Divide-and-Conquer. Divide-and-Conquer 1
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1 Divide-d-Coquer Divide-d-Coquer 1
2 Outie d Redig Divide-d-coquer prdigm 5. Review Merge-sort Recurrece Equtios 5..1 tertive sustitutio Recursio trees Guess-d-test Te mster metod teger Mutipictio 5.. Divide-d-Coquer
3 Divide-d-Coquer Divide-d coquer is geer goritm desig prdigm: Divide: divide te iput dt S i two or more disjoit susets S 1, S, Recur: sove te suproems recursivey Coquer: comie te soutios or S 1,S,, ito soutio or S Te se cse or te recursio re suproems o costt size Aysis c e doe usig recurrece equtios Divide-d-Coquer 3
4 Merge-Sort Review Merge-sort o iput sequece S wit eemets cosists o tree steps: Divide: prtitio S ito two sequeces S 1 d S o out eemets ec Recur: recursivey sort S 1 d S Coquer: merge S 1 d S ito uique sorted sequece Agoritm mergesorts, C put sequece S wit eemets, comprtor C Output sequece S sorted ccordig to C S.size > 1 S 1, S prtitios, mergesorts 1, C mergesorts, C S merges 1, S Divide-d-Coquer 4
5 Recurrece Equtio Aysis Te coquer step o merge-sort cosists o mergig two sorted sequeces, ec wit eemets d impemeted y mes o douy iked ist, tkes t most steps, or some costt. Likewise, te sis cse < wi tke t most steps. Tereore, we et T deote te ruig time o merge-sort: T T < We c tereore yze te ruig time o merge-sort y idig cosed orm soutio to te ove equtio. Tt is, soutio tt s T oy o te et-d side. Divide-d-Coquer 5
6 tertive Sustitutio te itertive sustitutio, or pug-d-cug, tecique, we itertivey ppy te recurrece equtio to itse d see we c id ptter: T T T T 3 3 T T 4... i i T i Note tt se, T, cse occurs we i. Tt is, i. So, T Tus, T is O. Divide-d-Coquer 6
7 Te Recursio Tree Drw te recursio tree or te recurrece retio d ook or ptter: < T T dept 0 T s 1 size time 1 i i i Tot time st eve pus previous eves Divide-d-Coquer 7
8 Guess-d-Test Metod te guess-d-test metod, we guess cosed orm soutio d te try to prove it is true y iductio: < T T Guess: T < c. T T c c c c Wrog: we cot mke tis st ie e ess t c Divide-d-Coquer 8
9 Guess-d-Test Metod, Prt Rec te recurrece equtio: T T Guess #: T < c. T T c c c < c c c c >. So, T is O. geer, to use tis metod, you eed to ve good guess d you eed to e good t iductio proos. Divide-d-Coquer 9
10 Divide-d-Coquer 10 Mster Metod My divide-d-coquer recurrece equtios ve te orm: Te Mster Teorem: < d T d c T 1. or some provided, is te, is 3. is te, is. is te, is 1. 1 < Θ Ω Θ Θ Θ δ δ ε ε T T T O k k
11 Mster Metod, Exmpe 1 Te orm: Te Mster Teorem: provided Exmpe: T is O c T is Θ is Ω ε ε δ, te T is Θ k or some δ < 1. < d d, te T is Θ, te T is Θ, T 4T k 1 Soutio:, so cse 1 sys T is O. Divide-d-Coquer 11
12 Mster Metod, Exmpe Te orm: Te Mster Teorem: provided Exmpe: T is O c T is Θ is Ω ε ε δ, te T is Θ k or some δ < 1. < d d, te T is Θ, te T is Θ, T T k 1 Soutio: 1, so cse sys T is O. Divide-d-Coquer 1
13 Mster Metod, Exmpe 3 Te orm: Te Mster Teorem: provided Exmpe: T is O c T is Θ is Ω ε ε δ, te T is Θ T T 3 k or some δ < 1. < d d, te T is Θ, te T is Θ, k 1 Soutio: 0, so cse 3 sys T is O. Divide-d-Coquer 13
14 Mster Metod, Exmpe 4 Te orm: Te Mster Teorem: provided Exmpe: T is O c T is Θ is Ω ε ε δ, te T is Θ k or some δ < 1. < d d, te T is Θ, te T is Θ, T 8T Soutio: 3, so cse 1 sys T is O 3. k 1 Divide-d-Coquer 14
15 Mster Metod, Exmpe 5 Te orm: Te Mster Teorem: provided Exmpe: T is O c T is Θ is Ω ε ε δ, te T is Θ k or some δ < 1. < d d, te T is Θ, te T is Θ, T 9T 3 3 Soutio:, so cse 3 sys T is O 3. k 1 Divide-d-Coquer 15
16 Mster Metod, Exmpe 6 Te orm: Te Mster Teorem: provided Exmpe: T is O c T is Θ is Ω ε ε δ, te T is Θ k or some δ < 1. < d d, te T is Θ, te T is Θ, T T 1 k 1 iry serc Soutio: 0, so cse sys T is O. Divide-d-Coquer 16
17 Mster Metod, Exmpe 7 Te orm: Te Mster Teorem: provided Exmpe: T is O c T is Θ is Ω ε ε δ, te T is Θ k or some δ < 1. < d d, te T is Θ, te T is Θ, T T k 1 ep costructio Soutio: 1, so cse 1 sys T is O. Divide-d-Coquer 17
18 Divide-d-Coquer 18 tertive Proo o te Mster Teorem Usig itertive sustitutio, et us see we c id ptter: We te distiguis te tree cses s Te irst term is domit Ec prt o te summtio is equy domit Te summtio is geometric series i i i i i i T T T T T T T
19 Divide-d-Coquer 19 teger Mutipictio Agoritm: Mutipy two -it itegers d. Divide step: Spit d ito ig-order d ow-order its We c te deie * y mutipyig te prts d ddig: So, T 4T, wic impies T is O. But tt is o etter t te goritm we ered i grde scoo. * *
20 Divide-d-Coquer 0 A mproved teger Mutipictio Agoritm Agoritm: Mutipy two -it itegers d. Divide step: Spit d ito ig-order d ow-order its Oserve tt tere is deret wy to mutipy prts: So, T 3T, wic impies T is O 3, y te Mster Teorem. Tus, T is O ] [ ] [ *
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