❡❛❝❤ ❝✉rs♦r✱ ✐t ❝❤❡❝❦s ✐❢ t❤❡ ♣♦s✐t✐♦♥ ♦♥❡ t♦ t❤❡ ❧❡❢t ♠❛t❝❤❡s t❤❡ ♣♦s✐t✐♦♥ ♦♥❡ t♦ t❤❡ r✐❣❤t✳ ■❢ t❤❡② ❞♦ ♥♦t ♠❛t❝❤✱ t❤❡♥ t❤❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t t❤❛t ♣♦s✐t✐♦♥ ✐s ♦♥❡ ✭✐❢ ✐t✬s ❛ ❝❤❛r❛❝t❡r ♣♦s✐t✐♦♥✮ ♦r

✷❆❝t✉❛❧❧②✱ ▼❛♥❛❝❤❡r ✇r✐t❡s t❤❛t t❤❡ ❛❧❣♦r✐t❤♠ ✜♥❞s t❤❡ ✐♥✐t✐❛❧ ♣❛❧✐♥❞r♦♠❡ ♦❢ t❤❡ str✐♥❣✳

❖t❤❡rs ❧❛t❡r ♥♦t❡ t❤❛t ✇❤❡♥ ❤✐s ❛❧❣♦r✐t❤♠ ✐s ✐t❡r❛t✐✈❡❧② ♣❡r❢♦r♠❡❞ ♦♥ t❤❡ r❡♠❛✐♥✐♥❣ s✉❜str✐♥❣✱ ✐t ✜♥❞s t❤❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t ❡❛❝❤ ♣♦s✐t✐♦♥✳ ❲❡ ✇✐❧❧ ❞❡s❝r✐❜❡ ✐t t❤✐s ✇❛②✳

✶✾

♣♦s✐t✐♦♥ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ❝❤❛r❛❝t❡r ❜ ❛ ❜ ❝ ❜ ❛ ❜ ❝ ❜ ❛ ❝ ❝ ❜ ❛ ❛❧❣♦r✐t❤♠ ✵ ✶ ✸ ✶ ✼ ✶ ✾ ✵ ❚❛❜❧❡ ✺✿ ❚❤✐s ✐s ❛ s❛♠♣❧❡ r✉♥ ♦❢ ▼❛♥❛❝❤❡r✬s ❛❧❣♦r✐t❤♠✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ✐s ✉♣ t♦ ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ✈❛❧✉❡ ❢♦r ♣♦s✐t✐♦♥ ✼✳ ③❡r♦ ✭✐❢ ✐t✬s ❛♥ ✐♥✲❜❡t✇❡❡♥ ♣♦s✐t✐♦♥✮❀ ❛♥❞ t❤❡♥ t❤❡ ♥❡①t ✐t❡r❛t✐♦♥ ❜❡❣✐♥s ✇✐t❤ t❤❡ ❝✉rs♦r ♠♦✈✐♥❣ t♦ t❤❡ r✐❣❤t ❜② ♦♥❡ ♣♦s✐t✐♦♥✳ ❇✉t ✐❢ t❤♦s❡ t✇♦ ♣♦s✐t✐♦♥s ♠❛t❝❤✱ t❤❡♥ t❤✐s ♣♦s✐t✐♦♥ ✐s ❦♥♦✇♥ ❛s ❛ t❡♥t❛t✐✈❡ ❝❡♥t❡r✱ ♦r ❚❈✳ ❆t t❤✐s ♣♦✐♥t✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜♥❞s ❤♦✇ ♠❛♥② ♠♦r❡ ♣♦s✐t✐♦♥s t♦ t❤❡ ❧❡❢t ♦❢ t❤❡ ❚❈ ♠❛t❝❤ ♣♦s✐t✐♦♥s t♦ ✐ts r✐❣❤t✳ ■t ❞♦❡s s♦ ❜② ❧♦♦❦✐♥❣ ❛t t❤❡ ✈❛❧✉❡ ❛t t❤❡ ♣♦s✐t✐♦♥ t❤❛t ✐s ✐ts ♠✐rr♦r ✐♠❛❣❡✱ ❛♥❞ ✐❢ t❤❛t ♣♦s✐t✐♦♥✬s ♣❛❧✐♥❞r♦♠❡ ❡①t❡♥❞s ✉♥t✐❧ t❤✐s ♣♦s✐t✐♦♥✱ t❤❡♥ t❤✐s ♣♦s✐t✐♦♥✬s s❡❛r❝❤ ♥❡❡❞ ♥♦t st❛rt ✉♥t✐❧ ❛❢t❡r ✉♣❚♦✳ ❚❤❡ r❡❛s♦♥ ✐s t❤❛t t❤♦s❡ ♣♦s✐t✐♦♥s ❛r❡ ❢♦r s✉r❡ ♣❛rt ♦❢ t❤❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t t❤❡ ❝✉rs♦r✳ ■t✬s ♣♦ss✐❜❧❡ t❤❛t t❤✐s ♣❛❧✐♥❞r♦♠❡ ✐s ❧♦♥❣❡r✱ s♦ t❤❡ ❛❧❣♦r✐t❤♠ ❛tt❡♠♣ts t♦ ♠❛t❝❤ ♠♦r❡✱ ❛♥❞ ✐t ❞♦❡s s♦ ✉♥t✐❧ t❤❡r❡✬s ❛ ♠✐s♠❛t❝❤✳ ❆t t❤❛t ♣♦✐♥t✱ ✐t st♦♣s ❛♥❞ r❡❝♦r❞s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t t❤❡ ❝✉rs♦r❀ t❤❡♥ t❤❡ ❝✉rs♦r ♠♦✈❡s t♦ t❤❡ r✐❣❤t ❜② ♦♥❡ ♣♦s✐t✐♦♥✳ ❲❡ ✇✐❧❧ ♥♦✇ ✉s❡ ❚❛❜❧❡ ✺ t♦ s❤♦✇ t✇♦ st❡♣s ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✿ ♣♦s✐t✐♦♥s ✼ ❛♥❞ ✽✳ ■♥ t❤❡ ✐♥✐t✐❛❧✐③❛t✐♦♥ st❡♣ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✱ t❤❡ ♣♦s✐t✐♦♥s ❜❡❢♦r❡ t❤❡ ✜rst ❝❤❛r❛❝t❡r ❛♥❞ ❢♦❧❧♦✇✐♥❣ t❤❡ ❧❛st ❝❤❛r❛❝t❡r ✇❡r❡ ✐♥✐t✐❛❧✐③❡❞ t♦ ③❡r♦✳ ❚❤❡♥✱ t❤❡ ✈❛❧✉❡s ❢♦r ♣♦s✐t✐♦♥s ✶✲✻ ✇❡r❡ ❝❛❧❝✉❧❛t❡❞❀ t❤♦s❡ ✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ t❛❜❧❡✳ ❚❤❡ ❝✉rs♦r ✐s ♥♦✇ ✉♣ t♦ ♣♦s✐t✐♦♥ ✼✳ ◆♦t❡✱ t❤♦✉❣❤✱ t❤❛t ✇❡ ✇✐❧❧ ❧♦♦❦ ❛t ❥✉st t❤❡ ❝❤❛r❛❝t❡r ♣♦s✐t✐♦♥s✱ ❛♥❞ ✐❣♥♦r❡ t❤❡ ✐♥✲❜❡t✇❡❡♥ ♣♦s✐t✐♦♥s✱ ❢♦r s✐♠♣❧✐❝✐t②✳ ❋♦r ♣♦s✐t✐♦♥ ✼✱ ✐t ✜rst ❝❤❡❝❦s ✐❢ ✐t✬s ❛ ❚❈✳ ❙✐♥❝❡ t❤❡ ♣♦s✐t✐♦♥ t♦ t❤❡ ❧❡❢t ❛♥❞ t❤❡ ♣♦s✐t✐♦♥ t♦ t❤❡ r✐❣❤t ❞♦ ♥♦t ❝♦♥t❛✐♥ t❤❡ s❛♠❡ ❝❤❛r❛❝t❡r✱ ✐t ✐s ♥♦t ❛ ❚❈ ❛♥❞ s♦ t❤❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t t❤✐s ♣♦s✐t✐♦♥ ✐s ♦♥❡✳ ❚❤❡ ❝✉rs♦r ♠♦✈❡s t♦ ♣♦s✐t✐♦♥ ✽✳ ❋♦r t❤✐s ♣♦s✐t✐♦♥✱ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡ ♣♦s✐t✐♦♥ t♦ t❤❡ ❧❡❢t ❛♥❞ t❤❡ ♣♦s✐t✐♦♥ t♦ t❤❡ r✐❣❤t ❝♦♥t❛✐♥ t❤❡ s❛♠❡ ❝❤❛r❛❝t❡r✱ ❛♥❞ s♦ ✐t ✐s ❛ ❚❈✳ ❲❤❡♥ t❤❡ ❛❧❣♦✲ r✐t❤♠ ❝❛❧❝✉❧❛t❡❞ t❤❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t ♣♦s✐t✐♦♥ ✻✱ ✐t ❢♦✉♥❞ t❤❛t ♣♦s✐t✐♦♥s ✷ t❤r♦✉❣❤ ✶✵ ♣❛rt✐❝✐♣❛t❡ ✐♥ t❤❡ ♣❛❧✐♥❞r♦♠❡ ♦❢ t❤❛t ♣♦s✐t✐♦♥✳ ❚❤❡r❡❢♦r❡✱ ✇❤❡♥ ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t t❤✐s ♣♦s✐t✐♦♥✱ ♣♦s✐t✐♦♥ ✽✱ t❤❡ ❛❧❣♦r✐t❤♠ ✜rst ❝❤❡❝❦s t❤❡ ✈❛❧✉❡ ♦❢ ✐ts ♠✐rr♦r ✐♠❛❣❡ ❛r♦✉♥❞ ♣♦s✐t✐♦♥ ✻✱ ✇❤✐❝❤ ✐s ♣♦s✐t✐♦♥ ✹✳ ❚❤❛t ♣♦s✐t✐♦♥ ❤❛s ❛ ✈❛❧✉❡ ♦❢ ✼✱ s♦ ✐t✬s ❦♥♦✇♥ t❤❛t ♣♦s✐t✐♦♥ ✽✬s ✈❛❧✉❡ ✐s ❛t ❧❡❛st ✺✱ s✐♥❝❡ ✐t ♦♥❧② ❦♥♦✇s ✉♥t✐❧ ♣♦s✐t✐♦♥s ✻ t❤r♦✉❣❤ ✶✵ ✭❛ ✈❛❧✉❡ ♦❢ ✺✮ ❛r❡ ♣❛❧✐♥❞r♦♠✐❝✳ ■t ❛tt❡♠♣ts ♠♦r❡ ♠❛t❝❤❡s✱ st❛rt✐♥❣ ✇✐t❤ ♣♦s✐t✐♦♥ ✶✶ ✭t❤❡ ♣♦s✐t✐♦♥ ❛❢t❡r ✉♣❚♦✮✱ ❝♦♠♣❛r✐♥❣ ✐t t♦ ♣♦s✐t✐♦♥ ✺✳ ■❢ t❤❡② ✇♦✉❧❞ ♠❛t❝❤✱ t❤❡♥ t❤❡ ❛❧❣♦r✐t❤♠ ✇♦✉❧❞ ❝♦♥✲ t✐♥✉❡ t♦ tr② t♦ ♠❛t❝❤❀ ❜✉t s✐♥❝❡ t❤❡② ❞♦ ♥♦t ♠❛t❝❤✱ t❤❡ ❛❧❣♦r✐t❤♠ st♦♣s✱ ❛♥❞ ✐t ❦♥♦✇s t❤❛t t❤❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t ♣♦s✐t✐♦♥ ✽ ✐s ♦❢ ❧❡♥❣t❤ ✺✳ ❆❧s♦✱ ✐t ♥♦t❡s ❢♦r t❤❡ ❢✉t✉r❡ t❤❛t ✉♣❚♦ ♣♦✐♥ts t♦ ♣♦s✐t✐♦♥ ✶✵✳ ❚❤❡ ❝✉rs♦r ♠♦✈❡s t♦ ♣♦s✐t✐♦♥ ✾✳ ❚❤❡ r✉♥t✐♠❡ ♦❢ t❤✐s ❛❧❣♦r✐t❤♠ ✐s O(n)✳ ■t ❛❝❤✐❡✈❡s t❤✐s r✉♥t✐♠❡ s✐♥❝❡ ❝❛❧❝✉❧❛✲ t✐♦♥s ♦❢ ❧❛t❡r ✈❛❧✉❡s✱ t❤♦s❡ ✐♥ r✐❣❤t ❛r♠s✱ ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ♣r❡✈✐♦✉s❧②✲❝❛❧❝✉❧❛t❡❞ ✈❛❧✉❡s ♦❢ ❧❡❢t ❛r♠s✳ ✷✵

q ✶ ✷ ✸ ✹ ✺ r[q + 1] r[2] r[3] r[4] r[5] r[6] rR[n − q + 1] rR[6] rR[5] rR[4] rR[3] rR[2] ❛❜❝❝❜❛ bccba ccba cba ba a ❛❜❝❝❜❛ a ba cba ccba bccba ▲❈P ✵ ✵ ✸ ✵ ✵ ❛❜❜❛❜❛ bbaba baba aba ba a ❛❜❛❜❜❛ a ba bba abba babba ▲❈P ✵ ✷ ✵ ✵ ✵ ❝❛❜❜❛❞ abbad bbad bad ad d ❞❛❜❜❛❝ c ac bac bbac abbac ▲❈P ✵ ✵ ✷ ✵ ✵ ❛❜❝❞❜❛ bcdba cdba dba ba a ❛❜❞❝❜❛ a ba cba dcba bdcba ▲❈P ✵ ✵ ✵ ✵ ✵ ❚❛❜❧❡ ✻✿ ●❙❚ P❛❧✐♥❞r♦♠❡ ❆❧❣♦r✐t❤♠

✸✳✷ ▲✐♥❡❛r✲t✐♠❡ ❆❧❣♦r✐t❤♠ ★✷ ❢♦r ✭❊①❛❝t✮ P❛❧✐♥❞r♦♠❡s

❆♥♦t❤❡r ❧✐♥❡❛r✲t✐♠❡ ❛❧❣♦r✐t❤♠ ❢♦r ✭❡①❛❝t✮ ♣❛❧✐♥❞r♦♠❡s✱ ♦♥❡ t❤❛t ✐s ❜❛s❡❞ ♦♥ ❙❚s ✐s ❬✶❪❀ ✐t ✐s ❛❧s♦ ❞❡s❝r✐❜❡❞ ✐♥ s❡❝t✐♦♥ ✾✳✷✳✶ ♦❢ ❬✶✹❪ ❛♥❞ ✐♥ ❬✷✾❪✳ ❚❤❡ ✐❞❡❛ ❜❡❤✐♥❞ t❤❡ ❛❧❣♦r✐t❤♠ ✐s t♦ ♣❡r❢♦r♠✱ ❢♦r ❡❛❝❤ ♣♦s✐t✐♦♥✱ ❛ ❧♦♥❣❡st ❝♦♠♠♦♥ ❡①t❡♥s✐♦♥ ✭♦r ▲❈❊✮ q✉❡r②✳ ❆♥ ▲❈❊ q✉❡r② ✐s ❛♥ ▲❈P q✉❡r② ♦❢ t❤❡ s✉❜str✐♥❣ t♦ t❤❡ r✐❣❤t ♦❢ ❛ ♣♦s✐t✐♦♥ ✇✐t❤ t❤❡ s✉❜str✐♥❣ t♦ t❤❡ ❧❡❢t ♦❢ ❛ ♣♦s✐t✐♦♥ ✲ ❡ss❡♥t✐❛❧❧②✱ ✐t r❡t✉r♥s ❤♦✇ ♠❛♥② ♣♦s✐t✐♦♥s ♦♥ t❤❡ r✐❣❤t ♠❛t❝❤ ♣♦s✐t✐♦♥s ♦♥ t❤❡ ❧❡❢t✳ ❚♦ ✐♠♣❧❡♠❡♥t t❤❛t ✐❞❡❛✱ t❤❡ ❛❧❣♦r✐t❤♠ ✐s✿ ❝r❡❛t❡✱ ✐♥ ❧✐♥❡❛r t✐♠❡✱ ❛ ●❙❚ ♦❢ r ❛♥❞ ♦❢ rR✳ Pr❡♣r♦❝❡ss t❤❡ tr❡❡ ✐♥ ♦r❞❡r t♦ ❜❡ ❛❜❧❡ t♦ ❛♥s✇❡r ❛♥ ▲❈❆ q✉❡r② ✭✇❤✐❝❤✱ ❢♦r str✐♥❣s✱ ✐s ❛♥ ▲❈P q✉❡r②✮ ✐♥ ❝♦♥st❛♥t t✐♠❡✳ ❚♦ ❧♦♦❦ ❢♦r ♣❛❧✐♥❞r♦♠❡s ♦❢ ❡✈❡♥ ❧❡♥❣t❤✱ s♦❧✈❡ t❤❡ ▲❈P ❢♦r ❡✈❡r② ♣♦s✐t✐♦♥ q✱ 1 ≤ q ≤ n − 1✱ ♦❢ r[q + 1] ✇✐t❤ rR[n − q + 1]✳ ■❢ ❛♥② ▲❈P q✉❡r② r❡s✉❧ts ✐♥ ❛ ♥♦♥✲③❡r♦ ✈❛❧✉❡✱ t❤❡♥ t❤❡r❡ ✐s ❛ ♣❛❧✐♥❞r♦♠❡✱ ✇✐t❤ ❛ r❛❞✐✉s s✐③❡ ♦❢ t❤❛t ♥♦♥✲③❡r♦ ✈❛❧✉❡✱ ❝❡♥t❡r❡❞ t❤❡r❡✳ ❚♦ ❧♦♦❦ ❢♦r ♣❛❧✐♥❞r♦♠❡s ♦❢ ♦❞❞ ❧❡♥❣t❤✱ t❤❡ ♣r♦❝❡❞✉r❡ ✐s t❤❡ s❛♠❡✱ ❡①❝❡♣t t♦ s♦❧✈❡ t❤❡ ▲❈P q✉❡r② ❜❡t✇❡❡♥ r[q + 1] ❛♥❞ rR[n − q + 2]✳ ❚❤❡ r✉♥t✐♠❡ ✐s O(n) t✐♠❡✱ s✐♥❝❡ t❤❡r❡ ❛r❡ t❤❛t ♠❛♥② ♣♦ss✐❜❧❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡rs ❛♥❞ ❡❛❝❤ ♦♥❡ ✐s ❝❤❡❝❦❡❞ ✇✐t❤ ♦♥❡ ❝♦♥st❛♥t✲t✐♠❡ ▲❈P q✉❡r②✳ ❚❛❜❧❡ ✻ s❤♦✇s t❤❡ ❛❧❣♦r✐t❤♠✬s ♦✉t♣✉t ❢♦r t❤❡ str✐♥❣s ❛❜❝❝❜❛✱ ❛❜❜❛❜❛✱ ❝❛❜❜❛❞✱ ❛♥❞ ❛❜❞❝❜❛✳ ❚❤❡ ✜rst ✐s ❢♦✉♥❞ t♦ ❜❡ ❛ ♣❛❧✐♥❞r♦♠❡ ✇✐t❤ ❛ r❛❞✐✉s ♦❢ t❤r❡❡ ✭❞✐❛♠❡t❡r ♦❢ s✐①✮✱ ❛s t❤❡ ▲❈P ♦❢ r[4] ✇✐t❤ rR[4] ✐s t❤r❡❡❀ t❤❡ s❡❝♦♥❞ ✐s ❢♦✉♥❞ t♦ ❝♦♥t❛✐♥ ❛ ♣❛❧✐♥❞r♦♠❡ ✇✐t❤ ❛ r❛❞✐✉s ♦❢ t✇♦ ✭❞✐❛♠❡t❡r ♦❢ ❢♦✉r✮✱ ❛s t❤❡ ▲❈P ♦❢ r[3] ✇✐t❤ rR[5] ✐s t✇♦❀ t❤❡ t❤✐r❞ ✐s ❢♦✉♥❞ t♦ ❝♦♥t❛✐♥ ❛ ♣❛❧✐♥❞r♦♠❡ ✇✐t❤ ❛ r❛❞✐✉s ♦❢ t✇♦ ✭❞✐❛♠❡t❡r ♦❢ ❢♦✉r✮✱ ❛s t❤❡ ▲❈P ♦❢ r[4] ✇✐t❤ rR[4] ✐s t✇♦❀ ❛♥❞ t❤❡ ❧❛st ✐s ♥♦t ❢♦✉♥❞ t♦ ♥♦t ❝♦♥t❛✐♥ ❛ ♣❛❧✐♥❞r♦♠❡✱ s✐♥❝❡ ❡✈❡r② ▲❈P q✉❡r② r❡t✉r♥s ❛ ③❡r♦✳ ✷✶

✸✳✸ ❱❛r✐❛t✐♦♥s ♦❢ P❛❧✐♥❞r♦♠❡s

✸✳✸✳✶ P❛❧st❛r ❬✶✽❪ ❛♥❞ ❬✶✷❪ ❢♦❝✉s ♦♥ ✜♥❞✐♥❣ ♣❛❧st❛rs✱ ✇❤✐❝❤✱ ❧♦♦s❡❧②✱ ✐s t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ♣❛❧✐♥❞r♦♠❡s✳ ■♥ t❤❡ ❢♦r♠❡r✱ ❛ ♣❛❧st❛r ✐s t❤❡ ❝♦♥❝❛t❡♥❛t✐♦♥ ♦❢ ❡✈❡♥✲❧❡♥❣t❤ ♣❛❧✐♥✲ ❞r♦♠❡s✿ ❛ str✐♥❣ ✐s ❛ ♣❛❧st❛r ✐✛ r = a1, . . . , an✱ ✇❤❡r❡ ❡❛❝❤ ai✱ 1 ≤ i ≤ n✱ ✐s ❛ ♣❛❧✐♥❞r♦♠❡ ♦❢ ❡✈❡♥ ❧❡♥❣t❤✳ ◆♦t❡ t❤❛t t❤❡ str✐♥❣ ♦❢ ❧❡♥❣t❤ ③❡r♦ ✐s ❛ ♣❛❧st❛r✳ ■♥ t❤❡ ❧❛tt❡r✱ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ❛ ♣❛❧st❛r ✐s ❞❡✜♥❡❞ ❞✐✛❡r❡♥t❧②✳ ❆ str✐♥❣ ✐s ❛ ♣❛❧st❛r ✐✛ r = a1, . . . , an✱ ✇❤❡r❡ ❡❛❝❤ ai✱ 1 ≤ i ≤ n✱ ✐s ❛ ♣❛❧✐♥❞r♦♠❡✱ ♦❢ ❛♥② ❧❡♥❣t❤ ✭❡✈❡♥ ♦r ♦❞❞✮✱ ❛♥❞ |r| ≥ 2✳ ✸✳✸✳✷

• ❛♣♣❡❞ P❛❧✐♥❞r♦♠❡

❆♥♦t❤❡r r❡s❡❛r❝❤ t♦♣✐❝ ✐s t❤❛t ♦❢ ❣❛♣♣❡❞ ♣❛❧✐♥❞r♦♠❡s✳ ❆ ❣❛♣♣❡❞ ♣❛❧✐♥❞r♦♠❡ ✐s ♦❢ t❤❡ ❢♦r♠ r = uauR✱ ✇❤❡r❡ u ✐s ❛ str✐♥❣ ❛♥❞ uR ✐s ✐ts r❡✈❡rs❡❀ |a| ≥ 2✳ ❚❤❡r❡ ❛r❡ t✇♦ ❝❧❛ss❡s ♦❢ ❣❛♣♣❡❞ ♣❛❧✐♥❞r♦♠❡s✿ ✶✳ ❧♦♥❣✲❛r♠❡❞✿ |a| ≤ |u| ✲ ✐♥ ♦t❤❡r ✇♦r❞s✱ ✇❤❡r❡ t❤❡ ❧❡♥❣t❤ ♦❢ ❡❛❝❤ ❛r♠ ♠✉st ❡①❝❡❡❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❣❛♣ ✷✳ ❧❡♥❣t❤✲❝♦♥str❛✐♥❡❞✿ ❛ss✉♠✐♥❣ ♣r❡✲❞❡✜♥❡❞ ❝♦♥st❛♥ts ▼✐♥●❛♣✱ ▼❛①●❛♣ ✭▼✐♥●❛♣ ≤ ▼❛①●❛♣✮✱ ❛♥❞ ▼✐♥❆r♠✱ ▼✐♥●❛♣ ≤ |a| ≤ ▼❛①●❛♣ ❛♥❞ ▼✐✲ ♥❆r♠ ≤ |u| ✲ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❣❛♣ ✐s ❝♦♥str❛✐♥❡❞ ❜② t✇♦ s✐③❡s ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ ❡❛❝❤ ❛r♠ ✐s ❛t ❧❡❛st ❛ ❝❡rt❛✐♥ s✐③❡ ▲✐❦❡ ❜❡❢♦r❡✱ ❣❛♣♣❡❞ ♣❛❧✐♥❞r♦♠❡s t❤❛t ❛r❡ ♠❛①✐♠❛❧ ❛r❡ ♦❢ ✐♥t❡r❡st✳ ❬✷✷❪ ❤❛s ♦♥❡ ❛❧❣♦r✐t❤♠ ❢♦r ❡❛❝❤ ♦❢ t❤❡ t✇♦ ❝❧❛ss❡s❀ ❜♦t❤ r✉♥ ✐♥ t✐♠❡ O(n+S)✱ ✇❤❡r❡ n ✐s t❤❡ s✐③❡ ♦❢ t❤❡ ✐♥♣✉t ❛♥❞ S ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♦✉t♣✉t ❣❛♣♣❡❞ ♣❛❧✐♥❞r♦♠❡s✳ ❍♦✇✲ ❡✈❡r✱ t❤❡ ❛❧❣♦r✐t❤♠ ❢♦r ❧♦♥❣✲❛r♠❡❞ ♣❛❧✐♥❞r♦♠❡s ✐s ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ❛❧♣❤❛❜❡t s✐③❡ ❜❡✐♥❣ ❝♦♥st❛♥t✱ ✇❤✐❧❡ t❤❡ ❛❧❣♦r✐t❤♠ ❢♦r ❧❡♥❣t❤✲❝♦♥str❛✐♥❡❞ ♣❛❧✐♥❞r♦♠❡s ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❛❧♣❤❛❜❡t s✐③❡✳ ✸✳✸✳✸ ❆♣♣r♦①✐♠❛t❡ ●❛♣♣❡❞ P❛❧✐♥❞r♦♠❡ ❆ ❢✉rt❤❡r st❡♣ ✐s ❛♣♣r♦①✐♠❛t❡ ❣❛♣♣❡❞ ♣❛❧✐♥❞r♦♠❡s✳ ❚❤✐s ✐s ✇❤❡♥ t❤❡ ♣❛❧✐♥❞r♦♠❡ ✐s ❣❛♣♣❡❞✱ ❜✉t t❤❡ ❛r♠s ❛r❡ ❛❧❧♦✇❡❞ k✱ ❛ ✜①❡❞ ♥✉♠❜❡r✱ ♦❢ ❡❞✐t ❞✐st❛♥❝❡ ❡rr♦rs ❢r♦♠ ♦♥❡ ❛r♠ t♦ t❤❡ ♦t❤❡r✳ ❚❤❡ ✜rst ❛❧❣♦r✐t❤♠ t❤❛t ❞✐s❝✉ss❡s ❛♣♣r♦①✐♠❛t❡ ❣❛♣♣❡❞ ♣❛❧✐♥❞r♦♠❡s ✐s ❬✸✵❪✳ ■t r❡q✉✐r❡s t❤❛t a✱ t❤❡ ❣❛♣✱ ✐s ❛ ♣r❡❞❡t❡r♠✐♥❡❞ s✐③❡✱ ❛♥❞ ✐t ❤❛s ❛ r✉♥t✐♠❡ ♦❢ O(k2n)✱ ✇❤❡r❡ n ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ str✐♥❣✳ ❆♥ ❛❧❣♦r✐t❤♠ t❤❛t s♦❧✈❡s t❤❡ s❛♠❡ ♣r♦❜❧❡♠✱ ❜✉t ✇✐t❤ ❛♥ ✐♠♣r♦✈❡❞ r✉♥t✐♠❡✱ ✐s ❬✶✼❪❀ ✐ts r✉♥t✐♠❡ ✐s O(kn)✳

✸✳✹ ❆♣♣r♦①✐♠❛t❡ P❛❧✐♥❞r♦♠❡s ✐♥ ❘▲❊✲❝♦♠♣r❡ss❡❞ ❉❛t❛

❉✉❡ t♦ t❤❡ ❡①♣❧♦s✐♦♥ ✐♥ t❤❡ ❛♠♦✉♥t ♦❢ ❞❛t❛ t❤❛t ✐s st♦r❡❞✱ t❤❡r❡ ♥♦✇ ❡①✐sts t❤❡ ♥❡❡❞ t♦ st♦r❡ ❞❛t❛ ✐♥ ❛ ❝♦♠♣r❡ss❡❞ ❢♦r♠✱ s✉❝❤ ❛s ✐♥ t❤❡ ❢♦r♠ ♦❢ r✉♥✲❧❡♥❣t❤ ✷✷

♣♦s✐t✐♦♥ ✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶✵ ✶✶ ✶✷ ✶✸ ✶✹ ✶✺ ✶✻ ✶✼ ✶✽ ❝❤❛r❛❝t❡r ❛ ❛ ❛ ❜ ❛ ❛ ❝ ❝ ❛ ❛ ❛ ❜ ❝ ❝ ❛ ❛ ❜ ❜ r✉♥ X1 X2 X3 X4 X5 X6 X7 X8 X9 ❚❛❜❧❡ ✼✿ A ❢♦r r ❂ ❛❛❛❜❛❛❝❝❛❛❛❜❝❝❛❛❜❜ ❡♥❝♦❞✐♥❣ ✭❘▲❊✮✱ ▲❡♠♣❡❧✲❩✐✈ ❡♥❝♦❞✐♥❣✱ ❛♥❞ str❛✐❣❤t✲❧✐♥❡ ♣r♦❣r❛♠s ✭❙▲P✮✳ ❍♦✇✲ ❡✈❡r✱ ❛ ♣♦t❡♥t✐❛❧ ❞r❛✇❜❛❝❦ ♦❢ ❞❛t❛ ❝♦♠♣r❡ss✐♦♥ ✐s t❤❛t ❝❡rt❛✐♥ ❛❧❣♦r✐t❤♠s ♠❛② ♥❡❡❞ t❤❡ ❞❛t❛ t♦ ❜❡ ❞❡❝♦♠♣r❡ss❡❞✱ ❜❡❢♦r❡ t❤❡ ❛❧❣♦r✐t❤♠ ❝❛♥ r✉♥✳ ❲❡ ✇✐❧❧ ❞✐s❝✉ss ❛ ♣❛❧✐♥❞r♦♠❡✲r❡❧❛t❡❞ ❛❧❣♦r✐t❤♠ t❤❛t ✇♦r❦s ✇✐t❤ ❞❛t❛ t❤❛t ✐s ❝♦♠♣r❡ss❡❞ ✉s✐♥❣ t❤❡ ❘▲❊ ❞❛t❛ ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡✳ ❘▲❊ ✐s ❛ s✐♠♣❧❡ s❝❤❡♠❡ ✇❤✐❝❤ ❡♥❝♦❞❡s ❛ str✐♥❣ ❛s r✉♥s✱ ♠❛❞❡ ♦❢ ❝♦♥s❡❝✉t✐✈❡ ❛♥❞ ✐❞❡♥t✐❝❛❧ s②♠❜♦❧s✳ ❋♦r ✐♥st❛♥❝❡✱ ✐♥ ❘▲❊✲❢♦r♠✱ ❜❜❝❝❝❝❡❞❞❛❛❛❛❛ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s b2c4e1d2a5 ❛♥❞ ❛s b2c2c2e1d2a2a2a1✱ t❤♦✉❣❤ t❤❡ s❡❝♦♥❞ ✇❛② ✐s ♥♦t ♦♣t✐♠❛❧✳ ❬✼❪ ♣r❡s❡♥ts t✇♦ ❛❧❣♦r✐t❤♠s ✇✐t❤ t❤❡ s❛♠❡ ❣♦❛❧✿ t♦ ✜♥❞ t❤❡ ❛♣♣r♦①✐♠❛t❡ ✲ ✉♥❞❡r t❤❡ ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ ♠❡tr✐❝ ✲ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t ❛ ❣✐✈❡♥ ♣♦s✐t✐♦♥✱ ✇❤❡♥ t❤❡ str✐♥❣ ✐s ✐♥ ♦♣t✐♠❛❧ ❘▲❊✲❢♦r♠✱ ✇✐t❤♦✉t ❞❡❝♦♠♣r❡ss✐♥❣ t❤❡ str✐♥❣✳ ❚❤❡ ✜rst ❛❧❣♦r✐t❤♠ ✐s ❜❡tt❡r ✇❤❡♥ k✱ t❤❡ ♥✉♠❜❡r ♦❢ ❍❛♠♠✐♥❣ ❞✐st❛♥❝❡ ♠✐s✲ ♠❛t❝❤❡s ❛❧❧♦✇❡❞ ❜❡t✇❡❡♥ t❤❡ ❛r♠s✱ ✐s s♠❛❧❧✱ ❛s ✐ts r✉♥t✐♠❡ r❡❧✐❡s ♦♥ k❀ t❤❡ s❡❝♦♥❞ ❛❧❣♦r✐t❤♠✬s r✉♥t✐♠❡ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ k✳ ❚♦ ❜❡ s♣❡❝✐✜❝✱ t❤❡ ✜rst ❛❧❣♦r✐t❤♠ ❤❛s ❛♥ O(n log n) ♣r❡♣r♦❝❡ss✐♥❣ t✐♠❡✱ O(log n + min{k, n}) r❡tr✐❡✈❛❧ t✐♠❡✱ ❛♥❞ ✉s❡s O(n) s♣❛❝❡✱ ✇❤✐❧❡ t❤❡ s❡❝♦♥❞ ❤❛s ❛♥ O(n2 log n) ♣r❡♣r♦❝❡ss✐♥❣ t✐♠❡✱ O(log2 n) r❡tr✐❡✈❛❧ t✐♠❡✱ ❛♥❞ ✉s❡s O(n2 log n) s♣❛❝❡✳ ❘❡❣❛r❞❧❡ss ♦❢ ✇❤✐❝❤ ❛❧❣♦r✐t❤♠ ✐s ❝❤♦s❡♥✱ t❤❡ ✐♥♣✉ts ❛r❡ t❤❡ s❛♠❡✿ ❛ str✐♥❣ r ♦❢ ❧❡♥❣t❤ n ✐♥ ✐ts ♦♣t✐♠❛❧ ❘▲❊✲❢♦r♠✱ ❛ s♣❡❝✐✜❡❞ ❝❡♥t❡r ♣♦s✐t✐♦♥✱ ❛♥❞ k✳ ❲❡ ✇✐❧❧ ❞✐s❝✉ss t❤❡ ✜rst ❛❧❣♦r✐t❤♠ ✐♥ ❞❡t❛✐❧✳ ■♥ t❤❡ ♣r❡♣r♦❝❡ss✐♥❣ st❛❣❡✱ ❛♥ ❛rr❛② A[1 · · · m] ✐s ❝♦♥str✉❝t❡❞✱ ✇❤❡r❡ m ✐s t❤❡ ♥✉♠❜❡r ♦❢ r✉♥s ✐♥ t❤❡ ❝♦♠♣r❡ss❡❞ str✐♥❣✳ ❚❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ ❛rr❛② ❛r❡ t❤❡ st❛rt ♣♦s✐t✐♦♥s ♦❢ t❤❡ r✉♥✱ ✇❤❡♥ ❞❡❝♦♠♣r❡ss❡❞✳ A ❢♦r r ❂ ❛❛❛❜❛❛❝❝❛❛❛❜❝❝❛❛❜❜ ✐s s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✼✳ r ✐s ❝♦♠♣r❡ss❡❞ ✐♥t♦ ❘▲❊ ✭r✮ ❂ X1X2 . . . X9✱ ✇❤✐❝❤ ❤❛s m = 9 r✉♥s✳ ❆ q✉❡r② ♦❢ ▲❈P ✭❘▲❊ ✭r✮✱ ✺✱ ✻✮ r❡t✉r♥s t❤❡ ✈❛❧✉❡ ♦❢ t✇♦✱ s✐♥❝❡ X4 = X7 ❛♥❞ X3 = X8 ✭❢♦r ❡❛❝❤ ♦❢ t❤❡s❡ ♣❛✐rs✱ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ❝❤❛r❛❝t❡r ✭❝ ❛♥❞ ❛✱ r❡s♣❡❝t✐✈❡❧②✮✱ ❛♥❞ t❤❡② ❤❛✈❡ t❤❡ s❛♠❡ ♥✉♠❜❡r ♦❢ t❤❛t ❝❤❛r❛❝t❡r✮✱ ❜✉t X2 = X9✳ A✬s ❝♦♥str✉❝t✐♦♥ ✐s ❜❛s❡❞ ♦♥ t❤❡ ❝♦♥str✉❝t✐♦♥ ♦❢ ❛ ●❙❚ ❢♦r ❘▲❊ ✭r✮ ❛♥❞ ♦❢ ❘▲❊ ✭rR✮✳ ❙✐♥❝❡ t❤❡s❡ ❛r❡ str✐♥❣s ♦✈❡r ❛♥ ✉♥❜♦✉♥❞❡❞ ❛❧♣❤❛❜❡t✱ ✐t t❛❦❡s O(n log n) t✐♠❡ ❛♥❞ O(n) s♣❛❝❡✳ ❆❧s♦ ✐♥ t❤✐s st❛❣❡✱ t❤❡ ❞❛t❛ ✐s ♣r❡♣r♦❝❡ss❡❞ t♦ s✉♣♣♦rt ❝♦♥st❛♥t✲t✐♠❡ ▲❈P q✉❡r✐❡s✳ ❚❤✐s t❛❦❡s O(n) t✐♠❡ ❛♥❞ O(n) s♣❛❝❡✳ ❖✈❡r❛❧❧ t❤❡ ♣r❡♣r♦❝❡ss✐♥❣ st❛❣❡ t❛❦❡s O(n log n) t✐♠❡ ❛♥❞ O(n) s♣❛❝❡✳ ❚❤❡ s❡❝♦♥❞ st❛❣❡ ✐s t❤❡ r✉♥♥✐♥❣ ♦❢ t❤❡ ❛❧❣♦r✐t❤♠✳ ■t r✉♥s ❛s ❢♦❧❧♦✇s✿ ♣❡r❢♦r♠ ❛ ❜✐♥❛r② s❡❛r❝❤ ♦♥ ❘▲❊ ✭r✮✱ ✐♥ ♦r❞❡r t♦ ✜♥❞ ✐♥ ✇❤✐❝❤ r✉♥ t❤❡ q✉❡r✐❡❞ ❝❡♥t❡r ❧✐❡s✳ ❚❤❡♥✱ t❤❡ ❛❧❣♦r✐t❤♠ ♣❡r❢♦r♠s ✐t❡r❛t✐♦♥s✱ ✐♥ ✇❤✐❝❤ ✐t tr✐❡s t♦ ❡①t❡♥❞ t❤❡ ♣❛❧✐♥❞r♦♠❡ ❢r♦♠ t❤❡ ❝❡♥t❡r ♣♦s✐t✐♦♥ ♦✉t✇❛r❞✱ ❛s ♠✉❝❤ ❛s ♣♦ss✐❜❧❡✳ ❆♥ ✐t❡r❛t✐♦♥ ❡♥❞s ✇❤❡♥ ✐t ❤✐ts ❛ r✉♥ ❜♦✉♥❞❛r② ♦♥ ❡✐t❤❡r t❤❡ ❧❡❢t ♦r t❤❡ r✐❣❤t✳ ■❢ t❤❡ t✇♦ ✷✸

❛r❡ ❤✐t s✐♠✉❧t❛♥❡♦✉s❧②✱ t❤❡♥ ❛♥ ❛❞❞✐t✐♦♥❛❧ ▲❈P q✉❡r② ✐s ♣❡r❢♦r♠❡❞✱ s♦ t❤❛t t❤❡ ❛❧❣♦r✐t❤♠ ❝❛♥ ✏❥✉♠♣✑ ♦✈❡r ♦t❤❡r r✉♥s✱ ✐♥ ♦r❞❡r t♦ ❡①t❡♥❞ t❤❡ ♣❛❧✐♥❞r♦♠❡ ❡✈❡♥ ♠♦r❡ t♦ ❜♦t❤ s✐❞❡s✳ ❚❤❡ ✐t❡r❛t✐♦♥s ❝♦♥t✐♥✉❡✱ ✉♥t✐❧ ❡✐t❤❡r k ♠✐s♠❛t❝❤❡s ❛r❡ ✉s❡❞ ♦r ✉♥t✐❧ t❤❡ ❡①t❡♥s✐♦♥ ❤✐ts t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ str✐♥❣ ✭♣♦s✐t✐♦♥ ✶ ♦r ♣♦s✐t✐♦♥ n✮✳ ❲❤❡♥ ✐♥♣✉t t♦ t❤❡ ❛❧❣♦r✐t❤♠ ✐s t❤❡ ❘▲❊ ✭r✮ ❢♦r ❛❛❛❜❛❛❝❝❛❛❛❜❝❝❛❛❜❜ ✭s❡❡ t❛❜❧❡ ✼✮✱ ❛ ❝❡♥t❡r ♣♦s✐t✐♦♥ ♦❢ ✶✵✳✺✱ ❛♥❞ k = 1 ❛❧❧♦✇❡❞ ♠✐s♠❛t❝❤❡s✱ t❤❡ ❛❧❣♦r✐t❤♠ ♣❡r❢♦r♠s ❛ ❜✐♥❛r② s❡❛r❝❤ ❛♥❞ ✜♥❞s t❤❛t ♣♦s✐t✐♦♥ ✶✵✳✺ ❧✐❡s ✐♥ X5❀ t❤❡♥ ✐t r✉♥s ❢♦r ❢♦✉r ✐t❡r❛t✐♦♥s✳ ❊❛❝❤ ✐t❡r❛t✐♦♥ ✐s ❝❧❛ss✐✜❡❞ ❛s ♠❛t❝❤ ✭s❛♠❡ ❝❤❛r❛❝t❡r ❛♥❞ s❛♠❡ ♥✉♠❜❡r ♦❢ t❤❛t ❝❤❛r❛❝t❡r✮ ♦r ♠✐s♠❛t❝❤ ✭s❛♠❡ ❝❤❛r❛❝t❡r ❜✉t ♥✉♠❜❡r ♦❢ t❤❛t ❝❤❛r❛❝t❡r ♦r ❞✐✛❡r❡♥t ❝❤❛r❛❝t❡r✮✳ ✶✳ ■t❡r❛t✐♦♥ ✶ ✲ ♠❛t❝❤✿ t❤❡ ❛❧❣♦r✐t❤♠ ❡①t❡♥❞s t♦ t❤❡ ❧❡❢t ❛♥❞ t♦ t❤❡ r✐❣❤t ♦❢ ♣♦s✐t✐♦♥ ✶✵✳✺✱ t♦ ♣♦s✐t✐♦♥s ✶✵ ❛♥❞ ✶✶✱ ❛♥❞ ❛ ✐s ✐♥ ❜♦t❤ t❤♦s❡ ♣♦s✐t✐♦♥s✳ ❚❤❡ ❛❧❣♦r✐t❤♠ ❝♦♥t✐♥✉❡s t♦ ❡①t❡♥❞ t♦ t❤❡ ❧❡❢t ❛♥❞ t♦ t❤❡ r✐❣❤t✱ ❜✉t t❤❡ r✐❣❤t ❡①t❡♥s✐♦♥ ❡♥❝♦✉♥t❡rs ❛ r✉♥ ❜♦✉♥❞❛r② ✜rst✳ ❙♦ ❢❛r✱ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t ♣♦s✐t✐♦♥ ✶✵✳✺ ✐s ✶✳ ✷✳ ■t❡r❛t✐♦♥ ✷ ✲ ♠✐s♠❛t❝❤✿ t❤❡ ❛❧❣♦r✐t❤♠ ❡①t❡♥❞s t♦ ♣♦s✐t✐♦♥ ✾ ♦♥ t❤❡ ❧❡❢t ❛♥❞ t♦ ♣♦s✐t✐♦♥ ✶✷ ♦♥ t❤❡ r✐❣❤t✳ ❚❤❡② ❞♦ ♥♦t ♠❛t❝❤✱ ❛♥❞ s♦ k = 1 ♠✐s♠❛t❝❤❡s ❛r❡ ❢♦✉♥❞✳ ❙✐♥❝❡ t❤❡ ❧❡❢t ❜♦✉♥❞❛r② ❤✐ts X5 ❛t t❤❡ s❛♠❡ t✐♠❡ t❤❛t t❤❡ r✐❣❤t ❜♦✉♥❞❛r② ❤✐ts X6✱ t❤❡r❡ ✐s ❛ q✉❡r② ♦❢ ▲❈P ✭❘▲❊ ✭r✮✱ ✺✱ ✻✮✱ ✇❤✐❝❤ r❡t✉r♥s t✇♦ ✭❛s ❡①♣❧❛✐♥❡❞ ❛❜♦✈❡✮✱ ♠❡❛♥✐♥❣ t❤❛t t✇♦ r✉♥s ✇✐❧❧ ❜❡ ❥✉♠♣❡❞ ♦✈❡r✱ ❛♥❞ t❤❡ ♥❡①t ✐t❡r❛t✐♦♥ ❧♦♦❦s ❛t X2 ❛♥❞ X9✳ ❙♦ ❢❛r✱ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t ♣♦s✐t✐♦♥ ✶✵✳✺ ✐s ✻❀ ❛❧❧ k ♠✐s♠❛t❝❤❡s ❛r❡ ✉s❡❞ ✉♣✳ ✸✳ ■t❡r❛t✐♦♥ ✸ ✲ ♠❛t❝❤✿ X2 ❛♥❞ X9 ❝♦♥t❛✐♥ t❤❡ s❛♠❡ ❝❤❛r❛❝t❡r✱ ❜✳ ❚❤❡♥ t❤❡ ❧❡❢t ❡①t❡♥s✐♦♥ ❤✐ts t❤❡ r✉♥ ❜♦✉♥❞❛r② ♦❢ X2 ❜❡❢♦r❡ t❤❡ r✐❣❤t ❡①t❡♥s✐♦♥ ❤✐ts ❛ r✉♥ ❜♦✉♥❞❛r②✳ ◆♦✇ t❤❡ r❛❞✐✉s ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t ♣♦s✐t✐♦♥ ✶✵✳✺ ✐s ♥♦✇ ❦♥♦✇♥ t♦ ❜❡ ✼✳ ✹✳ ■t❡r❛t✐♦♥ ✹ ✲ ♠✐s♠❛t❝❤✿ X1 ❛♥❞ X9 ❝♦♥t❛✐♥ ❞✐✛❡r❡♥t ❝❤❛r❛❝t❡rs✳ ❙✐♥❝❡ ♥♦ ♠♦r❡ ♠✐s♠❛t❝❤❡s ❛r❡ ❛❧❧♦✇❡❞ ✭❛❧❧ k ❤❛✈❡ ❜❡❡♥ ✉s❡❞ ✉♣ ❛❧r❡❛❞②✮✱ t❤❡ ❛❧❣♦r✐t❤♠ ♠✉st st♦♣✳ ❚❤❡ r❛❞✐✉s ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t❡ ♣❛❧✐♥❞r♦♠❡ ❝❡♥t❡r❡❞ ❛t ♣♦s✐t✐♦♥ ✶✵✳✺ r❡♠❛✐♥s ❛t ✼✳ ❚❤❡ r✉♥t✐♠❡ ♦❢ t❤❡ s❡❝♦♥❞ st❛❣❡ ✐s O(log n + min{k, n})✳ ❚❤❡ ✜rst t❡r♠ ✐s ❢♦r t❤❡ ❜✐♥❛r② s❡❛r❝❤❀ t❤❡ s❡❝♦♥❞ t❡r♠ ✐s ♣r♦✈❡♥ s❡♣❛r❛t❡❧② ❢♦r ✇❤❡♥ ❛♥ ✐t❡r❛t✐♦♥ ❤❛s ❛ ♠✐s♠❛t❝❤ ♦r ❛ ♠❛t❝❤✳ ❋♦r t❤❡ ❢♦r♠❡r✿ ✐t❡r❛t✐♦♥s ❝♦♥t✐♥✉❡✱ ✉♥t✐❧ ❡✐t❤❡r k ♠✐s♠❛t❝❤❡s ❛r❡ ✉s❡❞ ♦r ✉♥t✐❧ t❤❡ ❡①t❡♥s✐♦♥ ❤✐ts t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ str✐♥❣ ✭♣♦s✐t✐♦♥ ✶ ♦r ♣♦s✐t✐♦♥ n✮✳ ❋♦r t❤❡ ❧❛tt❡r✱ t❤❡ ❛❧❣♦r✐t❤♠ ✐s ❝♦♥str✉❝t❡❞ ✐♥ ❛ ♠❛♥♥❡r s✉❝❤ t❤❛t t❤❡r❡ ❝❛♥♥♦t ❜❡ t✇♦ ♠❛t❝❤ ✐t❡r❛t✐♦♥s ❝♦♥s❡❝✉t✐✈❡❧②✳

✹ ❘❡s❡❛r❝❤ ❉✐r❡❝t✐♦♥s

❚❤✐s s✉r✈❡② ❞✐s❝✉ss❡s t♦♣✐❝s r❡❧❛t❡❞ t♦ r❡♣❡❛ts ❛♥❞ ♣❛❧✐♥❞r♦♠❡s✳ ❬✸✼❪ ♠❛❦❡s t❤❡ ❝♦♥♥❡❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ t✇♦✿ ❛ ♣❛❧✐♥❞r♦♠❡ ✐s s✐♠♣❧② ❛♥ ✐♥✈❡rt❡❞ r❡♣❡❛t ✭♦r ✷✹

❋✐❣✉r❡ ✻✿ ❆ ✷❉ ♣❛❧✐♥❞r♦♠❡ ■❘✮✳ ❆♥ ■❘ ✐s ✇❤❡♥ t❤❡r❡ ✐s ❛ s✉❜str✐♥❣ t❤❛t ✐s ❢♦❧❧♦✇❡❞ ❜② ❛ ❣❛♣ ❛♥❞ t❤❡♥ ❜② t❤❡ s✉❜str✐♥❣✬s r❡✈❡rs❡ ✲ ✐♥ ♦t❤❡r ✇♦r❞s✱ ❛♥ ■❘ ✐s ❛ ❣❛♣♣❡❞ ♣❛❧✐♥❞r♦♠❡ ✭❛♥❞ ✇❤❡♥ t❤❡ ❣❛♣ ✐s ♦❢ ❧❡♥❣t❤ ③❡r♦✱ t❤❡♥ ❛♥ ■❘ ✐s ❛ ♣❛❧✐♥❞r♦♠❡✮✳ ❚❤✉s✱ r❡♣❡❛ts ❛♥❞ ♣❛❧✐♥❞r♦♠❡s r❡❛❧❧② ❛r❡ s✐♠✐❧❛r ♣r♦❜❧❡♠s✳ ❚❤❛t ❜❡✐♥❣ s❛✐❞✱ ✇❡ ✇❛♥t t♦ ❛♣♣❧② r❡s❡❛r❝❤ ❢r♦♠ r❡♣❡❛ts t♦✇❛r❞s ♣❛❧✐♥✲ ❞r♦♠❡s✳ ▼♦r❡♦✈❡r✱ ✇❡ ✇❛♥t t♦ ❡①♣❧♦r❡ t❤❡ ❝♦♥❝❡♣t ♦❢ ❛ ✷❉ ♣❛❧✐♥❞r♦♠❡✱ ✇❤✐❝❤ ✐s ✇❤❡♥ ❛ sq✉❛r❡ ❜❧♦❝❦ ♦❢ t❡①t r♦t❛t❡❞ ❛r♦✉♥❞ ✐ts ❝❡♥t❡r r❡s✉❧ts ✐♥ t❤❡ s❛♠❡ sq✉❛r❡ ❜❧♦❝❦ ♦❢ t❡①t✸❀ s❡❡ ✜❣✉r❡ ✻ ✭❝♦♣✐❡❞ ❢r♦♠ ❬❄❪✮ ❢♦r ❛♥ ❡①❛♠♣❧❡ ❢r♦♠ t❤❡ ❡❛r❧② ❘♦♠❛♥s✳ ❲❡ ❞✐s❝✉ss❡❞ ❛❧❣♦r✐t❤♠s t♦ ✜♥❞ ♣❛❧✐♥❞r♦♠❡s ✐♥ ❘▲❊✲❝♦♠♣r❡ss❡❞ ❞❛t❛✳ ❖✉r ❧❛st r❡s❡❛r❝❤ ❣♦❛❧ ✐s t♦ ✜♥❞ ♣❛❧✐♥❞r♦♠❡s ✲ ✇❤❡t❤❡r r❡❣✉❧❛r ♣❛❧✐♥❞r♦♠❡s ♦r ✷❉ ♦♥❡s ✲ ✐♥ ❞❛t❛ t❤❛t ✇❛s ❝♦♠♣r❡ss❡❞ ✉s✐♥❣ ♦t❤❡r ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s✱ ✇✐t❤♦✉t ❞❡❝♦♠♣r❡ss✐♥❣✳

✸❲❡ ✉♥❞❡rst❛♥❞ t❤❛t ❬✻❪ r❡❝❡♥t❧② ❞❡✜♥❡❞ ❛ ✷❉ ♣❛❧✐♥❞r♦♠❡ t♦ ❜❡ ✇❤❡♥ ❛ r❡❝t❛♥❣✉❧❛r ❛r❡❛

❤❛s ❛ ♣❛rt✐❝✉❧❛r t②♣❡ ♦❢ s②♠♠❡tr②✳ ❲❡ ❝♦♥t✐♥✉❡ t♦ ❢♦❧❧♦✇ t❤❛t ♦❢ t❤❡ ❡❛r❧② ❘♦♠❛♥s✳

✷✺

❘❡❢❡r❡♥❝❡s

❬✶❪ ❙r✐♥✐✈❛s ❆❧✉r✉ ❛♥❞ P❛♥❣ ❑♦✳ ❙✉✣① tr❡❡s ❛♥❞ s✉✣① ❛rr❛②s✳ ■♥ ❍❛♥❞❜♦♦❦ ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼♦❧❡❝✉❧❛r ❇✐♦❧♦❣②✱ ♣❛❣❡ ✶✕✷✷✳ ❈❘❈ Pr❡ss✱ ▲▲❈✱ ✷✵✵✶✳ ❬✷❪ ❆♠✐❤♦♦❞ ❆♠✐r✳ P❛tt❡r♥ ▼❛t❝❤✐♥❣ ❆❧❣♦r✐t❤♠s✱ ❚❤❡ ■♥❞❡①✐♥❣ Pr♦❜❧❡♠✳ ❇❛r✲ ■❧❛♥ ❯♥✐✈❡rs✐t②✱ ❇❛r✲■❧❛♥ ❯♥✐✈❡rs✐t②✱ ✷✵✶✶✳ ❬✸❪ ❆♠✐❤♦♦❞ ❆♠✐r✱ ❊str❡❧❧❛ ❊✐s❡♥❜❡r❣✱ ❆✈✐✈✐t ▲❡✈②✱ ❊❧② P♦r❛t✱ ❛♥❞ ◆❛t❛❧✐❡ ❙❤❛♣✐r❛✳ ❈②❝❧❡ ❞❡t❡❝t✐♦♥ ❛♥❞ ❝♦rr❡❝t✐♦♥✳ ■♥ ❙❛♠s♦♥ ❆❜r❛♠s❦②✱ ❈②r✐❧

• ❛✈♦✐❧❧❡✱ ❈❧❛✉❞❡ ❑✐r❝❤♥❡r✱ ❋r✐❡❞❤❡❧♠ ▼❡②❡r ❛✉❢ ❞❡r ❍❡✐❞❡✱ ❛♥❞ P❛✉❧ ●✳

❙♣✐r❛❦✐s✱ ❡❞✐t♦rs✱ ❆✉t♦♠❛t❛✱ ▲❛♥❣✉❛❣❡s ❛♥❞ Pr♦❣r❛♠♠✐♥❣✱ ♥✉♠❜❡r ✻✶✾✽ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡ ✹✸✕✺✹✳ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧✲ ❜❡r❣✱ ❏❛♥✉❛r② ✷✵✶✵✳ ❬✹❪ ▼✐❦❛ ❆♠✐t✱ ▼❛①✐♠❡ ❈r♦❝❤❡♠♦r❡✱ ❛♥❞ ●❛❞ ▼✳ ▲❛♥❞❛✉✳ ▲♦❝❛t✐♥❣ ❛❧❧ ♠❛①✐✲ ♠❛❧ ❛♣♣r♦①✐♠❛t❡ r✉♥s ✐♥ ❛ str✐♥❣✳ ■♥ ❏♦❤❛♥♥❡s ❋✐s❝❤❡r ❛♥❞ P❡t❡r ❙❛♥❞❡rs✱ ❡❞✐t♦rs✱ ❈♦♠❜✐♥❛t♦r✐❛❧ P❛tt❡r♥ ▼❛t❝❤✐♥❣✱ ♥✉♠❜❡r ✼✾✷✷ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡ ✶✸✕✷✼✳ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❏❛♥✉❛r② ✷✵✶✸✳ ❬✺❪ ❆❧❜❡rt♦ ❆♣♦st♦❧✐❝♦ ❛♥❞ ❋r❛♥❝♦ P✳ Pr❡♣❛r❛t❛✳ ❖♣t✐♠❛❧ ♦✛✲❧✐♥❡ ❞❡t❡❝t✐♦♥ ♦❢ r❡♣❡t✐t✐♦♥s ✐♥ ❛ str✐♥❣✳ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✷✷✭✸✮✿✷✾✼✕✸✶✺✱ ❋❡❜r✉❛r② ✶✾✽✸✳ ❬✻❪ ❱❛❧ér✐❡ ❇❡rt❤é ❛♥❞ ▲❛✉r❡♥t ❱✉✐❧❧♦♥✳ P❛❧✐♥❞r♦♠❡s ❛♥❞ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ st✉r♠✐❛♥ s❡q✉❡♥❝❡s✳ ❏♦✉r♥❛❧ ♦❢ ❆✉t♦♠❛t❛✱ ▲❛♥❣✉❛❣❡s ❛♥❞ ❈♦♠❜✐♥❛t♦r✐❝s✱ ✻✭✷✮✿✶✷✶✕✶✸✽✱ ✷✵✵✶✳ ❬✼❪ ❑✉❛♥✲❨✉ ❈❤❡♥✱ P✐♥❣✲❍✉✐ ❍s✉✱ ❛♥❞ ❑✉♥✲▼❛♦ ❈❤❛♦✳ ❊✣❝✐❡♥t r❡tr✐❡✈❛❧ ♦❢ ❛♣♣r♦①✐♠❛t❡ ♣❛❧✐♥❞r♦♠❡s ✐♥ ❛ r✉♥✲❧❡♥❣t❤ ❡♥❝♦❞❡❞ str✐♥❣✳ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠✲ ♣✉t❡r ❙❝✐❡♥❝❡✱ ✹✸✷✿✷✽✕✸✼✱ ✷✵✶✷✳ ❬✽❪ ▼❛♥♦❧✐s ❈❤r✐st♦❞♦✉❧❛❦✐s✱ ❈♦st❛s ❙✳ ■❧✐♦♣♦✉❧♦s✱ ❑✉♥s♦♦ P❛r❦✱ ❛♥❞ ❏❡♦♥❣ ❙❡♦♣ ❙✐♠✳ ❆♣♣r♦①✐♠❛t❡ s❡❡❞s ♦❢ str✐♥❣s✳ ■♥ ❚❤❡ Pr❛❣✉❡ ❙tr✐♥❣♦❧♦❣② ❈♦♥❢❡r❡♥❝❡ ✷✵✵✸✱ ♣❛❣❡ ✷✺✕✸✻✱ Pr❛❣✉❡✱ ❈③❡❝❤ ❘♣✉❜❧✐❝✱ ✷✵✵✸✳ ❬✾❪ ▼❛①✐♠❡ ❈r♦❝❤❡♠♦r❡✳ ❆♥ ♦♣t✐♠❛❧ ❛❧❣♦r✐t❤♠ ❢♦r ❝♦♠♣✉t✐♥❣ t❤❡ r❡♣❡t✐t✐♦♥s ✐♥ ❛ ✇♦r❞✳ ■♥❢♦r♠❛t✐♦♥ Pr♦❝❡ss✐♥❣ ▲❡tt❡rs✱ ✶✷✭✺✮✿✷✹✹✕✷✺✵✱ ✶✾✽✶✳ ❬✶✵❪ ❊str❡❧❧❛ ❊✐s❡♥❜❡r❣✳ ❆♣♣r♦①✐♠❛t❡ P❛tt❡r♥ ▼❛t❝❤✐♥❣✳ P❤❉ t❤❡s✐s✱ ❇❛r✲■❧❛♥ ❯♥✐✈❡rs✐t②✱ ❘❛♠❛t✲●❛♥✱ ■sr❛❡❧✱ ✷✵✶✷✳ ❬✶✶❪ ▼❛rt✐♥ ❋❛r❛❝❤✲❈♦❧t♦♥✱ P❛♦❧♦ ❋❡rr❛❣✐♥❛✱ ❛♥❞ ❙✳ ▼✉t❤✉❦r✐s❤♥❛♥✳ ❖♥ t❤❡ s♦rt✐♥❣✲❝♦♠♣❧❡①✐t② ♦❢ s✉✣① tr❡❡ ❝♦♥str✉❝t✐♦♥✳ ❏✳ ❆❈▼✱ ✹✼✭✻✮✿✾✽✼✕✶✵✶✶✱ ◆♦✈❡♠❜❡r ✷✵✵✵✳ ❬✶✷❪ ❩✈✐ ●❛❧✐❧ ❛♥❞ ❏♦❡❧ ❙❡✐❢❡r❛s✳ ❆ ❧✐♥❡❛r✲t✐♠❡ ♦♥✲❧✐♥❡ r❡❝♦❣♥✐t✐♦♥ ❛❧❣♦r✐t❤♠ ❢♦r ✧♣❛❧st❛r✧✳ ❏✳ ❆❈▼✱ ✷✺✭✶✮✿✶✵✷✕✶✶✶✱ ❏❛♥✉❛r② ✶✾✼✽✳ ✷✻

❬✶✸❪ ❘♦❜❡rt ●✐❡❣❡r✐❝❤ ❛♥❞ ❙t❡❢❛♥ ❑✉rt③✳ ❋r♦♠ ✉❦❦♦♥❡♥ t♦ ▼❝❈r❡✐❣❤t ❛♥❞ ✇❡✐♥❡r✿ ❆ ✉♥✐❢②✐♥❣ ✈✐❡✇ ♦❢ ❧✐♥❡❛r✲t✐♠❡ s✉✣① tr❡❡ ❝♦♥str✉❝t✐♦♥✳ ❆❧❣♦r✐t❤✲ ♠✐❝❛✱ ✶✾✭✸✮✿✸✸✶✕✸✺✸✱ ◆♦✈❡♠❜❡r ✶✾✾✼✳ ❬✶✹❪ ❉❛♥ ●✉s✜❡❧❞✳ ❆❧❣♦r✐t❤♠s ♦♥ ❙tr✐♥❣s✱ ❚r❡❡s✱ ❛♥❞ ❙❡q✉❡♥❝❡s ✲ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥❛❧ ❇✐♦❧♦❣②✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ✶✾✾✼✳ ❬✶✺❪ ❉♦✈ ❍❛r❡❧ ❛♥❞ ❘♦❜❡rt ❊♥❞r❡ ❚❛r❥❛♥✳ ❋❛st ❛❧❣♦r✐t❤♠s ❢♦r ✜♥❞✐♥❣ ♥❡❛r❡st ❝♦♠♠♦♥ ❛♥❝❡st♦rs✳ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❈♦♠♣✉t✐♥❣✱ ✶✸✭✷✮✱ ▼❛② ✶✾✽✹✳ ❬✶✻❪ ❆♠② ▼ ❍❛✉t❤ ❛♥❞ ❉❡❜♦r❛❤ ❆ ❏♦s❡♣❤✳ ❇❡②♦♥❞ t❛♥❞❡♠ r❡♣❡❛ts✿ ❈♦♠♣❧❡① ♣❛tt❡r♥ str✉❝t✉r❡s ❛♥❞ ❞✐st❛♥t r❡❣✐♦♥s ♦❢ s✐♠✐❧❛r✐t②✳ ❇✐♦✐♥❢♦r♠❛t✐❝s✱ ✶✽ ❙✉♣♣❧ ✶✿❙✸✶✕✸✼✱ ✷✵✵✷✳ ❬✶✼❪ P✐♥❣✲❍✉✐ ❍s✉✱ ❑✉❛♥✲❨✉ ❈❤❡♥✱ ❛♥❞ ❑✉♥✲▼❛♦ ❈❤❛♦✳ ❋✐♥❞✐♥❣ ❛❧❧ ❛♣♣r♦①✲ ✐♠❛t❡ ❣❛♣♣❡❞ ♣❛❧✐♥❞r♦♠❡s✳ ■♥ ❨✐♥❣❢❡✐ ❉♦♥❣✱ ❉✐♥❣✲❩❤✉ ❉✉✱ ❛♥❞ ❖s❝❛r ■❜❛rr❛✱ ❡❞✐t♦rs✱ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣✉t❛t✐♦♥✱ ♥✉♠❜❡r ✺✽✼✽ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡ ✶✵✽✹✕✶✵✾✸✳ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❏❛♥✉❛r② ✷✵✵✾✳ ❬✶✽❪ ❉♦♥❛❧❞ ❊✳ ❑♥✉t❤✱ ❏❛♠❡s ❍✳ ▼♦rr✐s✱ ❏r✳✱ ❛♥❞ ❱❛✉❣❤❛♥ ❘✳ Pr❛tt✳ ❋❛st ♣❛tt❡r♥ ♠❛t❝❤✐♥❣ ✐♥ str✐♥❣s✳ ❙■❆▼ ❏♦✉r♥❛❧ ♦♥ ❈♦♠♣✉t✐♥❣✱ ✻✭✷✮✿✸✷✸✕✸✺✵✱ ❏✉♥❡ ✶✾✼✼✳ ❬✶✾❪ ❘♦♠❛♥ ❑♦❧♣❛❦♦✈✱ ●❤✐③❧❛♥❡ ❇❛♥❛✱ ❛♥❞ ●r❡❣♦r② ❑✉❝❤❡r♦✈✳ ♠r❡♣s✿ ❡✣❝✐❡♥t ❛♥❞ ✢❡①✐❜❧❡ ❞❡t❡❝t✐♦♥ ♦❢ t❛♥❞❡♠ r❡♣❡❛ts ✐♥ ❉◆❆✳ ◆✉❝❧❡✐❝ ❆❝✐❞s ❘❡s❡❛r❝❤✱ ✸✶✭✶✸✮✿✸✻✼✷✕✸✻✼✽✱ ❏✉❧② ✷✵✵✸✳ ❬✷✵❪ ❘♦♠❛♥ ❑♦❧♣❛❦♦✈ ❛♥❞ ●r❡❣♦r② ❑✉❝❤❡r♦✈✳ ❋✐♥❞✐♥❣ ♠❛①✐♠❛❧ r❡♣❡t✐t✐♦♥s ✐♥ ❛ ✇♦r❞ ✐♥ ❧✐♥❡❛r t✐♠❡✳ ■♥ ✹✵t❤ ❆♥♥✉❛❧ ❙②♠♣♦s✐✉♠ ♦♥ ❋♦✉♥❞❛t✐♦♥s ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✶✾✾✾✱ ♣❛❣❡ ✺✾✻✕✻✵✹✱ ✶✾✾✾✳ ❬✷✶❪ ❘♦♠❛♥ ❑♦❧♣❛❦♦✈ ❛♥❞ ●r❡❣♦r② ❑✉❝❤❡r♦✈✳ ❋✐♥❞✐♥❣ ❛♣♣r♦①✐♠❛t❡ r❡♣❡t✐t✐♦♥s ✉♥❞❡r ❤❛♠♠✐♥❣ ❞✐st❛♥❝❡✳ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✸✵✸✭✶✮✿✶✸✺✕✶✺✻✱ ❏✉♥❡ ✷✵✵✸✳ ❬✷✷❪ ❘♦♠❛♥ ❑♦❧♣❛❦♦✈ ❛♥❞ ●r❡❣♦r② ❑✉❝❤❡r♦✈✳ ❙❡❛r❝❤✐♥❣ ❢♦r ❣❛♣♣❡❞ ♣❛❧✐♥✲ ❞r♦♠❡s✳ ■♥ P❛♦❧♦ ❋❡rr❛❣✐♥❛ ❛♥❞ ●❛❞ ▼✳ ▲❛♥❞❛✉✱ ❡❞✐t♦rs✱ ❈♦♠❜✐♥❛t♦r✐❛❧ P❛tt❡r♥ ▼❛t❝❤✐♥❣✱ ♥✉♠❜❡r ✺✵✷✾ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡ ✶✽✕✸✵✳ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❏❛♥✉❛r② ✷✵✵✽✳ ❬✷✸❪ ❱✳ ■✳ ▲❡✈❡♥s❤t❡✐♥✳ ❇✐♥❛r② ❝♦❞❡s ❝❛♣❛❜❧❡ ♦❢ ❝♦rr❡❝t✐♥❣ ❞❡❧❡t✐♦♥s✱ ✐♥s❡rt✐♦♥s ❛♥❞ r❡✈❡rs❛❧s✳ ❙♦✈✐❡t P❤②s✐❝s ❉♦❦❧❛❞②✱ ✶✵✿✼✵✼✱ ❋❡❜r✉❛r② ✶✾✻✻✳ ❬✷✹❪ ▼✐❝❤❛❡❧ ●✳ ▼❛✐♥ ❛♥❞ ❘✐❝❤❛r❞ ❏✳ ▲♦r❡♥t③✳ ❆♥ ✵ ✭♥ ❧♦❣ ♥✮ ❛❧❣♦r✐t❤♠ ❢♦r ✜♥❞✐♥❣ ❛❧❧ r❡♣❡t✐t✐♦♥s ✐♥ ❛ str✐♥❣✳ ❏✳ ❆❧❣♦r✐t❤♠s✱ ✺✭✸✮✿✹✷✷✕✹✸✷✱ ❙❡♣t❡♠❜❡r ✶✾✽✹✳ ❬✷✺❪ ●❧❡♥♥ ▼❛♥❛❝❤❡r✳ ❆ ♥❡✇ ❧✐♥❡❛r✲t✐♠❡ ♦♥✲❧✐♥❡ ❛❧❣♦r✐t❤♠ ❢♦r ✜♥❞✐♥❣ t❤❡ s♠❛❧❧✲ ❡st ✐♥✐t✐❛❧ ♣❛❧✐♥❞r♦♠❡ ♦❢ ❛ str✐♥❣✳ ❏✳ ❆❈▼✱ ✷✷✭✸✮✿✸✹✻✕✸✺✶✱ ❏✉❧② ✶✾✼✺✳ ✷✼

❬✷✻❪ ❆t❤❡❡r ❆✳ ▼❛tr♦✉❞✱ ▼✐❝❤❛❡❧ ❉✳ ❍❡♥❞②✱ ❛♥❞ ❈❤r✐st♦♣❤❡r P✳ ❚✉✤❡②✳ ❆♥ ❛❧❣♦r✐t❤♠ t♦ s♦❧✈❡ t❤❡ ♠♦t✐❢ ❛❧✐❣♥♠❡♥t ♣r♦❜❧❡♠ ❢♦r ❛♣♣r♦①✐♠❛t❡ ♥❡st❡❞ t❛♥❞❡♠ r❡♣❡❛ts✳ ■♥ ❊r✐❝ ❚❛♥♥✐❡r✱ ❡❞✐t♦r✱ ❈♦♠♣❛r❛t✐✈❡ ●❡♥♦♠✐❝s✱ ♥✉♠❜❡r ✻✸✾✽ ✐♥ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ♣❛❣❡ ✶✽✽✕✶✾✼✳ ❙♣r✐♥❣❡r ❇❡r❧✐♥ ❍❡✐❞❡❧❜❡r❣✱ ❏❛♥✉❛r② ✷✵✶✶✳ ❬✷✼❪ ❆t❤❡❡r ❆✳ ▼❛tr♦✉❞✱ ▼✐❝❤❛❡❧ ❉✳ ❍❡♥❞②✱ ❛♥❞ ❈❤r✐st♦♣❤❡r P✳ ❚✉✤❡②✳ ◆❚❘❋✐♥❞❡r✿ ❛ s♦❢t✇❛r❡ t♦♦❧ t♦ ✜♥❞ ♥❡st❡❞ t❛♥❞❡♠ r❡♣❡❛ts✳ ◆✉❝❧❡✐❝ ❆❝✐❞s ❘❡s❡❛r❝❤✱ ✹✵✭✸✮✿❡✶✼✱ ✷✵✶✷✳ ❬✷✽❪ ❊❞✇❛r❞ ▼✳ ▼❝❈r❡✐❣❤t✳ ❆ s♣❛❝❡✲❡❝♦♥♦♠✐❝❛❧ s✉✣① tr❡❡ ❝♦♥str✉❝t✐♦♥ ❛❧❣♦✲ r✐t❤♠✳ ❏✳ ❆❈▼✱ ✷✸✭✷✮✿✷✻✷✕✷✼✷✱ ❆♣r✐❧ ✶✾✼✻✳ ❬✷✾❪ ❙❤✐❤ ❏❛♥❣ P❛♥ ❛♥❞ ❘✳ ❈✳ ❚✳ ▲❡❡✳ ▲♦♦❦✐♥❣ ❢♦r ❛❧❧ ♣❛❧✐♥❞r♦♠❡s ✐♥ ❛ str✐♥❣✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✷✸r❞ ❲♦r❦s❤♦♣ ♦♥ ❈♦♠❜✐♥❛t♦r✐❛❧ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ❚❤❡♦r②✱ ♣❛❣❡ ✶✻✻✕✶✼✸✱ ❈❤❛♥❣✲❍✉❛✱ ❚❛✐✇❛♥✱ ✷✵✵✻✳ ❬✸✵❪ ❆❧❡①❛♥❞r❡ ❍✳ ▲✳ P♦rt♦ ❛♥❞ ❱❛❧♠✐r ❈✳ ❇❛r❜♦s❛✳ ❋✐♥❞✐♥❣ ❛♣♣r♦①✐♠❛t❡ ♣❛❧✐♥✲ ❞r♦♠❡s ✐♥ str✐♥❣s✳ P❛tt❡r♥ ❘❡❝♦❣♥✐t✐♦♥✱ ✸✺✭✶✶✮✿✷✺✽✶✕✷✺✾✶✱ ✷✵✵✷✳ ❬✸✶❪ ▼❛r✐❡✲❋r❛♥❝❡ ❙❛❣♦t ❛♥❞ ❊✉❣❡♥❡ ❲✳ ▼②❡rs✳ ■❞❡♥t✐❢②✐♥❣ s❛t❡❧❧✐t❡s ❛♥❞ ♣❡r✐✲ ♦❞✐❝ r❡♣❡t✐t✐♦♥s ✐♥ ❜✐♦❧♦❣✐❝❛❧ s❡q✉❡♥❝❡s✳ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♣✉t❛t✐♦♥❛❧ ❇✐♦❧♦❣②✱ ✺✿✺✸✾✕✺✺✸✱ ✶✾✾✽✳ ❬✸✷❪ ▼❛r✐❡✲❋r❛♥❝❡ ❙❛❣♦t✱ ❆❧❛✐♥ ❱✐❛r✐✱ ❛♥❞ ❍❡♥r✐ ❙♦❧❞❛♥♦✳ ▼✉❧t✐♣❧❡ ❝♦♠♣❛r✐s♦♥✿ ❛ ♣❡♣t✐❞❡ ♠❛t❝❤✐♥❣ ❛♣♣r♦❛❝❤✳ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✶✽✵✿✶✶✺✕✶✸✼✱ ✶✾✾✼✳ ❬✸✸❪ ❚❡♠♣❧❡ ❋✳ ❙♠✐t❤ ❛♥❞ ▼✐❝❤❛❡❧ ❙✳ ❲❛t❡r♠❛♥✳ ■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❝♦♠♠♦♥ ♠♦❧❡❝✉❧❛r s✉❜s❡q✉❡♥❝❡s✳ ❏♦✉r♥❛❧ ♦❢ ▼♦❧❡❝✉❧❛r ❇✐♦❧♦❣②✱ ✶✹✼✭✶✮✿✶✾✺✕✶✾✼✱ ▼❛r❝❤ ✶✾✽✶✳ ❬✸✹❪ ❉✐♥❛ ❙♦❦♦❧✱ ●❛r② ❇❡♥s♦♥✱ ❛♥❞ ❏✉st✐♥ ❚♦❥❡✐r❛✳ ❚❛♥❞❡♠ r❡♣❡❛ts ♦✈❡r t❤❡ ❡❞✐t ❞✐st❛♥❝❡✳ ❇✐♦✐♥❢♦r♠❛t✐❝s✱ ✷✸✭✷✮✿❡✸✵✕❡✸✺✱ ❏❛♥✉❛r② ✷✵✵✼✳ ❬✸✺❪ ❉✐♥❛ ❙♦❦♦❧ ❛♥❞ ❏✉st✐♥ ❚♦❥❡✐r❛✳ ❙♣❡❡❞✐♥❣ ✉♣ t❤❡ ❞❡t❡❝t✐♦♥ ♦❢ t❛♥❞❡♠ r❡♣❡❛ts ♦✈❡r t❤❡ ❡❞✐t ❞✐st❛♥❝❡✳ ❚❤❡♦r❡t✐❝❛❧ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡✱ ✷✵✶✸✳ ❬✸✻❪ ❊s❦♦ ❯❦❦♦♥❡♥✳ ❖♥✲❧✐♥❡ ❝♦♥str✉❝t✐♦♥ ♦❢ s✉✣① tr❡❡s✳ ❆❧❣♦r✐t❤♠✐❝❛✱ ✶✹✭✸✮✿✷✹✾✕✷✻✵✱ ❙❡♣t❡♠❜❡r ✶✾✾✺✳ ❬✸✼❪ P❡t❡r ❊✳ ❲❛r❜✉rt♦♥✱ ❏♦t✐ ●✐♦r❞❛♥♦✱ ❋❛♥♥② ❈❤❡✉♥❣✱ ❨❡❢❣❡♥✐② ●❡❧❢❛♥❞✱ ❛♥❞

• ❛r② ❇❡♥s♦♥✳

■♥✈❡rt❡❞ r❡♣❡❛t str✉❝t✉r❡ ♦❢ t❤❡ ❤✉♠❛♥ ❣❡♥♦♠❡✿ ❚❤❡ ①✲ ❝❤r♦♠♦s♦♠❡ ❝♦♥t❛✐♥s ❛ ♣r❡♣♦♥❞❡r❛♥❝❡ ♦❢ ❧❛r❣❡✱ ❤✐❣❤❧② ❤♦♠♦❧♦❣♦✉s ✐♥✈❡rt❡❞ r❡♣❡❛ts t❤❛t ❝♦♥t❛✐♥ t❡st❡s ❣❡♥❡s✳ ●❡♥♦♠❡ ❘❡s❡❛r❝❤✱ ✶✹✭✶✵❛✮✿✶✽✻✶✕✶✽✻✾✱ ❖❝t♦❜❡r ✷✵✵✹✳ ❬✸✽❪ P❡t❡r ❲❡✐♥❡r✳ ▲✐♥❡❛r ♣❛tt❡r♥ ♠❛t❝❤✐♥❣ ❛❧❣♦r✐t❤♠s✳ ■♥ ■❊❊❊ ❈♦♥❢❡r❡♥❝❡ ❘❡❝♦r❞ ♦❢ ✶✹t❤ ❆♥♥✉❛❧ ❙②♠♣♦s✐✉♠ ♦♥ ❙✇✐t❝❤✐♥❣ ❛♥❞ ❆✉t♦♠❛t❛ ❚❤❡♦r②✱ ✶✾✼✸✳ ❙❲❆❚ ✬✵✽✱ ♣❛❣❡ ✶✕✶✶✱ ❲❛s❤✐♥❣t♦♥✱ ❉❈✱ ✶✾✼✸✳ ✷✽