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 Mergesort Mergesort divide-d-coquer) Divide rry ito two hlves. A L G O R I T H M S A L G O R I T H M S divide CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
4 Mergesort Mergesort divide-d-coquer) Divide rry ito two hlves. Recursively sort ech hlf. A L G O R I T H M S A L G O R I T H M S A G L O R H I M S T divide sort CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
5 Mergesort Mergesort divide-d-coquer) Divide rry ito two hlves. Recursively sort ech hlf. Merge two hlves to me sorted whole. A L G O R I T H M S A L G O R I T H M S A G L O R H I M S T divide sort A G H I L M O R S T merge CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
6 Merge. Mergig Keep trc of smllest elemet i ech sorted hlf. Isert smllest of two elemets ito uxiliry rry. Repet util doe. smllest smllest A G L O R H I M S T A uxiliry rry CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
7 Merge. Mergig Keep trc of smllest elemet i ech sorted hlf. Isert smllest of two elemets ito uxiliry rry. Repet util doe. smllest smllest A G L O R H I M S T A G uxiliry rry CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
8 Merge. Mergig Keep trc of smllest elemet i ech sorted hlf. Isert smllest of two elemets ito uxiliry rry. Repet util doe. smllest smllest A G L O R H I M S T A G H uxiliry rry CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
9 Merge. Mergig Keep trc of smllest elemet i ech sorted hlf. Isert smllest of two elemets ito uxiliry rry. Repet util doe. smllest smllest A G L O R H I M S T A G H I uxiliry rry CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
10 Merge. Mergig Keep trc of smllest elemet i ech sorted hlf. Isert smllest of two elemets ito uxiliry rry. Repet util doe. smllest smllest A G L O R H I M S T A G H I L uxiliry rry CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
11 Merge. Mergig Keep trc of smllest elemet i ech sorted hlf. Isert smllest of two elemets ito uxiliry rry. Repet util doe. smllest smllest A G L O R H I M S T A G H I L M uxiliry rry CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
12 Merge. Mergig Keep trc of smllest elemet i ech sorted hlf. Isert smllest of two elemets ito uxiliry rry. Repet util doe. smllest smllest A G L O R H I M S T A G H I L M O uxiliry rry CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
13 Merge. Mergig Keep trc of smllest elemet i ech sorted hlf. Isert smllest of two elemets ito uxiliry rry. Repet util doe. smllest smllest A G L O R H I M S T A G H I L M O R uxiliry rry CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
14 Mergig Merge. Keep trc of smllest elemet i ech sorted hlf. Isert smllest of two elemets ito uxiliry rry. Repet util doe. first hlf exhusted smllest A G L O R H I M S T A G H I L M O R S uxiliry rry CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
15 Mergig Merge. Keep trc of smllest elemet i ech sorted hlf. Isert smllest of two elemets ito uxiliry rry. Repet util doe. first hlf exhusted smllest A G L O R H I M S T A G H I L M O R S T uxiliry rry CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
16 Merge Sort: Exmple Show MergeSort) ruig o the rry A {, 5, 7, 6,, 4, 8, 3, 2, 9}; CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
17 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) Θ) whe, d 2T/2) Θ) whe > So wht more succictly) is T)? Effort T) Θ) Θ) T/2) T/2) Θ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
18 Recurreces The expressio: T ) c 2T 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
19 Recurrece Exmples > ) ) s c s > ) ) s s c CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov > 2 2 ) c T c T > ) c T c T
20 Solvig Recurreces Sustitutio method Itertio method Mster method CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
21 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) Θ) T) Θ lg ) T) 2T /2 )??? CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
22 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) Θ) T) Θ lg ) T) 2T /2 ) T) Θ lg ) T) 2T /2 ) 7)??? CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
23 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) Θ) T) Θ lg ) T) 2T /2 ) T) Θ lg ) T) 2T /2 7) Θ lg ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
24 Solvig Recurreces Aother optio is wht the oo clls the 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
25 Solvig Recurreces exmple) s ) s) c s-) c c c s-2) 2c s-2) 2c c s-3) 3c s-3) s ) c s-) c s-) > c T ) T c > CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
26 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
27 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
28 Solvig Recurreces exmple) s) s ) s-) - s-2) s ) - -2 s-3) s-4) ) s-) > CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
29 Solvig Recurreces exmple) s ) s) s-) s ) > - s-2) - -2 s-3) s-4) ) s-) i s ) i CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
30 > ) ) s s So fr for > we hve ) s i Solvig Recurreces exmple) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov ) s i i
31 > ) ) s s So fr for > we hve ) s i Solvig Recurreces exmple) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov Wht if? ) s i i
32 > ) ) s s So fr for > we hve ) s i i Solvig Recurreces exmple) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov Wht if? i 2 ) i s i i i
33 > ) ) s s So fr for > we hve ) s i i Solvig Recurreces exmple) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov Wht if? Thus i geerl 2 ) i s i i i 2 ) s
34 Solvig Recurreces exmple) T) T ) c 2T c > 2 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 ) 2 - )c CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
35 Solvig Recurreces exmple) T ) 2T c 2 So fr for > 2 we hve T) 2 T/2 ) 2 - )c Wht if lg? 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
36 Solvig Recurreces exmple) T) T/) c T ) T//) c/) c 2 T/ 2 ) c/ c 2 T/ 2 ) c/ ) T c 2 T/ 2 /) c/ 2 ) c/ ) 3 T/ 3 ) c 2 / 2 ) c/ ) 3 T/ 3 ) c 2 / 2 / ) c > T/ ) c - / - -2 / -2 2 / 2 / ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
37 Solvig Recurreces exmple) So we hve T ) T c c > T) T/ ) c - / / 2 / ) For log T) T) c - / / 2 / ) c c - / / 2 / ) c c - / / 2 / ) c / c - / / 2 / ) c /... 2 / 2 / ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
38 Solvig Recurreces exmple) T ) So with log T c T) c /... 2 / 2 / ) Wht if? T) c ) clog ) Θ log ) c > CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
39 Solvig Recurreces exmple) T ) So with log T c T) c /... 2 / 2 / ) Wht if <? c > CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
40 Solvig Recurreces exmple) T ) So with log T c T) c /... 2 / 2 / ) Wht if <? c > Recll tht Σx x - x ) x -)/x-) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
41 So with log T) c /... 2 / 2 / ) > ) c T c T Solvig Recurreces exmple) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov Wht if <? Recll tht x x - x ) x -)/x-) So: ) ) ) ) < L
42 So with log T) c /... 2 / 2 / ) > ) c T c T Solvig Recurreces exmple) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov Wht if <? Recll tht Σx x - x ) x -)/x-) So: T) c Θ) Θ) ) ) ) ) < L
43 Solvig Recurreces exmple) T ) So with log T c T) c /... 2 / 2 / ) Wht if >? c > CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
44 So with log T) c /... 2 / 2 / ) > ) c T c T Solvig Recurreces exmple) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov T) c /... / / ) Wht if >? ) ) ) ) Θ L
45 Solvig Recurreces exmple) T ) So with log T c T) c /... 2 / 2 / ) Wht if >? L T) c Θ / ) c ) ) > Θ ) ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
46 Solvig Recurreces exmple) T ) So with log T c T) c /... 2 / 2 / ) Wht if >? L T) c Θ / ) c ) ) > c Θ log / log ) c Θ log / ) Θ ) ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
47 Solvig Recurreces exmple) T ) So with log T c T) c /... 2 / 2 / ) Wht if >? L T) c Θ / ) c ) ) > c Θ log / log ) c Θ log / ) recll logrithm fct: log log Θ ) ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
48 Solvig Recurreces exmple) T ) So with log T c T) c /... 2 / 2 / ) Wht if >? L T) c Θ / ) c ) ) > c Θ log / log ) c Θ log / ) recll logrithm fct: log log c Θ log / ) Θc log / ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov Θ ) )
49 T ) So with log T) c /... 2 / 2 / ) Wht if >? T c c ) ) L T) c Θ / ) > c Θ log / log ) c Θ log / ) recll logrithm fct: log log c Θ log / ) Θc log / ) Θ log ) Θ ) ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
50 So > ) c T c T Solvig Recurreces exmple) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov ) ) ) > Θ Θ < Θ T log log )
51 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
52 The Mster Theorem T ) if T) T/) f) the Θ Θ Θ log ) log log ) f ) ) f ) O f ) f ) Θ Ω log ) ε log ) f / ) < cf log ) ε ) AND for lrge ε > c < CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
53 Usig The Mster Method T) 9T/3) 9, 3, f) log log 3 9 Θ 2 ) Sice f) O log ε ), where ε, cse pplies: T ) Θ log ) ε ) log whe f ) O Thus the solutio is T) Θ 2 ) CS 447, Algorithms, Uiversity College Cor, Gregory M. Prov
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