Introduction to Algorithms 6.046J/18.401J LECTURE 3 Divide and conquer Binary search Powering a number Fibonacci numbers Matrix multiplication
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1 Itroductio to Algorithms 6.046J/8.40J LECTURE 3 Divide ad coquer Biary search Powerig a umber Fiboacci umbers Matrix multiplicatio Strasse s algorithm VLSI tree layout Prof. Charles E. Leiserso
2 The divide-ad-coquer desig paradigm. Divide the problem (istace) ito subproblems. 2. Coquer the subproblems by solvig them recursively. 3. Combie subproblem solutios. September 5, 2004 Itroductio to Algorithms L3.2
3 Merge sort. Divide: Trivial. 2. Coquer: Recursively sort 2 subarrays. 3. Combie: Liear-time merge. September 5, 2004 Itroductio to Algorithms L3.3
4 Merge sort. Divide: Trivial. 2. Coquer: Recursively sort 2 subarrays. 3. Combie: Liear-time merge. T() = 2 T(/2) + Θ() # subproblems subproblem size work dividig ad combiig September 5, 2004 Itroductio to Algorithms L3.4
5 Master theorem (reprise) T() = at(/b) + f () CASE : f () = O( log ba ε ), costat ε > 0 T() = Θ( log ba ). CASE 2: f () = Θ( log ba lg k ), costat k 0 T() = Θ( log ba lg k+ ). CASE 3: f () = Ω( log ba + ε ), costat ε > 0, ad regularity coditio T() = Θ( f ()). September 5, 2004 Itroductio to Algorithms L3.5
6 September 5, 2004 Master theorem (reprise) T() = at(/b) + f () CASE : f () = O( log ba ε ), costat ε > 0 T() = Θ( log ba ). CASE 2: f () = Θ( log ba lg k ), costat k 0 T() = Θ( log ba lg k+ ). CASE 3: f () = Ω( log ba + ε ), costat ε > 0, ad regularity coditio T() = Θ( f ()). Merge sort: a = 2, b = 2 log ba = log 22 = CASE 2 (k = 0) T() = Θ( lg ). Itroductio to Algorithms L3.6
7 Biary search Fid a elemet i a sorted array:. Divide: Check middle elemet. 2. Coquer: Recursively search subarray. 3. Combie: Trivial. September 5, 2004 Itroductio to Algorithms L3.7
8 Biary search Fid a elemet i a sorted array:. Divide: Check middle elemet. 2. Coquer: Recursively search subarray. 3. Combie: Trivial. Example: Fid September 5, 2004 Itroductio to Algorithms L3.8
9 Biary search Fid a elemet i a sorted array:. Divide: Check middle elemet. 2. Coquer: Recursively search subarray. 3. Combie: Trivial. Example: Fid September 5, 2004 Itroductio to Algorithms L3.9
10 Biary search Fid a elemet i a sorted array:. Divide: Check middle elemet. 2. Coquer: Recursively search subarray. 3. Combie: Trivial. Example: Fid September 5, 2004 Itroductio to Algorithms L3.0
11 Biary search Fid a elemet i a sorted array:. Divide: Check middle elemet. 2. Coquer: Recursively search subarray. 3. Combie: Trivial. Example: Fid September 5, 2004 Itroductio to Algorithms L3.
12 Biary search Fid a elemet i a sorted array:. Divide: Check middle elemet. 2. Coquer: Recursively search subarray. 3. Combie: Trivial. Example: Fid September 5, 2004 Itroductio to Algorithms L3.2
13 Biary search Fid a elemet i a sorted array:. Divide: Check middle elemet. 2. Coquer: Recursively search subarray. 3. Combie: Trivial. Example: Fid September 5, 2004 Itroductio to Algorithms L3.3
14 Recurrece for biary search T() = T(/2) + Θ() # subproblems subproblem size work dividig ad combiig September 5, 2004 Itroductio to Algorithms L3.4
15 Recurrece for biary search T() = T(/2) + Θ() # subproblems subproblem size work dividig ad combiig log ba = log 2 = 0 = CASE 2 (k = 0) T() = Θ(lg ). September 5, 2004 Itroductio to Algorithms L3.5
16 Powerig a umber Problem: Compute a, where N. Naive algorithm: Θ(). September 5, 2004 Itroductio to Algorithms L3.6
17 Powerig a umber Problem: Compute a, where N. Naive algorithm: Θ(). Divide-ad-coquer algorithm: a = a /2 a /2 a ( )/2 a ( )/2 a if is eve; if is odd. September 5, 2004 Itroductio to Algorithms L3.7
18 Powerig a umber Problem: Compute a, where N. Naive algorithm: Θ(). Divide-ad-coquer algorithm: a = a /2 a /2 a ( )/2 a ( )/2 a if is eve; if is odd. T() = T(/2) + Θ() T() = Θ(lg ). September 5, 2004 Itroductio to Algorithms L3.8
19 Fiboacci umbers Recursive defiitio: 0 if = 0; F = if = ; F + F 2 if L September 5, 2004 Itroductio to Algorithms L3.9
20 Fiboacci umbers Recursive defiitio: F = 0 if = 0; if = ; F + F 2 if L Naive recursive algorithm: Ω(φ ) (expoetial time), where φ = ( + is the golde ratio. 5)/2 September 5, 2004 Itroductio to Algorithms L3.20
21 Computig Fiboacci umbers Bottom-up: Compute F 0, F, F 2,, F i order, formig each umber by summig the two previous. Ruig time: Θ(). September 5, 2004 Itroductio to Algorithms L3.2
22 Computig Fiboacci umbers Bottom-up: Compute F 0, F, F 2,, F i order, formig each umber by summig the two previous. Ruig time: Θ(). Naive recursive squarig: F = φ / 5 rouded to the earest iteger. Recursive squarig: Θ(lg ) time. This method is ureliable, sice floatig-poit arithmetic is proe to roud-off errors. September 5, 2004 Itroductio to Algorithms L3.22
23 Recursive squarig F + F Theorem: = F F. 0 September 5, 2004 Itroductio to Algorithms L3.23
24 Recursive squarig F + F Theorem: = F F. 0 Algorithm: Recursive squarig. Time = Θ(lg ). September 5, 2004 Itroductio to Algorithms L3.24
25 September 5, 2004 Itroductio to Algorithms L3.25 Recursive squarig F F F F = + 0 Theorem:. Proof of theorem. (Iductio o.) Base ( = ): = F F F F Algorithm: Recursive squarig. Time = Θ(lg ).
26 September 5, 2004 Itroductio to Algorithms L3.26 Recursive squarig.. Iductive step ( 2): F F F F F F F F = = =
27 September 5, 2004 Itroductio to Algorithms L3.27 Matrix multiplicatio = b b b b b b b b b a a a a a a a a a c c c c c c c c c L M O M M L L L M O M M L L L M O M M L L = = k kj ik ij b a c Iput: A = [a ij ], B = [b ij ]. Output: C = [c ij ] = A B. i, j =, 2,,.
28 Stadard algorithm for i to do for j to do c ij 0 for k to do c ij c ij + a ik b kj September 5, 2004 Itroductio to Algorithms L3.28
29 Stadard algorithm for i to do for j to do c ij 0 for k to do c ij c ij + a ik b kj Ruig time = Θ( 3 ) September 5, 2004 Itroductio to Algorithms L3.29
30 Divide-ad-coquer algorithm IDEA: matrix = 2 2 matrix of (/2) (/2) submatrices: r=ae+bg s=af+bh t =ce+dg u=cf+dh r t s u = a c e g September 5, 2004 Itroductio to Algorithms L3.30 b d C = A B 8 mults of (/2) (/2) submatrices 4 adds of (/2) (/2) submatrices f h
31 Divide-ad-coquer algorithm IDEA: matrix = 2 2 matrix of (/2) (/2) submatrices: r=ae+bg s=af+bh t =ce+dh u=cf+dg r t s u = a c e g September 5, 2004 Itroductio to Algorithms L3.3 b d C = A B recursive 8 mults of (/2) (/2) submatrices ^ 4 adds of (/2) (/2) submatrices f h
32 Aalysis of D&C algorithm T() = 8 T(/2) + Θ( 2 ) # submatrices submatrix size work addig submatrices September 5, 2004 Itroductio to Algorithms L3.32
33 Aalysis of D&C algorithm T() = 8 T(/2) + Θ( 2 ) # submatrices submatrix size work addig submatrices log ba = log 2 8 = 3 CASE T() = Θ( 3 ). September 5, 2004 Itroductio to Algorithms L3.33
34 Aalysis of D&C algorithm T() = 8 T(/2) + Θ( 2 ) # submatrices submatrix size work addig submatrices log ba = log 2 8 = 3 CASE T() = Θ( 3 ). No better tha the ordiary algorithm. September 5, 2004 Itroductio to Algorithms L3.34
35 Strasse s idea Multiply 2 2 matrices with oly 7 recursive mults. September 5, 2004 Itroductio to Algorithms L3.35
36 Strasse s idea Multiply 2 2 matrices with oly 7 recursive mults. P = a ( f h) P 2 = (a + b) h P 3 = (c + d) e P 4 = d (g e) P 5 = (a + d) (e + h) P 6 = (b d) (g + h) P 7 = (a c) (e + f ) September 5, 2004 Itroductio to Algorithms L3.36
37 Strasse s idea Multiply 2 2 matrices with oly 7 recursive mults. P = a ( f h) P 2 = (a + b) h P 3 = (c + d) e P 4 = d (g e) P 5 = (a + d) (e + h) P 6 = (b d) (g + h) P 7 = (a c) (e + f ) r = P 5 + P 4 P 2 + P 6 s = P + P 2 t = P 3 + P 4 u = P 5 + P P 3 P 7 September 5, 2004 Itroductio to Algorithms L3.37
38 Strasse s idea Multiply 2 2 matrices with oly 7 recursive mults. P = a ( f h) P 2 = (a + b) h P 3 = (c + d) e P 4 = d (g e) P 5 = (a + d) (e + h) P 6 = (b d) (g + h) P 7 = (a c) (e + f ) r = P 5 + P 4 P 2 + P 6 s = P + P 2 t = P 3 + P 4 u = P 5 + P P 3 P 7 7 mults, 8 adds/subs. Note: No reliace o commutativity of mult! September 5, 2004 Itroductio to Algorithms L3.38
39 Strasse s idea Multiply 2 2 matrices with oly 7 recursive mults. P = a ( f h) P 2 = (a + b) h P 3 = (c + d) e P 4 = d (g e) P 5 = (a + d) (e + h) P 6 = (b d) (g + h) P 7 = (a c) (e + f ) r = P 5 + P 4 P 2 + P 6 =(a + d)(e + h) + d (g e) (a + b) h + (b d)(g + h) = ae + ah + de + dh + dg de ah bh + bg + bh dg dh = ae + bg September 5, 2004 Itroductio to Algorithms L3.39
40 Strasse s algorithm. Divide: Partitio A ad B ito (/2) (/2) submatrices. Form terms to be multiplied usig + ad. 2. Coquer: Perform 7 multiplicatios of (/2) (/2) submatrices recursively. 3. Combie: Form C usig + ad o (/2) (/2) submatrices. September 5, 2004 Itroductio to Algorithms L3.40
41 Strasse s algorithm. Divide: Partitio A ad B ito (/2) (/2) submatrices. Form terms to be multiplied usig + ad. 2. Coquer: Perform 7 multiplicatios of (/2) (/2) submatrices recursively. 3. Combie: Form C usig + ad o (/2) (/2) submatrices. T() = 7 T(/2) + Θ( 2 ) September 5, 2004 Itroductio to Algorithms L3.4
42 Aalysis of Strasse T() = 7 T(/2) + Θ( 2 ) September 5, 2004 Itroductio to Algorithms L3.42
43 Aalysis of Strasse T() = 7 T(/2) + Θ( 2 ) log ba = log CASE T() = Θ( lg 7 ). September 5, 2004 Itroductio to Algorithms L3.43
44 Aalysis of Strasse T() = 7 T(/2) + Θ( 2 ) log ba = log CASE T() = Θ( lg 7 ). The umber 2.8 may ot seem much smaller tha 3, but because the differece is i the expoet, the impact o ruig time is sigificat. I fact, Strasse s algorithm beats the ordiary algorithm o today s machies for 32 or so. September 5, 2004 Itroductio to Algorithms L3.44
45 Aalysis of Strasse T() = 7 T(/2) + Θ( 2 ) log ba = log CASE T() = Θ( lg 7 ). The umber 2.8 may ot seem much smaller tha 3, but because the differece is i the expoet, the impact o ruig time is sigificat. I fact, Strasse s algorithm beats the ordiary algorithm o today s machies for 32 or so. Best to date (of theoretical iterest oly): Θ( 2.376L ). September 5, 2004 Itroductio to Algorithms L3.45
46 VLSI layout Problem: Embed a complete biary tree with leaves i a grid usig miimal area. September 5, 2004 Itroductio to Algorithms L3.46
47 VLSI layout Problem: Embed a complete biary tree with leaves i a grid usig miimal area. W() H() September 5, 2004 Itroductio to Algorithms L3.47
48 VLSI layout Problem: Embed a complete biary tree with leaves i a grid usig miimal area. W() H() H() = H(/2) + Θ() = Θ(lg ) September 5, 2004 Itroductio to Algorithms L3.48
49 VLSI layout Problem: Embed a complete biary tree with leaves i a grid usig miimal area. W() H() H() = H(/2) + Θ() = Θ(lg ) W() = 2W(/2) + Θ() = Θ() September 5, 2004 Itroductio to Algorithms L3.49
50 VLSI layout Problem: Embed a complete biary tree with leaves i a grid usig miimal area. W() H() H() = H(/2) + Θ() W() = 2W(/2) + Θ() = Θ(lg ) = Θ() Area = Θ( lg ) September 5, 2004 Itroductio to Algorithms L3.50
51 H-tree embeddig L() L() September 5, 2004 Itroductio to Algorithms L3.5
52 H-tree embeddig L() L() L(/4) Θ() L(/4) September 5, 2004 Itroductio to Algorithms L3.52
53 H-tree embeddig L() L() L() = 2L(/4) + Θ() = Θ( ) Area = Θ() L(/4) Θ() L(/4) September 5, 2004 Itroductio to Algorithms L3.53
54 Coclusio Divide ad coquer is just oe of several powerful techiques for algorithm desig. Divide-ad-coquer algorithms ca be aalyzed usig recurreces ad the master method (so practice this math). The divide-ad-coquer strategy ofte leads to efficiet algorithms. September 5, 2004 Itroductio to Algorithms L3.54
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