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Realistic analysis of algorithms Application to some popular algorithms Julien Clment (GREYC, CNRS, Univ. Caen, France) with Thu Hien Nguyen Thi and Brigitte Valle (and initiated with Philippe Flajolet) CanaDAM 2013

  1. “Realistic” analysis of algorithms Application to some popular algorithms Julien Clément (GREYC, CNRS, Univ. Caen, France) with Thu Hien Nguyen Thi and Brigitte Vallée (and initiated with Philippe Flajolet) CanaDAM 2013

  2. ❆♥❛❧②s✐s ♦❢ ❛❧❣♦r✐t❤♠s ❊①tr❡♠❛❧ ❝❛s❡s ✭✇♦rst✲❝❛s❡✱ ❜❡st✲❝❛s❡✮ ❖♥ t❤❡ ❛✈❡r❛❣❡ ✭♠♦r❡ r❡♣r❡s❡♥t❛t✐✈❡ ❢r♦♠ t❤❡ ✏❣❡♥❡r✐❝ ❝❛s❡✑❄✮ ■♥ ❞✐str✐❜✉t✐♦♥ ✭■♥♣✉ts ❞✐str✐❜✉t✐♦♥❄✮ ❲❤②❄ ❱❛❧✐❞❛t❡ ❛♥ ❛❧❣♦r✐t❤♠ ✭s❝❛❧✐♥❣✮✱ ❡st✐♠❛t❡ r✉♥♥✐♥❣ t✐♠❡ ❉❡s✐❣♥✐♥❣ ❛♥❞ ✉♥❞❡rst❛♥❞✐♥❣ ❛❧❣♦r✐t❤♠s ❖♣t✐♠✐③❛t✐♦♥ ❉❡✈❡❧♦♣ t♦♦❧s ❛♥❞ t❡❝❤♥✐q✉❡s ❍♦✇❄ ❆ ❧♦t ♦❢ t♦♦❧s ❛♥❞ t❡❝❤♥✐q✉❡s ✭❝♦♠❜✐♥❛t♦r✐❝s✱ ♣r♦❜❛❜✐❧✐t✐❡s✮ ❬❝❢✳ ❉✳ ❑♥✉t❤✱ P❤✳ ❋❧❛❥♦❧❡t❪ ❆♥❛❧②t✐❝ ❝♦♠❜✐♥❛t♦r✐❝s ✭❋❧❛❥♦❧❡t✲❙❡❞❣❡✇✐❝❦✮ ❋r❡❡❧② ❞♦✇♥❧♦❛❞❛❜❧❡✦ ✭❝❢✳ ❙❡❞❣❡✇✐❝❦✬s t❛❧❦✮ Introduction ❙t✉❞② t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛❧❣♦r✐t❤♠s

  3. ❲❤②❄ ❱❛❧✐❞❛t❡ ❛♥ ❛❧❣♦r✐t❤♠ ✭s❝❛❧✐♥❣✮✱ ❡st✐♠❛t❡ r✉♥♥✐♥❣ t✐♠❡ ❉❡s✐❣♥✐♥❣ ❛♥❞ ✉♥❞❡rst❛♥❞✐♥❣ ❛❧❣♦r✐t❤♠s ❖♣t✐♠✐③❛t✐♦♥ ❉❡✈❡❧♦♣ t♦♦❧s ❛♥❞ t❡❝❤♥✐q✉❡s ❍♦✇❄ ❆ ❧♦t ♦❢ t♦♦❧s ❛♥❞ t❡❝❤♥✐q✉❡s ✭❝♦♠❜✐♥❛t♦r✐❝s✱ ♣r♦❜❛❜✐❧✐t✐❡s✮ ❬❝❢✳ ❉✳ ❑♥✉t❤✱ P❤✳ ❋❧❛❥♦❧❡t❪ ❆♥❛❧②t✐❝ ❝♦♠❜✐♥❛t♦r✐❝s ✭❋❧❛❥♦❧❡t✲❙❡❞❣❡✇✐❝❦✮ ❋r❡❡❧② ❞♦✇♥❧♦❛❞❛❜❧❡✦ ✭❝❢✳ ❙❡❞❣❡✇✐❝❦✬s t❛❧❦✮ Introduction ❙t✉❞② t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛❧❣♦r✐t❤♠s ◮ ❆♥❛❧②s✐s ♦❢ ❛❧❣♦r✐t❤♠s ◮ ❊①tr❡♠❛❧ ❝❛s❡s ✭✇♦rst✲❝❛s❡✱ ❜❡st✲❝❛s❡✮ ◮ ❖♥ t❤❡ ❛✈❡r❛❣❡ ✭♠♦r❡ r❡♣r❡s❡♥t❛t✐✈❡ ❢r♦♠ t❤❡ ✏❣❡♥❡r✐❝ ❝❛s❡✑❄✮ ◮ ■♥ ❞✐str✐❜✉t✐♦♥ ✭■♥♣✉ts ❞✐str✐❜✉t✐♦♥❄✮

  4. ❍♦✇❄ ❆ ❧♦t ♦❢ t♦♦❧s ❛♥❞ t❡❝❤♥✐q✉❡s ✭❝♦♠❜✐♥❛t♦r✐❝s✱ ♣r♦❜❛❜✐❧✐t✐❡s✮ ❬❝❢✳ ❉✳ ❑♥✉t❤✱ P❤✳ ❋❧❛❥♦❧❡t❪ ❆♥❛❧②t✐❝ ❝♦♠❜✐♥❛t♦r✐❝s ✭❋❧❛❥♦❧❡t✲❙❡❞❣❡✇✐❝❦✮ ❋r❡❡❧② ❞♦✇♥❧♦❛❞❛❜❧❡✦ ✭❝❢✳ ❙❡❞❣❡✇✐❝❦✬s t❛❧❦✮ Introduction ❙t✉❞② t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛❧❣♦r✐t❤♠s ◮ ❆♥❛❧②s✐s ♦❢ ❛❧❣♦r✐t❤♠s ◮ ❊①tr❡♠❛❧ ❝❛s❡s ✭✇♦rst✲❝❛s❡✱ ❜❡st✲❝❛s❡✮ ◮ ❖♥ t❤❡ ❛✈❡r❛❣❡ ✭♠♦r❡ r❡♣r❡s❡♥t❛t✐✈❡ ❢r♦♠ t❤❡ ✏❣❡♥❡r✐❝ ❝❛s❡✑❄✮ ◮ ■♥ ❞✐str✐❜✉t✐♦♥ ✭■♥♣✉ts ❞✐str✐❜✉t✐♦♥❄✮ ◮ ❲❤②❄ ◮ ❱❛❧✐❞❛t❡ ❛♥ ❛❧❣♦r✐t❤♠ ✭s❝❛❧✐♥❣✮✱ ❡st✐♠❛t❡ r✉♥♥✐♥❣ t✐♠❡ ◮ ❉❡s✐❣♥✐♥❣ ❛♥❞ ✉♥❞❡rst❛♥❞✐♥❣ ❛❧❣♦r✐t❤♠s ◮ ❖♣t✐♠✐③❛t✐♦♥ ◮ ❉❡✈❡❧♦♣ t♦♦❧s ❛♥❞ t❡❝❤♥✐q✉❡s

  5. Introduction ❙t✉❞② t❤❡ ♣❡r❢♦r♠❛♥❝❡ ♦❢ ❛❧❣♦r✐t❤♠s ◮ ❆♥❛❧②s✐s ♦❢ ❛❧❣♦r✐t❤♠s ◮ ❊①tr❡♠❛❧ ❝❛s❡s ✭✇♦rst✲❝❛s❡✱ ❜❡st✲❝❛s❡✮ ◮ ❖♥ t❤❡ ❛✈❡r❛❣❡ ✭♠♦r❡ r❡♣r❡s❡♥t❛t✐✈❡ ❢r♦♠ t❤❡ ✏❣❡♥❡r✐❝ ❝❛s❡✑❄✮ ◮ ■♥ ❞✐str✐❜✉t✐♦♥ ✭■♥♣✉ts ❞✐str✐❜✉t✐♦♥❄✮ ◮ ❲❤②❄ ◮ ❱❛❧✐❞❛t❡ ❛♥ ❛❧❣♦r✐t❤♠ ✭s❝❛❧✐♥❣✮✱ ❡st✐♠❛t❡ r✉♥♥✐♥❣ t✐♠❡ ◮ ❉❡s✐❣♥✐♥❣ ❛♥❞ ✉♥❞❡rst❛♥❞✐♥❣ ❛❧❣♦r✐t❤♠s ◮ ❖♣t✐♠✐③❛t✐♦♥ ◮ ❉❡✈❡❧♦♣ t♦♦❧s ❛♥❞ t❡❝❤♥✐q✉❡s ◮ ❍♦✇❄ ◮ ❆ ❧♦t ♦❢ t♦♦❧s ❛♥❞ t❡❝❤♥✐q✉❡s ✭❝♦♠❜✐♥❛t♦r✐❝s✱ ♣r♦❜❛❜✐❧✐t✐❡s✮ ❬❝❢✳ ❉✳ ❑♥✉t❤✱ P❤✳ ❋❧❛❥♦❧❡t❪ ◮ ❆♥❛❧②t✐❝ ❝♦♠❜✐♥❛t♦r✐❝s ✭❋❧❛❥♦❧❡t✲❙❡❞❣❡✇✐❝❦✮ ❋r❡❡❧② ❞♦✇♥❧♦❛❞❛❜❧❡✦ ✭❝❢✳ ❙❡❞❣❡✇✐❝❦✬s t❛❧❦✮

  6. ❜❛s❡❞ ✉♣♦♥ ❝♦♠♣❛r✐s♦♥s ✭❛♥❞ s✇❛♣s✮ ❜❡t✇❡❡♥ ❦❡②s✳ ❚❤❡ ✉♥✐t❛r② ❝♦st ✐s t❤❡ ❦❡② ❝♦♠♣❛r✐s♦♥✳ ❚❤❡ ❜❡❤❛✈✐♦✉r ♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ r❡❧❛t✐✈❡ ♦r❞❡r ❜❡t✇❡❡♥ ❦❡②s✳ ❲❡ ❝❛♥ r❡str✐❝t t♦ ✶ ♥ ✳ ❚❤❡ s❡t ♦❢ ✐♥♣✉ts ✐s t❤❡♥ ♥ ✭♣❡r♠✉t❛t✐♦♥s✮ ✇✐t❤ t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥✳ ■♥ t❤✐s ❝♦♥t❡①t✱ ♠♦st t❤✐♥❣s ❛r❡ ✇❡❧❧ ✉♥❞❡rst♦♦❞ ✭❜❡st✱ ✇♦rst✱ ❛✈❡r❛❣❡ ❝❛s❡s✮✳ ❆❧❣♦r✐t❤♠s ❑ ♥ ◗✉✐❝❦❙♦rt ✷ ♥ ❧♦❣ ♥ ♥ ✷ ■♥s❙♦rt ✹ ♥ ✷ ❇✉❜❙♦rt ✷ ◗✉✐❝❦▼✐♥ ✷ ♥ ❙❡❧▼✐♥ ♥ ❑ ♥ ♠❡❛♥ ♥✉♠❜❡r ♦❢ ❦❡② ❝♦♠♣❛r✐s♦♥s Classic framework for sorting and searching algorithms ◮ ❚❤❡ ♠❛✐♥ s♦rt✐♥❣ ❛♥❞ s❡❛r❝❤✐♥❣ ❛❧❣♦r✐t❤♠s✱ ❡✳❣✳✱ ◗✉✐❝❦❙♦rt✱ ❇❙❚✲❙❡❛r❝❤✱ ■♥s❡rt✐♦♥❙♦rt✱✳✳✳ ❝♦♥s✐❞❡r ♥ ✭❞✐st✐♥❝t✮ ❦❡②s ❯ ✶ , ❯ ✷ , . . . , ❯ ♥ ❢r♦♠ t❤❡ s❛♠❡ ♦r❞❡r❡❞ s❡t Ω ✳

  7. ❚❤❡ ❜❡❤❛✈✐♦✉r ♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ r❡❧❛t✐✈❡ ♦r❞❡r ❜❡t✇❡❡♥ ❦❡②s✳ ❲❡ ❝❛♥ r❡str✐❝t t♦ ✶ ♥ ✳ ❚❤❡ s❡t ♦❢ ✐♥♣✉ts ✐s t❤❡♥ ♥ ✭♣❡r♠✉t❛t✐♦♥s✮ ✇✐t❤ t❤❡ ✉♥✐❢♦r♠ ❞✐str✐❜✉t✐♦♥✳ ■♥ t❤✐s ❝♦♥t❡①t✱ ♠♦st t❤✐♥❣s ❛r❡ ✇❡❧❧ ✉♥❞❡rst♦♦❞ ✭❜❡st✱ ✇♦rst✱ ❛✈❡r❛❣❡ ❝❛s❡s✮✳ ❆❧❣♦r✐t❤♠s ❑ ♥ ◗✉✐❝❦❙♦rt ✷ ♥ ❧♦❣ ♥ ♥ ✷ ■♥s❙♦rt ✹ ♥ ✷ ❇✉❜❙♦rt ✷ ◗✉✐❝❦▼✐♥ ✷ ♥ ❙❡❧▼✐♥ ♥ ❑ ♥ ♠❡❛♥ ♥✉♠❜❡r ♦❢ ❦❡② ❝♦♠♣❛r✐s♦♥s Classic framework for sorting and searching algorithms ◮ ❚❤❡ ♠❛✐♥ s♦rt✐♥❣ ❛♥❞ s❡❛r❝❤✐♥❣ ❛❧❣♦r✐t❤♠s✱ ❡✳❣✳✱ ◗✉✐❝❦❙♦rt✱ ❇❙❚✲❙❡❛r❝❤✱ ■♥s❡rt✐♦♥❙♦rt✱✳✳✳ ❝♦♥s✐❞❡r ♥ ✭❞✐st✐♥❝t✮ ❦❡②s ❯ ✶ , ❯ ✷ , . . . , ❯ ♥ ❢r♦♠ t❤❡ s❛♠❡ ♦r❞❡r❡❞ s❡t Ω ✳ ◮ ❜❛s❡❞ ✉♣♦♥ ❝♦♠♣❛r✐s♦♥s ✭❛♥❞ s✇❛♣s✮ ❜❡t✇❡❡♥ ❦❡②s✳ ❚❤❡ ✉♥✐t❛r② ❝♦st ✐s t❤❡ ❦❡② ❝♦♠♣❛r✐s♦♥✳

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