Introduction to Algorithms 6.046J/18.401J

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Itrodutio to Algorithms.04J/8.40J The divide-d-oquer desig prdigm. Divide the problem (iste) ito subproblems.. Coquer the subproblems by solvig them reursively. 3. Combie subproblem solutios.

1 Itrodutio to Algorithms.04J/8.40J The divide-d-oquer desig prdigm. Divide the problem (iste) ito subproblems.. Coquer the subproblems by solvig them reursively. 3. Combie subproblem solutios. Leture 3 Prof. Molis Kellis L3. Exmple : merge sort. Divide: Trivil.. Coquer: Reursively sort subrrys. 3. Combie: Lier-time merge. T() = T() + O() work dividig d ombiig L3.3 L R log b vs. leves Mster theorem (reprise) root T() = T() + > < = = O( logb ε ) CASE : T() = Θ( log b ). L domites (most work doe t the leves) = Θ( logb lg k ) CASE : T() = Θ( log b lg k+ ). L,R bled (bled work doe throughout) = Ω( logb + ε ) d f() f() CASE 3: T() = Θ( ). R domites (most work doe t the root) : =, b=, = (L = log b ) = (R = ) CASE (k = 0) T() = Θ( lg ). L3.4 Exmple : Biry Serh Powerig um. Polyomil mult. Mtrix multipli. Fiboi Nive Fid elemet i sorted rry:. Divide: Chek middle elemet.. Coquer: Reursively serh subrry. 3. Combie: Trivil. Exmple: Fid L3.5 L3.

2 Fid elemet i sorted rry:. Divide: Chek middle elemet.. Coquer: Reursively serh subrry. 3. Combie: Trivil. Exmple: Fid Fid elemet i sorted rry:. Divide: Chek middle elemet.. Coquer: Reursively serh subrry. 3. Combie: Trivil. Exmple: Fid L3.7 L3.8 Fid elemet i sorted rry:. Divide: Chek middle elemet.. Coquer: Reursively serh subrry. 3. Combie: Trivil. Exmple: Fid Fid elemet i sorted rry:. Divide: Chek middle elemet.. Coquer: Reursively serh subrry. 3. Combie: Trivil. Exmple: Fid L3.9 L3.0 Fid elemet i sorted rry:. Divide: Chek middle elemet.. Coquer: Reursively serh subrry. 3. Combie: Trivil. Exmple: Fid L3. Reurree for biry serh T() = T() + Θ() work dividig d ombiig L( log b = log = 0 )= R() CASE (k = 0) T() =. Mster method L3.

3 Exmple 3: Reursive squrig Powerig umber Powerig um. Polyomil mult. Mtrix multipli. Nive Θ() Problem: Compute, where N. Nive lgorithm:. Divide-d-oquer lgorithm: if is eve; = ( )/ ( )/ if is odd. Fiboi T() = T() + Θ() T() =. L3.3 L3.4 Fiboi Exmple 4: Polyomil mult. Powerig um. Polyomil mult. Mtrix multipli. Nive L3.5 Θ() Θ() Polyomil multiplitio Iput: (x)= 0 + x+ + x, b(x)=b 0 +b x+ +b x, Output: (x)= 0 + x+ + x = (x)*b(x) x 0 x x x 3 x 4 x i = 0 b i + b i- + + i- b + i b 0 i =Σ k k b i-k b 0 0 b b * b 3 b 4 b Ruig time: x + x + x - x - Exmple: ( 0 + x) * (b 0 +b x) = 0 b 0 +( 0 b + b 0 )x+ b x L3. Motivtio (more i reittios) Essetilly equivlet to multiplyig lrge itegers: 04*00 = (* *0 + 0*0 + *0 3 ) * (* * 0 + 0*0 + *0 3 ) = 3 (0) * b(0) = (0), where (x)=(x)*b(x) (0) = The oeffiiets of form the digits of the produt (0) L Divide d oquer multiplitio (x) = 0 + x+ + x = = ( x ) + x ( x x ) = x 0 x x - x x + x = x 0 x x - x 0 x x = 0 - ( - + x ) = p(x) + x q(x) Shift opertios (x) = p + x q b(x) = s + x t *b = (p+x q) * (s+x t) = p*s + x (p*t+q*s) + x q*t 4 multiplitios of legth- polyomils (p,q,s,t) multiplitio of legth- polyomils (,b) ompute the produts reursively! (d the perform dditios) L3.8 3

4 The gret momet T() = 4 T() + ompute the produts reursively work dividig d ombiig the perform dditios) (L= log b = log 4 = ) > (R=) CASE T() =. No better th the ordiry lgorithm??? Need to be more lever Need to ompute this: (4 multiplitios) *b = (p+x q) * (s+x t) = p*s + x (p*t+q*s) + x q*t Note tht: (p+q)*(s+t) = (p*s)+(p*t)+(q*s)+(q*t) C isted ompute this: (3 multiplitios) p*s q*t p*t + q*s = (p+q) * (s+t) - p*s - q*t C ompute *b with oly 3 multiplitios! L3.9 L3.0 The truly gret momet Exmple 5: mtrix multiplitio T() = 3T() + work ddig d subtrtig Powerig um. Nive Θ() Θ() (L= log b = log 3 =.5849 ) > (R=) CASE T() = Θ(.5849 ). Polyomil mult. Mtrix multipli. Θ(.58 ) 4 3 Muh better th! Fiboi L3. L3. M Mtrix multiplitio Iput: A = [ ij ], B = [b ij ]. Output: C = [ ij ] = A B. M L L = O M M L ij = M L b L b O M M L b L3.3 ik k= b kj i, j =,,,. b b b M L b L b O M L b Stdrd lgorithm for i to do for j to do ij 0 for k to do ij ij + ik b kj Ruig time = Θ( 3 ) L3.4 4

5 Divide-d-oquer lgorithm IDEA: mtrix = mtrix of () () submtries: r s b = e f t u d g h r=e+bg s=f+bh t =e+dg u=f+dh C = A B 8 mults of () () submtries 4 dds of () () submtries L3.5 Alysis of D&C lgorithm T() = 8 T() + # submtries submtrix size work ddig submtries (L= log b = log 8 = 3 ) > (R= ) CASE T() = Θ( 3 ). No better th the ordiry lgorithm. L3. Strsse s ide (99) r=e+bg s=f+bh t=e+dg u=f +dh Strsse s ide r=e+bg s=f+bh t=e+dg u=f +dh Multiply mtries with oly 7 reursive mults. P = ( f h) P = ( + b) h P 3 = ( + d) e P 4 = d (g e) P 5 = ( + d) (e + h) P = (b d) (g + h) P 7 = ( ) (e + f ) r = P 5 + P 4 P + P s = P + P t = P 3 + P 4 u = P 5 + P P 3 P 7 7 mults, 8 dds/subs. Note: No relie o ommuttivity of mult! L3.7 Multiply mtries with oly 7 reursive mults. P = ( f h) P = ( + b) h P 3 = ( + d) e P 4 = d (g e) P 5 = ( + d) (e + h) P = (b d) (g + h) P 7 = ( ) (e + f ) r = P 5 + P 4 P + P =( + d)(e + h) + d (g e) ( + b) h + (b d)(g + h) = e + h + de + dh + dg de h bh + bg + bh dg dh = e + bg L3.8 Strsse s lgorithm. Divide: Prtitio A d B ito () () submtries. Form terms to be multiplied usig + d.. Coquer: Perform 7 multiplitios of () () submtries reursively. 3. Combie: Form C usig + d o () () submtries. T() = 7 T() + L3.9 Alysis of Strsse T() = 7 T() + (L = log b = log 7.8 ) > (R = ) CASE T() = Θ( lg 7 ) = Θ( lg 7 ). The umber.8 my ot seem muh smller th 3, but beuse the differee is i the expoet, the impt o ruig time is sigifit. I ft, Strsse s lgorithm bets the ordiry lgorithm o tody s mhies for 30 or so. Best to dte (of theoretil iterest oly): Θ(.37 ). L3.30 5

6 Exmple : Fiboi umbers Fiboi umbers Powerig um. Polyomil mult. Nive Θ(.58 ) 4 3 Θ() Θ() Reursive defiitio: F = 0 if = 0; if = ; F + F if L Mtrix multipli. Fiboi Θ( 3 ) Θ(.8 ) 8 7 Nive reursive lgorithm: Ω(φ ) (expoetil time), where φ = ( + 5)/ is the golde rtio. L3.3 L3.3 Computig Fiboi umbers Nive reursive squrig: F = φ / 5 rouded to the erest iteger. Reursive squrig: time. This method is urelible, sie flotig-poit rithmeti is proe to roud-off errors. Bottom-up: Compute F 0, F, F,, F i order, formig eh umber by summig the two previous. Ruig time:. L3.33 Reursive squrig + Theorem: = F. Algorithm: Reursive squrig. Time =. Proof of theorem. (Idutio o.) F F Bse ( = ): =. 0 0 F F L3.34 Reursive squrig Divide d oquer lgorithms Idutive step ( ):. + F = =. = Powerig um. Polyomil mult. Mtrix multipli. Fiboi Nive Θ( 3 ) φ Θ(.58 ) Θ(.8 ) Θ() Θ() Θ() L3.35 L3.3

7 Colusio Mster Theorem T() = T() + Divide d oquer is just oe of severl powerful tehiques for lgorithm desig. Divide-d-oquer lgorithms be lyzed usig reurrees d the mster method (so prtie this mth). C led to more effiiet lgorithms L3.37 Februry,

Algorithms Design & Analysis. Divide & Conquer

Algorithms Design & Analysis. Divide & Conquer Algorithms Desig & Aalysis Divide & Coquer Recap Direct-accessible table Hash tables Hash fuctios Uiversal hashig Perfect Hashig Ope addressig 2 Today s topics The divide-ad-coquer desig paradigm Revised