Recursion. Algorithm : Design & Analysis [3]

Transcription

1 Recusio Algoithm : Desig & Aalysis []

2 I the last class Asymptotic gowth ate he Sets Ο, Ω ad Θ Complexity Class A Example: Maximum Susequece Sum Impovemet of Algoithm Compaiso of Asymptotic Behavio Aothe Example: Biay Seach Biay Seach Is Optimal

3 Recusio Recusive Pocedues Povig Coectess of Recusive Pocedues Deivig ecuece equatios Solutio of the Recuece equatios Guess ad povig Recusio tee Maste theoem Divide-ad-coque

4 Recusio as a hiig Way Cuttig the plae How may sectios ca e geeated at most y staight lies with ifiite legth. Itesectig all - existig lies to get as most sectios as possile L0 L L- Lie

5 Recusio fo Algoithm Computig! if the etu else etu Fac-* M0 ad MM- fo >0 citical opeatio: multiplicatio Haoi owe Recuece elatios if the move d to peg else { Haoi-, peg, peg; move d to peg; Haoi-, peg, peg M ad MM- fo > citical opeatio: move

6 Coutig the Nume of Bit Iput: a positive decimal itege Output: the ume of iay digits i s iay epesetatio It BiCoutig it.. if if etu ; ;.. else.. etu BiCoutig div ;

7 Coectess of BiCoutig Poof y iductio Base case: if, tivial y lie. Iductive hypothesis: fo ay 0<<, BiCoutig etu the coect esult. Iductio If the lie is excuted div is a positive decimal itege so, the pecoditio fo BiCoutig is still hold, ad 0< div <, so, the iductive hypothesis applies So, the coectess the ume of it of is oe moe the that of div

8 Complexity Aalysis of BiCoutig he citical opeatio: additio he ecuece elatio 0 / >

9 Solutio y acwad sustitutios log log log that is, oegative itege, a is let Fo simplicity, ecusio equatio By the : 0

10 Smooth Fuctios Let f e a oegative evetually odeceasig fuctio defied o the set of atual umes, f is called smooth if f Θf. Note: log,, log ad α α 0 ae all smooth. Fo example: log loglog Θlog

11 Eve Smoothe Let f e a smooth fuctio, the, fo ay fixed itege, f Θf. hat is, thee exist positive costats c ad d ad a oegative itege 0 such that d f f c f fo 0. It is easy to pove that the esult holds fo fo the secod iequality : f c f fo,,... ad Fo a aitay itege, he, f f c - f, we ca use c 0., as c.

12 Smoothess Rule Let e a evetually odeceasig fuctio ad f e a smooth fuctio. If Θf fo values of that ae powes of, the Θf., Let. fo : By the pio esult. fo By the hypothsis: : Just povig the ig - Oh pat f cc f cc cf cf f c f cf No-deceasig hypothesis Pio esult No-deceasig

13 Computig the th Fioacci Nume ff 0 0 ff ff ff - - ff - - a a L 0,,,,, 5, 8,,, 4,... a m a is called liea homogeeous elatio of degee. Fo the special case of Fioacci: a a - a -,

14 Chaacteistic Equatio Fo a liea homogeeous ecuece elatio of degee the polyomial of degee is called its chaacteistic equatio. he chaacteistic equatio of liea homogeeous ecuece elatio of degee is: a a a a L x x x L 0 x x

15 Solutio of Recuece Relatio If the chaacteistic equatio x x 0 of the ecuece elatio has two distict oots s ad s, the a a a a us vs whee u ad v deped o the iitial coditios, is the explicit fomula fo the sequece. If the equatio has a sigle oot s, the, oth s ad s i the fomula aove ae eplaced y s

16 Poof of the Solutio : that pove eed We 0 : equatio Rememe a a vs us vs us vs vs us us s vs s us s vs s us vs us a a vs us x x the

17 Retu to Fioacci Sequece ff ff 0,,,,, 5, 8,,, 4,... ff ff - - ff - - Explicit fomula fo Fioacci Sequece he chaacteistic equatio is x -x-0, which has oots: s 5 ad s 5 Note: y iitial coditios f us vs ad f us vs which esults: f

18 Guess the Solutios Example: / Guess y to pove c: / / c / Howeve: c/ c / log c, / Fail! / clog / c / c, to e poved fo c lage eough c[-/] c log c log c-c c c log c c, to e poved fo c c lage log eough fo c O? O? O maye, Olog? clog, to e poved fo c lage eough

19 Recusio ee size oecusive cost / / / / /4 /4 /4 /4 /4 /4 /4 /4 he ecusio tee fo //

20 Recusio ee Rules Costuctio of a ecusio tee wo copy: use auxiliay vaiale oot ode expasio of a ode: ecusive pats: childe oecusive pats: oecusive cost the ode with ase-case size

21 Recusio tee equatio Fo ay sutee of the ecusio tee, size field of oot Σ oecusive costs of expaded odes Σ size fields of icomplete odes Example: divide-ad-coque: /c f Afte th expasio: c i i f c i

22 Evaluatio of a Recusio ee Computig the sum of the oecusive costs of all odes. Level y level though the tee dow. Kowledge of the maximum depth of the ecusio tee, that is the depth at which the size paamete educe to a ase case.

23 Recusio ee Wo copy: // Wo copy: // lg / / / / / d size /4 /4 /4 /4 /4 /4 /4 /4 At At this level: /4/44/4

24 Recusio ee fo /4 Θ c c c¼ c¼ c¼ c 6 log 4 c/6 c/6 c/6 c/6 c/6 6 c Note: log 4 log 4 log 4 Θ otal: Θ

25 Veifyig Guess y Recusive ee log log 0 log log 0 log O c c c c i i i i Θ Θ Θ < Θ c d whe d c d c d c d c / 4 / 4 / Iductive hypothesis Iductive hypothesis

26 Recusio ee fo /cf f f f/c f/c f/c f / c log c f/c f/c f/c f/c f/c f/c f/c f/c f/c f / c Θ log c Note: log c log c otal?

27 Solvig the Divide-ad-Coque he ecusio equatio fo divide-ad-coque, the geeal case:/cf Osevatios: Let ase-cases occu at depth Dleaf, the /c D, that is Dlg/lgc Let the ume of leaves of the tee e L, the L D, that is L lg/lgc. By a little algea: L E, whee Elg/lgc, called citical expoet. L lg lg c lg lg lg c lg lg c lg lg lg lg c

28 Divide-ad-Coque: the Solutio he ecusio tee has depth Dlg/ lgc, so thee ae aout that may ow-sums. he 0 th ow-sum is f, the oecusive cost of the oot. he Dth ow-sum is E, assumig ase cases cost, o Θ E i ay evet. he solutio of divide-ad-coque equatio is the o-ecusive costs of all odes i the tee, which is the sum of the ow-sums.

29 Solutio y Row-sums [Little Maste heoem] Row-sums decide the solutio of the equatio fo divide-ad-coque: Iceasig geometic seies: Θ E Costat: Θ f log Deceasig geometic seies: Θ f his ca e geealized to get a esult ot usig explicitly ow-sums.

30 Maste heoem he positive ε is is citical, esultig gaps etwee cases as as well Looseig the estictios o f Case : f O E-ε, ε>0, the: Θ E Case : f Θ E, as all ode depth cotiute aout equally: Θflog case : f Ω Eε, ε>0, ad f O Eδ, δ ε, the: Θf

31 Usig Maste heoem lg,, 0,,,,,,, 9, 9 0 applies case f E c Example applies case O f E c Example E Θ Θ Θ

32 Usig Maste heoem lg, lg 0.79, log 4,, lg applies case O f E c Example E E Θ Ω

33 Looig at the Gap /lg a,, E, flg We have fω E, ut o ε>0 satisfies fω Eε, sice lg gows slowe that ε fo ay small positive ε. So, case does t apply. Howeve, eithe case applies.

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