Presentation for use with the textbook, Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, Divide-and-Conquer
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1 Presettio or use wit te textook, Agoritm Desig Appictios, y M. T. Gooric R. Tmssi, Wiey, 015 Divie--Coquer 015 Gooric Tmssi Divie--Coquer 1
2 Appictio: Mxim Sets We c visuize te vrious tre-os or optimizig twoimesio t, suc s poits represetig otes ccorig to teir poo size resturt quity, y pottig ec s twoimesio poit, x, y, were x is te poo size y is te resturt quity score. We sy tt suc poit is mximum poit i set tere is o oter poit, x, y, i tt set suc tt x x y y. Te mximum poits re te est poteti coices se o tese two imesios iig o tem is te mxim set proem. We c eiciety i te mxim poits y ivie--coquer. Here te set is {A,H,,G,D}. 015 Gooric Tmssi Divie--Coquer
3 Divie--Coquer Divie- coquer is geer goritm esig prigm: Divie: ivie te iput t S i two or more isjoit susets S 1, S, Coquer: sove te suproems recursivey Comie: 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 oe usig recurrece equtios 015 Gooric Tmssi Divie--Coquer 3
4 Merge-Sort Review Merge-sort o iput sequece S wit eemets cosists o tree steps: Divie: prtitio S ito two sequeces S 1 S o out eemets ec Coquer: recursivey sort S 1 S Comie: merge S 1 S ito uique sorte sequece Agoritm mergesorts put sequece S wit eemets Output sequece S sorte ccorig to C S.size > 1 S 1, S prtitios, mergesorts 1 mergesorts S merges 1, S 015 Gooric Tmssi Divie--Coquer 4
5 Recurrece Equtio Aysis Te coquer step o merge-sort cosists o mergig two sorte sequeces, ec wit eemets impemete y mes o ouy ike ist, tkes t most steps, or some costt. Likewise, te sis cse < wi tke t most steps. Tereore, we et T eote te ruig time o merge-sort: T T < We c tereore yze te ruig time o merge-sort y iig cose orm soutio to te ove equtio. Tt is, soutio tt s T oy o te et- sie. 015 Gooric Tmssi Divie--Coquer 5
6 tertive Sustitutio te itertive sustitutio, or pug--cug, tecique, we itertivey ppy te recurrece equtio to itse see we c i 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. 015 Gooric Tmssi Divie--Coquer 6
7 Te Recursio Tree Drw te recursio tree or te recurrece retio ook or ptter: < T T ept T s size 0 1 time 1 i i i Tot time st eve pus previous eves 015 Gooric Tmssi Divie--Coquer 7
8 Guess--Test Meto te guess--test meto, we guess cose orm soutio te try to prove it is true y iuctio: < T T Guess: T < c. T T c c c c Wrog: we cot mke tis st ie e ess t c 015 Gooric Tmssi Divie--Coquer 8
9 Guess--Test Meto, cot. 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 meto, you ee to ve goo guess you ee to e goo t iuctio proos. 015 Gooric Tmssi Divie--Coquer 9 <
10 Mster Meto My ivie--coquer recurrece equtios ve te orm: T T c < Te Mster Teorem: is O is Ω ε ε, te T, te T, te T, provie δ or someδ < 1. k k Gooric Tmssi Divie--Coquer 10
11 Mster Meto, Exmpe 1 Te orm: Te Mster Teorem: provie Exmpe: T is O is Ω c T ε ε δ, te T k, te T, te T or someδ < 1. T 4T <, k 1 Soutio:, so cse 1 sys T is O. 015 Gooric Tmssi Divie--Coquer 11
12 Mster Meto, Exmpe Te orm: Te Mster Teorem: provie Exmpe: T is O is Ω c T ε ε δ, te T, te T T T k, te T or someδ < 1. <, Soutio: 1, so cse sys T is O. k Gooric Tmssi Divie--Coquer 1
13 Mster Meto, Exmpe 3 Te orm: Te Mster Teorem: provie Exmpe: T is O is Ω c T ε ε δ, te T, te T T T 3 k, te T or someδ < 1. <, Soutio: 0, so cse 3 sys T is O. k Gooric Tmssi Divie--Coquer 13
14 Mster Meto, Exmpe 4 Te orm: Te Mster Teorem: provie Exmpe: T is O is Ω c T ε ε δ, te T k, te T, te T or someδ < 1. T 8T <, Soutio: 3, so cse 1 sys T is O 3. k Gooric Tmssi Divie--Coquer 14
15 Mster Meto, Exmpe 5 Te orm: Te Mster Teorem: provie Exmpe: T is O is Ω c T ε ε δ, te T k, te T, te T or someδ < 1. T 9T 3 <, 3 Soutio:, so cse 3 sys T is O 3. k Gooric Tmssi Divie--Coquer 15
16 Mster Meto, Exmpe 6 Te orm: Te Mster Teorem: provie Exmpe: T is O is Ω c T ε ε δ, te T k, te T, te T or someδ < 1. T T 1 <, k 1 iry serc Soutio: 0, so cse sys T is O. 015 Gooric Tmssi Divie--Coquer 16
17 Mster Meto, Exmpe 7 Te orm: Te Mster Teorem: provie Exmpe: T is O is Ω c T ε ε δ, te T k, te T, te T or someδ < 1. T T <, k 1 ep costructio Soutio: 1, so cse 1 sys T is O. 015 Gooric Tmssi Divie--Coquer 17
18 015 Gooric Tmssi Divie--Coquer 18 Sketc o Proo o te Mster Teorem Usig itertive sustitutio, et us see we c i ptter: We te istiguis te tree cses s Te irst term is omit Ec prt o te summtio is equy omit Te summtio is geometric series i i i i i i T T T T T T T
19 015 Gooric Tmssi Divie--Coquer 19 teger Mutipictio Agoritm: Mutipy two -it itegers. Divie step: Spit ito ig-orer ow-orer its We c te eie * y mutipyig te prts ig: So, T 4T, wic impies T is O. But tt is o etter t te goritm we ere i gre scoo. * *
20 015 Gooric Tmssi Divie--Coquer 0 A mprove teger Mutipictio Agoritm Agoritm: Mutipy two -it itegers. Divie step: Spit ito ig-orer ow-orer its Oserve tt tere is eret wy to mutipy prts: So, T 3T, wic impies T is O 3, y te Mster Teorem. Tus, T is O ] [ ] [ *
21 Sovig te Mxim Set Proem Let us ow retur to te proem o iig mxim set or set, S, o poits i te pe. Tis proem is motivte rom muti-ojective optimiztio, were we re itereste i optimizig coices tt epe o mutipe vries. For istce, i te itrouctio we use te exmpe o someoe wisig to optimize otes se o te two vries o poo size resturt quity. A poit is mximum poit i S tere is o oter poit, x, y, i S suc tt x x y y. 015 Gooric Tmssi Divie--Coquer 1
22 Divie--Coquer Soutio Give set, S, o poits i te pe, tere is simpe ivie--coquer goritm or costructig te mxim set o poits i S. 1, te mxim set is just S itse. Oterwise, et p e te mei poit i S ccorig to exicogrpic orerig o te poits i S, tt is, were we orer se primriy o x- coorites te y y-coorites tere re ties. Next, we recursivey sove te mxim-set proem or te set o poits o te et o tis ie so or te poits o te rigt. Give tese soutios, te mxim set o poits o te rigt re so mxim poits or S. But some o te mxim poits or te et set migt e omite y poit rom te rigt, mey te poit, q, tt is etmost. So te we o sc o te et set o mxim, removig y poits tt re omite y q, uti recig te poit were q s omice extes. Te uio o remiig set o mxim rom te et te mxim set rom te rigt is te set o mxim or S. 015 Gooric Tmssi Divie--Coquer
23 Exmpe or te Comie Step 015 Gooric Tmssi Divie--Coquer 3
24 Pseuo-coe 015 Gooric Tmssi Divie--Coquer 4
25 A Litte mpemettio Deti Beore we yze te ivie--coquer mxim-set goritm, tere is itte impemettio eti tt we ee to work out. Nmey, tere is te issue o ow to eiciety i te poit, p, tt is te mei poit i exicogrpic orerig o te poits i S ccorig to teir x, y-coorites. Tere re two immeite possiiities: Oe coice is to use ier-time mei-iig goritm, suc s tt give i Sectio 9.. Tis cieves goo symptotic ruig time, ut s some impemettio compexity. Aoter coice is to sort te poits i S exicogrpicy y teir x, y-coorites s preprocessig step, prior to cig te MxmSet goritm o S. Give tis preprocessig step, te mei poit is simpy te poit i te mie o te ist. 015 Gooric Tmssi Divie--Coquer 5
26 Aysis eiter cse, te rest o te o-recursive steps c e perorme i O time, so tis impies tt, igorig oor ceiig uctios s owe y te ysis o Exercise C-11.5, te ruig time or te ivie--coquer mxim-set goritm c e specie s oows were is costt: T T Tus, ccorig to te Mster Teorem, tis goritm rus i O time. < 015 Gooric Tmssi Divie--Coquer 6
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