Lecture 4 Recursive Algorithm Analysis. Merge Sort Solving Recurrences The Master Theorem
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1 Lecture 4 Recursive Algorithm Alysis Merge Sort Solvig Recurreces The Mster Theorem
2 Merge Sort MergeSortA, left, right) { if left < right) { mid = floorleft + right) / 2); MergeSortA, left, mid); MergeSortA, mid+, right); MergeA, left, mid, right); } } // Merge) tes two sorted surrys of A d // merges them ito sigle sorted surry of A // how log should this te?) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
3 Alysis of Merge Sort Sttemet MergeSortA, left, right) { if left < right) { mid = floorleft + right) / 2); MergeSortA, left, mid); MergeSortA, mid+, right); MergeA, left, mid, right); } } So T) = ) whe =, d 2T/2) + ) whe > So wht more succictly) is T)? Effort T) ) ) T/2) T/2) ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
4 Recurreces The expressio: T ) 2T c 2 c is recurrece. Recurrece: equtio tht descries fuctio i terms of its vlue o smller fuctios CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
5 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov Recurrece Exmples ) ) s c s ) ) s s 2 2 ) c T c T ) c T c T
6 Solvig Recurreces Itertio method Mster method Sustitutio method CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
7 Solvig Recurreces itertio method Expd the recurrece Wor some lger to express s summtio Evlute the summtio We will show severl exmples CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
8 Eg. - Lier Serch Recursively Loo t elemet costt wor, c), the serch the remiig elemets T) = T - ) + c The cost of serchig elemets is the cost of looig t elemet, plus the cost of serchig - elemets CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
9 Lier Serch cot.) We ll uwid few of these T) = T-) + c ) But, T-) = T-2) + c, from ove Sustitutig c i: T) = T-2) + c + c Gtherig lie terms T) = T-2) + 2c 2) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
10 Lier Serch cot.) Keep goig: T) = T-2) + 2c T-2) = T-3) + c T) = T-3) + c + 2c T) = T-3) + 3c 3) Oe more: T) = T-4) + 4c 4) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
11 Looig for Ptters Note, the itermedite results re eumerted We eed to pull out ptters, to write geerl expressio for the th uwidig This requires prctise Be creful while simplifyig fter sustitutio CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
12 Eg. list of itermedites Result t i th uwidig i T) = T-) + c T) = T-2) + 2c 2 T) = T-3) + 3c 3 T) = T-4) + 4c 4 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
13 Lier Serch cot.) A expressio for the th uwidig: T) = T-) + c We hve 2 vriles, d, ut we hve reltio Td) is costt c e determied) for some costt d we ow the lgorithm) Choose y coveiet # to stop. CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
14 Lier Serch cot.) Let s decide to stop t T). Whe the list to serch is empty, you re doe is coveiet, i this exmple Let - = => = Now, sustitute i everywhere for : T) = T-) + c T) = T) + c = c + c = O) T) is some costt, c ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
15 Biry Serch T) T/2) T/4) T/8) Algorithm split rry, the serch lower ½ or upper ½ T) = T/2) + c where c is some costt, the cost of fidig the middle for the split CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
16 Biry Serch cot) Let s do some quic sustitutios: T) = T/2) + c ) ut T/2) = T/4) + c, so T) = T/4) + c + c T) = T/4) + 2c 2) T/4) = T/8) + c T) = T/8) + c + 2c T) = T/8) + 3c 3) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
17 Biry Serch cot.) Result t i th uwidig i T) = T/2) + c T) = T/4) + 2c 2 T) = T/8) + 3c 3 T) = T/6) + 4c 4 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
18 Biry Serch cot) We eed to write expressio for the th uwidig i & ) Must fid ptters, chges, s i=, 2,, This c e the hrd prt Do ot get discourged! Try somethig else We ll re-write those equtios We will the eed to relte d CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
19 Biry Serch cot) Result t i th uwidig i T) = T/2) + c =T/2 ) + c T) = T/4) + 2c =T/2 2 ) + 2c 2 T) = T/8) + 3c =T/2 3 ) + 3c 3 T) = T/6) + 4c =T/2 4 ) + 4c 4 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
20 Biry Serch cot) After uwidigs: T) = T/2 ) + c Need coveiet plce to stop uwidig eed to relte & Let s pic T) = c So, /2 = => = Hmm. Esy, ut ot useful CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
21 Biry Serch cot) Oy, let s cosider T) = c So, let: /2 = => = 2 => = log 2 = lg CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
22 Biry Serch cot.) Sustitutig c i gettig rid of ): T) = T) + c lg) = c lg) + c = O lg) ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
23 T ) Itertio Method T ) f ) for for The itertio method or uwidig the recurrece): T) = f) + T ) = f) + f-) + T 2) = f) + f-) + f-2) + T 3) = = f) + f-) + f-2) + + f3) + f2) + T) = f) + f-) + f-2) + + f3) + f2) + f) + T) = f) + f-) + f-2) + + f3) + f2) + f). f) is the drivig fuctio of the recurrece Exmple: f) = ) T) = 2 ). f) = 2 ) T) = 2 ). CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
24 Itertio Method T ) 2T for for 2 T) = + 2T/2) = + 2 [/2 + 2T/2 2 )] = T/2 2 ) = [/ T/2 3 )] = T/2 3 ) = [/ T/2 4 )] = T/2 4 ) = = + 2 T/2 ) te = + log, T/2 ) = = + log ) = log ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
25 Itertio Method cotiued) for T ) 2 2T for T) = 2 + 2T/2) = [/2) 2 + 2T/2 2 )] = 2 + ½) T/2 2 ) = 2 + ½) [/2 2 ) T/2 3 )] = 2 + ½ + ½ 2 ) T/2 3 ) = 2 + ½ + ½ 2 ) [/2 3 ) 2 + 2T/2 3 )] = 2 + ½ + ½ 2 + ½ 3 ) T/2 4 ) = 2 = 2 + ½ + ½ 2 + ½ ½ ) T/2 + ) = 2 + ½ + ½ 3 + ½ ½ ) te = log, T/2 + ) = = 2 ) geometric decresig = 2 ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
26 Recursio Tree Method T ) 2T for f ) for 2 T) f) f) T/2) f/2) T/2) f/2) 2f/2) T/4) T/4) T/4) T/4) f/4) f/4) f/4) f/4) T/8) T/8) T/8) T/8) T/8) T/8) T/8) T/8) f/8) f/8) f/8) f/8) f/8) f/8) f/8) f/8) 4f/4) 8f/8) T/6) T ) f ) 2 f log 2 i f 2i 2 ) 4 f 4 ) 8 f ) i CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov 8 T/6)
27 log 4 log 2 Recursio Tree Method T ) T for T for CLAIM: T) = ). Lower oud: T) W) ovious T) T/4) /4 T/2) /2 3/4 T/6) T/8) T/8) T/4) /6 /8 /8 /4 T/64) T/32) T/32) T/6) T/32) T/6) T/6) T/8) /64 /32 /32 /6 /32 /6 /6 /8 3/4) 2 3/4) 3 T/256) T/6) T ) O ) Upper oud CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
28 FACT: Recursio Tree Method T ) T for T for 4 2 This is specil cse of the followig recurrece: T) = T) + T) + ) where < d < re rel costt prmeters. ) T) = ) [lier] 2) T) = log ) 3) T) = d ) [super-lier poly] where d > is the uique costt tht stisfies the equtio d + d =. 4 4 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
29 Solvig Recurreces exmple) s ) s) = c s ) c + s-) c + c + s-2) 2c + s-2) 2c + c + s-3) 3c + s-3) c + s-) = c + s-) T ) T c c CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
30 Solvig Recurreces exmple) s ) c s ) So fr for >= we hve s) = c + s-) Wht if =? s) = c + s) = c CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
31 Solvig Recurreces exmple) s ) c s ) So fr for >= we hve s) = c + s-) Wht if =? s) = c + s) = c So s ) c s ) Thus i geerl s) = c CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
32 Solvig Recurreces exmple) s) s ) = + s-) = s-2) s ) = s-3) = s-4) = = ) + s-) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
33 Solvig Recurreces exmple) s ) s ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov s) = + s-) = s-2) = s-3) = s-4) = = ) + s-) = i s ) i
34 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov ) ) s s So fr for >= we hve ) s i i Solvig Recurreces exmple)
35 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov ) ) s s So fr for >= we hve Wht if =? ) s i i Solvig Recurreces exmple)
36 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov ) ) s s So fr for >= we hve Wht if =? ) s i i 2 ) i s i i i Solvig Recurreces exmple)
37 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov ) ) s s So fr for >= we hve Wht if =? Thus i geerl ) s i i 2 ) i s i i i 2 ) s Solvig Recurreces exmple)
38 Solvig Recurreces exmple) T) = T ) 2T c 2 c 2T/2) + c 22T/2/2) + c) + c 2 2 T/2 2 ) + 2c + c 2 2 2T/2 2 /2) + c) + 3c 2 3 T/2 3 ) + 4c + 3c 2 3 T/2 3 ) + 7c 2 3 2T/2 3 /2) + c) + 7c 2 4 T/2 4 ) + 5c 2 T/2 ) )c CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
39 Solvig Recurreces exmple) T ) 2T So fr for > 2 we hve T) = 2 T/2 ) )c Wht if = lg? c 2 c T) = 2 lg T/2 lg ) + 2 lg - )c = T/) + - )c = T) + -)c = c + -)c = 2 - )c CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
40 Solvig Recurreces exmple) T) = T/) + c T//) + c/) + c 2 T/ 2 ) + c/ + c 2 T/ 2 ) + c/ + ) 2 T/ 2 /) + c/ 2 ) + c/ + ) 3 T/ 3 ) + c 2 / 2 ) + c/ + ) 3 T/ 3 ) + c 2 / 2 + / + ) T ) T c c T/ ) + c - / / / 2 + / + ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
41 Solvig Recurreces exmple) So we hve T) = T/ ) + c - / / 2 + / + ) For = log = T ) T c c T) = T) + c - / / 2 + / + ) = c + c - / / 2 + / + ) = c + c - / / 2 + / + ) = c / + c - / / 2 + / + ) = c / / 2 + / + ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
42 Solvig Recurreces exmple) T ) So with = log T) = c / / 2 + / + ) Wht if =? T) = c + ) = clog + ) = log ) T c c CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
43 Solvig Recurreces exmple) T ) So with = log T) = c / / 2 + / + ) Wht if <? T c c CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
44 Solvig Recurreces exmple) T ) So with = log T) = c / / 2 + / + ) Wht if <? T c c Recll tht x + x x + ) = x + -)/x-) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
45 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov So with = log T) = c / / 2 + / + ) Wht if <? Recll tht x + x x + ) = x + -)/x-) So: ) c T c T Solvig Recurreces exmple)
46 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov So with = log T) = c / / 2 + / + ) Wht if <? Recll tht x + x x + ) = x + -)/x-) So: T) = c ) = ) ) c T c T Solvig Recurreces exmple)
47 Solvig Recurreces exmple) T ) So with = log T) = c / / 2 + / + ) Wht if >? T c c CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
48 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov So with = log T) = c / / 2 + / + ) Wht if >? ) c T c T Solvig Recurreces exmple)
49 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov So with = log T) = c / / 2 + / + ) Wht if >? T) = c / ) ) c T c T Solvig Recurreces exmple)
50 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov So with = log T) = c / / 2 + / + ) Wht if >? T) = c / ) = c log / log ) = c log / ) ) c T c T Solvig Recurreces exmple)
51 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov So with = log T) = c / / 2 + / + ) Wht if >? T) = c / ) = c log / log ) = c log / ) recll logrithm fct: log = log ) c T c T Solvig Recurreces exmple)
52 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov So with = log T) = c / / 2 + / + ) Wht if >? T) = c / ) = c log / log ) = c log / ) recll logrithm fct: log = log = c log / ) = c log / ) ) c T c T Solvig Recurreces exmple)
53 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov So with = log T) = c / / 2 + / + ) Wht if >? T) = c / ) = c log / log ) = c log / ) recll logrithm fct: log = log = c log / ) = c log / ) = log ) ) c T c T
54 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov So ) c T c T T log log ) Solvig Recurreces exmple)
55 The Mster Theorem Give: divide d coquer lgorithm A lgorithm tht divides the prolem of size ito suprolems, ech of size / Let the cost of ech stge i.e., the wor to divide the prolem + comie solved suprolems) e descried y the fuctio f) The, the Mster Theorem gives us coooo for the lgorithm s ruig time: CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
56 CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov The Mster Theorem if T) = T/) + f) the W lrge for ) ) / AND ) ) ) ) log ) log log log log log c cf f f f O f f T
57 Usig The Mster Method T) = 9T/3) + =9, =3, f) = log = log 3 9 = 2 ) Sice f) = O log ), where =, cse pplies: T ) log log whe f ) O Thus the solutio is T) = 2 ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
58 Solvig Recurreces The sustitutio method CLR 4.) A... the mig good guess method Guess the form of the swer, the use iductio to fid the costts d show tht solutio wors Exmples: T) = 2T/2) + ) T) = lg ) T) = 2T/2) +??? CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
59 Solvig Recurreces The sustitutio method CLR 4.) A... the mig good guess method Guess the form of the swer, the use iductio to fid the costts d show tht solutio wors Exmples: T) = 2T/2) + ) T) = lg ) T) = 2T/2) + T) = lg ) T) = 2T/2 )+ 7) +??? CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
60 Solvig Recurreces The sustitutio method CLR 4.) A... the mig good guess method Guess the form of the swer, the use iductio to fid the costts d show tht solutio wors Exmples: T) = 2T/2) + ) T) = lg ) T) = 2T/2) + T) = lg ) T) = 2T/2+ 7) + lg ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
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