Divide and Conquer II
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1 Algorithms Divide ad Coquer II Divide ad Coquer II Desig ad Aalsis of Algorithms Adrei Bulatov
2 Algorithms Divide ad Coquer II 6- Closest Pair: The Problem The Closest Pair Problem Istace: poits i the plae Objective: Fid a pair of poits that are closest together mi
3 Algorithms Divide ad Coquer II 6-3 Algorithm Idea We have a algorithm that rus i O( time Divide ad coquer: Split the set of poits ito two halves Fid closest pairs i the halves Check if there is a closer pair crossig the border mi mi
4 Algorithms Divide ad Coquer II 6-4 Assumptios ad Costructios Assumptio: No two poits have the same -coordiate or the same -coordiate Let P be the set of poits. We create lists P ad P of poits sorted b the - ad -coordiate, respectivel This ca be doe usig the same divide ad coquer process, sortig, ad the mergig the two halves. Let Q ad R be the two halves, Q, Q ad R, R the halves ordered with respect to the - ad -coordiates Let q * *,q ad r * *,r be the closest pairs from the two halves
5 Algorithms Divide ad Coquer II 6-5 Crossig the Border δ L δ L is give b = * mi mi Let δ be the miimum of d ( q, q ad d( r, r Lemma * * If there eist q Q ad r R such that d(q,r < δ, the each of q ad r lies withi a distace δ of L * *
6 Algorithms Divide ad Coquer II 6-6 Crossig the Border (ctd Proof Suppose q ad r eist, sa, q = q, q ad We have The ad q * q r q d( q, r r * r * r q d( q, r < δ < δ ( ( r, r so each of q ad r has a -coordiate withi δ of * ad hece lies withi distace δ of the lie L r = QED
7 Algorithms Divide ad Coquer II 6-7 Crossig the Border (ctd Let S deote the set of poits belogig to the bad of width δ aroud L S Let also be the set S sorted b icreasig -coordiate Lemma There eist q Q ad r R for which d(q,r < δ if ad ol if there eist s,s S for which d(s,s < δ
8 Algorithms Divide ad Coquer II 6-8 Crossig the Border (ctd Lemma If s,s S have the propert that d(s,s < δ, the s ad s are withi 5 positios of each other i the sorted list S δ/ L δ/ boes
9 Algorithms Divide ad Coquer II 6-9 Crossig the Border (ctd Proof Deote b Z the bad of width δ aroud L We partitio Z ito boes: squares with side δ/ No two poits belog to the same bo, as it cotradicts the assumptio that miimal distace betwee two poits o the same side of L is δ Sice the distace betwee s, s is less the δ the also caot be more tha boes apart verticall QED
10 Algorithms Divide ad Coquer II 6- Algorithm Closest-Pair(P P P costruct ad /*I O( log time set ( p * *, p :=Closest-Pair-Rec( P, P
11 Algorithms Divide ad Coquer II 6- Algorithm P, Closest-Pair-Rec( P if P 3 the use brute force Q, Q, R, R costruct /*O( time * * ( q, q :=Closest-Pair-Rec( Q ( r * *, r, Q :=Closest-Pair-Rec( R, R * * * * set δ:= mi{ d( q, q, d( r, r set *:=ma -coord. of poits i Q, L:={=*} set S:={poits i P withi distace δ from L} costruct S for each s S compute dist. to et 5 poits i if s,s is the pair achievig the miimum ad d(s,s <δ retur (s,s * * * * * * else if d ( q, q < d( r, r the retur ( q, q * * else retur ( r, r } S
12 Algorithms Divide ad Coquer II 6- Closest Pair: Aalsis Theorem The Closest-Pair algorithm outputs a closest pair of poits i P Theorem The Closest-Pair algorithm rus i O( log time for a list of elemets
13 Algorithms Divide ad Coquer II 6-3 Iteger Multiplicatio How much time do we reall eed to multipl two umbers? Stadard algorithm takes O( time
14 Algorithms Divide ad Coquer II 6-4 Divide-ad-Coquer Algorithm Let ad be give umbers, digits each. Represet them as: The We eed to compute 4 smaller products Does it help? Let T( be the ruig time B the Master Theorem / /, + = + = / / / ( ( ( = + + = C T T + / ( 4 ( ( ( ( log4 O O T =
15 Algorithms Divide ad Coquer II 6-5 Divide-ad-Coquer Algorithm (ctd Reduce the umber of recursive calls = / + ( + + Compute ( + ( + ad the + = ( + ( + Does it help? Now we eed 3 recursive calls: ( + ( +,, Thus T ( 3T ( / + C B the Master Theorem log3 T ( O( = O(.59
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