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 for use with the textook, Algorithm Desig d Applictios, y M. T. Goodrich d R. Tmssi, Wiley, 25 Divide-d-Coquer

2 Divide-d-Coquer Divide-d coquer is geerl lgorithm desig prdigm: Divide: divide the iput dt S i two or more disjoit susets S, S 2, Coquer: solve the suprolems recursively Comie: comie the solutios for S, S 2,, ito solutio for S The se cse for the recursio re suprolems of costt size Alysis c e doe usig recurrece equtios 2

3 Mxim Set Prolem We c visulize vrious trde-offs for optimizig two-dimesiol dt, such s poits represetig hotels ccordig to their pool size d resturt qulity, y plottig ech s two-dimesiol poit, (x, y), where x is the pool size d y is the resturt qulity score. We sy tht such poit is mximum poit i set there is o other poit, (x, y ), i tht set such tht x x d y y. The mximum poits re the est potetil choices sed o these two dimesios d fidig ll of them is the mxim set prolem. We c efficietly fid ll the mxim poits y divide-d-coquer. Here the mxim set is {A,H,I,G,D}.

4 Solvig the Mxim Set Prolem A poit (x, y) is mximum poit i S there is o other poit, (x, y ), i S such tht x x d y y. To fid mxim set for set, S, of poits i the ple, we my divide S ito two equl prts. We compre two poits i S usig lexicogrphic orderig of the poits i S, tht is, where we order sed primrily o x- coordites d the y y-coordites there re ties.

5 Divide-d-Coquer Solutio Bse cse: If, the mxim set is just S itself. Divide: let p = (x p, y p ) e the medi poit i S ccordig to the lexicogrphic order. The x = x p is lie dividig S ito two hlves. Coquer: we recursively solve the mxim-set prolem for the set of poits o the left of this lie d lso for the poits o the right. Comie: The mxim set of poits o the right re lso mxim poits for S. 5

6 Exmple for the Comie Step 6

7 Divide-d-Coquer Solutio Bse cse: If, the mxim set is just S itself. Divide: let p = (x p, y p ) e the medi poit i S ccordig to the lexicogrphic order. The x = x p is lie dividig S ito two hlves. Coquer: we recursively solve the mxim-set prolem for the set of poits o the left of this lie d lso for the poits o the right. Comie: The mxim set of poits o the right re lso mxim poits for S. But some of the mxim poits for the left set might e domited y poit from the right, mely the poit, q, tht is leftmost. So the we do sc of the left set of mxim, removig y poits tht re domited y q. The uio of remiig set of mxim from the left d the mxim set from the right is the set of mxim for S. 7

8 Pseudo-code 8

9 A Little Implemettio Detil There is the issue of how to efficietly fid the poit, p, tht is the medi poit i lexicogrphicl orderig of the poits i S ccordig to their (x, y)-coordites. There re two immedite possiilities: Oe choice is to use lier-time medi-fidig lgorithm, such s tht give i Sectio 9.2. O() for ech recursive cll. Aother choice is to sort the poits i S lexicogrphiclly y their (x, y)-coordites s preprocessig step, prior to cllig the MxmSet lgorithm o S. O( ()) for preprocessig d O() for ech recursive cll, to fid the middle of the list. 9

10 Alysis I either cse, the rest of the o-recursive steps c e performed i O() time, so this implies tht, igorig floor d ceilig fuctios, the ruig time for the divide-d-coquer mxim-set lgorithm c e specied s follows (where is costt): T ( ) 2T ( / 2) Thus, ccordig to the merge sort exmple, this lgorithm rus i O( ) time. 2 2

11 Itertive Sustitutio I the itertive sustitutio, or plug-d-chug, techique, we itertively pply the recurrece equtio to itself d see we c fid ptter: T ( ) 2T ( / 2) 2 2(2T ( / 2 )) ( / 2)) T ( / 2 ) T ( / 2 ) T ( / 2 ) 4... i i 2 T ( / 2 ) i Note tht se, T()=, cse occurs whe 2 i =. Tht is, i =. So, T( ) Thus, T() is O( ).

12 The Recursio Tree Drw the recursio tree for the recurrece reltio d look for ptter: 2 T ( ) 2T ( / 2) 2 depth T s size time 2 /2 i 2 i /2 i Totl time = + (lst level plus ll previous levels) 2

13 Guess-d-Test Method I the guess-d-test method, we guess closed form solutio d the try to prove it is true y iductio: 2 T ( ) 2T ( / 2) 2 Guess: T() c. T ( ) 2T ( / 2) 2( c( / 2) ( / 2)) c( 2) c c c ( c ) We c coclude tht e e less th T() c c. 3

14 Guess-d-Test Method I the guess-d-test method, we guess closed form solutio d the try to prove it is true y iductio: 2 T ( ) 2T ( / 2) 2 Guess: T() c. T ( ) 2T ( / 2) 2( c( / 2) ( / 2)) c( 2) c c Wrog: we cot mke this lst lie e less th c 4

15 Guess-d-Test Method, (cot.) Recll the recurrece equtio: T ( ) 2T ( / 2) Guess #2: T() c T ( ) 2T ( / 2) 2( c( / 2) c( 2) c c ( / 2)) 2 2c c c. So, T() is O( 2 ). I geerl, to use this method, you eed to hve good guess d you eed to e good t iductio proofs. 5

16 6 Mster Method My divide-d-coquer recurrece equtios hve the form: The Mster Theorem: d f T d c T ) ( ) / ( ) (. for some ) ( ) / ( provided )), ( ( is ) ( the ), ( is ) ( 3. ) ( is ) ( the ), ( is ) ( 2. ) ( is ) ( the ), ( is ) (. f f f T f T f T O f k k

17 Mster Method, Exmple The form: The Mster Theorem: provided Exmple: T ( ) f ( ) is O( f ( ) is ( f ( ) is ( c T( / ) f ( / ) f ( ) T( ) f ( ) ), the T ( ) is ( k for some. ), the T ( ) is ( ), the T ( ) is ( f ( )), 4T ( / 2) ) d d k ) Solutio: =2, so cse sys T() is O( 2 ). 7

18 Mster Method, Exmple 2 The form: The Mster Theorem: provided Exmple: T ( ) f ( ) is O( f ( ) is ( f ( ) is ( c T( / ) f ( / ) f ( ) f ( ) ), the T ( ) is ( k for some. ), the T ( ) is ( ), the T ( ) is ( f ( )), T( ) 2T ( / 2) ) d d k ) Solutio: =, so cse 2 sys T() is O( 2 ). 8

19 Mster Method, Exmple 3 The form: The Mster Theorem: provided Exmple: T ( ) f ( ) is O( f ( ) is ( f ( ) is ( c T( / ) f ( / ) f ( ) f ( ) ), the T ( ) is ( T( ) T( /3) k for some. ), the T ( ) is ( ), the T ( ) is ( f ( )), ) d d k ) Solutio: =, so cse 3 sys T() is O( ). 9

20 Mster Method, Exmple 4 The form: The Mster Theorem: provided Exmple: T ( ) f ( ) is O( f ( ) is ( f ( ) is ( c T( / ) f ( / ) f ( ) f ( ) ), the T ( ) is ( k for some. ), the T ( ) is ( ), the T ( ) is ( f ( )), T( ) 8T ( / 2) Solutio: =3, so cse sys T() is O( 3 ). 2 ) d d k ) 2

21 Mster Method, Exmple 5 The form: The Mster Theorem: provided Exmple: T ( ) f ( ) is O( f ( ) is ( f ( ) is ( c T( / ) f ( / ) f ( ) f ( ) ), the T ( ) is ( k for some. ), the T ( ) is ( ), the T ( ) is ( f ( )), T( ) 9T ( /3) Solutio: =2, so cse 3 sys T() is O( 3 ). 3 ) d d k ) 2

22 Mster Method, Exmple 6 The form: The Mster Theorem: provided Exmple: T ( ) f ( ) is O( f ( ) is ( f ( ) is ( c T( / ) f ( / ) f ( ) f ( ) ), the T ( ) is ( k for some. ), the T ( ) is ( ), the T ( ) is ( f ( )), T( ) T( / 2) ) d d k ) (iry serch) Solutio: =, so cse 2 sys T() is O( ). 22

23 Mster Method, Exmple 7 The form: The Mster Theorem: provided Exmple: T ( ) f ( ) is O( f ( ) is ( f ( ) is ( c T( / ) f ( / ) f ( ) f ( ) ), the T ( ) is ( k for some. ), the T ( ) is ( ), the T ( ) is ( f ( )), T( ) 2T ( / 2) ) d d k ) (hep costructio) Solutio: =, so cse sys T() is O(). 23

24 Iteger Additio Additio. Give two -it itegers d, compute +. Grde-school. () it opertios. crry + Remrk: Grde-school dditio lgorithm is optiml. 24

25 25 Iteger Multiplictio Multiplictio. Give two -it itegers d, compute. Grde-school. ( 2 ) it opertios. Q. Is grde-school multiplictio lgorithm optiml?

26 Divide-d-Coquer Multiplictio: Wrmup To multiply two -it itegers d : Multiply four ½-it itegers, recursively. Add d sht to oti result. 2 /2 2 /2 2 /2 2 2 /2 2 /2 Ex. = = 26

27 Recursio T() Tree 4T(/2) otherwise T() lg 2 k 2 lg 2 2 k 2 T() T(/2) T(/2) T(/2) T(/2) 4(/2) T(/4) T(/4) T(/4) T(/4)... T(/4) T(/4) T(/4) T(/4) 6(/4) T( / 2 k ) 4 k ( / 2 k ) T(2) T(2) T(2) T(2)... T(2) T(2) T(2) T(2) 4 lg () 27

28 Krtsu Multiplictio To multiply two -it itegers d : Add two ½ it itegers. Multiply three ½-it itegers, recursively. Add, sutrct, d sht to oti result. 2 /2 2 /2 2 2 /2 2 2 /2 ( )( )

29 Krtsu Multiplictio To multiply two -it itegers d : Add two ½ it itegers. Multiply three ½-it itegers, recursively. Add, sutrct, d sht to oti result. 2 /2 2 /2 2 2 /2 2 2 /2 ( )( )

30 Dot Product Dot product. Give two legth vectors d, compute c =. Grde-school. () rithmetic opertios. i i i (.7.3) (.2.4) (..3).32 Remrk. Grde-school dot product lgorithm is optiml. 3

31 Mtrix Multiplictio Mtrix multiplictio. Give two -y- mtrices A d B, compute C = AB. Grde-school. ( 3 ) rithmetic opertios. c ij ik kj k c c 2 L c c 2 c 22 L c 2 M M O M c c 2 L c 2 L 2 22 L 2 M M O M 2 L 2 L 2 22 L 2 M M O M 2 L Q. Is grde-school mtrix multiplictio lgorithm optiml? 3

32 Block Mtrix Multiplictio C A A 2 B B 2 C A B A 2 B

33 Mtrix Multiplictio: Wrmup To multiply two -y- mtrices A d B: Divide: prtitio A d B ito ½-y-½ locks. Coquer: multiply 8 pirs of ½-y-½ mtrices, recursively. Comie: dd pproprite products usig 4 mtrix dditios. C C 2 A A 2 B B 2 C 2 C 22 A 2 A 22 B 2 B 22 A 2 B 2 A 2 B 22 A 22 B 2 A 22 B 22 C A B C 2 A B 2 C 2 A 2 B C 22 A 2 B 2 33

34 Fst Mtrix Multiplictio Key ide. multiply 2-y-2 locks with oly 7 multiplictios. C C 2 A A 2 B B 2 P A (B 2 B 22 ) C 2 C 22 A 2 A 22 B 2 B 22 P 2 ( A A 2 ) B 22 C P 5 P 4 P 2 P 6 C 2 P P 2 C 2 P 3 P 4 C 22 P 5 P P 3 P 7 P 3 ( A 2 A 22 ) B P 4 A 22 (B 2 B ) P 5 ( A A 22 ) (B B 22 ) P 6 ( A 2 A 22 ) (B 2 B 22 ) P 7 ( A A 2 ) (B B 2 ) 7 multiplictios. 8 = 8 + dditios d sutrctios. 34

35 Fst Mtrix Multiplictio To multiply two -y- mtrices A d B: [Strsse 969] Divide: prtitio A d B ito ½-y-½ locks. Compute: 4 ½-y-½ mtrices vi mtrix dditios. Coquer: multiply 7 pirs of ½-y-½ mtrices, recursively. Comie: 7 products ito 4 terms usig 8 mtrix dditios. Alysis. Assume is power of 2. T() = # rithmetic opertios. 35

36 Fst Mtrix Multiplictio To multiply two -y- mtrices A d B: [Strsse 969] 36

37 Fst Mtrix Multiplictio: Prctice Implemettio issues. Sprsity. Cchig effects. Numericl stility. Odd mtrix dimesios. Crossover to clssicl lgorithm roud = 28. Commo misperceptio. Strsse is oly theoreticl curiosity. Apple reports 8x speedup o G4 Velocity Egie whe 2,5. Rge of istces where it's useful is suject of cotroversy. Remrk. C "Strsseize" Ax =, determit, eigevlues,. 37

38 Fst Mtrix Multiplictio: Theory Q. Multiply two 2-y-2 mtrices with 7 sclr multiplictios? A. Yes! [Strsse 969] Q. Multiply two 2-y-2 mtrices with 6 sclr multiplictios? A. Impossile. [Hopcroft d Kerr 97] Q. Two 3-y-3 mtrices with 2 sclr multiplictios? A. Also impossile. ( 2 7 ) O( 2.87 ) ( 2 6 ) O( 2.59 ) ( 3 2 ) O( 2.77 ) Begu, the deciml wrs hve. [P, Bii et l, Schöhge, ] Two 2-y-2 mtrices with 4,46 sclr multiplictios. Two 48-y-48 mtrices with 47,27 sclr multiplictios. A yer lter. Decemer, 979. Jury, 98. O( 2.85 ) O( 2.78 ) O( ) O( ) O( ) 38

39 Fst Mtrix Multiplictio: Theory Best kow. O( ) [Coppersmith-Wiogrd, 987] Cojecture. O( 2+ ) for y >. Cvet. Theoreticl improvemets to Strsse re progressively less prcticl. 39

40 Exercise Give mxm mtrix M d positive iteger, how to compute M efficietly? 4

Merge Sort. Outline and Reading. Divide-and-Conquer. Divide-and-conquer paradigm ( 4.1.1) Merge-sort ( 4.1.1)

Merge Sort. Outline and Reading. Divide-and-Conquer. Divide-and-conquer paradigm ( 4.1.1) Merge-sort ( 4.1.1) Merge Sort 7 2 9 4 2 4 7 9 7 2 2 7 9 4 4 9 7 7 2 2 9 9 4 4 Merge Sort versio 1.3 1 Outlie d Redig Divide-d-coquer prdigm ( 4.1.1 Merge-sort ( 4.1.1 Algorithm Mergig two sorted sequeces Merge-sort tree