Algorithms Design & Analysis. Divide & Conquer

Transcription

1 Algorithms Desig & Aalysis Divide & Coquer

2 Recap Direct-accessible table Hash tables Hash fuctios Uiversal hashig Perfect Hashig Ope addressig 2

3 Today s topics The divide-ad-coquer desig paradigm Revised master theorem 3

4 The divide-ad-coquer desig paradigm. Divide the problem (istace) ito subproblems. 2. Coquer the subproblems by solvig them recursively. 3. Combie subproblem solutios. 4

5 Example: merge sort. Divide: Trivial. 2. Coquer: Recursively sort 2 subarrays. 3. Combie: Liear-time merge. T() = 2T(/2) + O() # subproblems subproblem size work dividig ad combiig 5

6 Biary search Fid a elemet i a sorted array: Divide: Check middle elemet. Coquer: Recursively search subarray. Combie: Trivial. Example: Fid

7 Biary search Fid a elemet i a sorted array: Divide: Check middle elemet. Coquer: Recursively search subarray. Combie: Trivial. Example: Fid

8 Biary search Fid a elemet i a sorted array: Divide: Check middle elemet. Coquer: Recursively search subarray. Combie: Trivial. Example: Fid

9 Biary search Fid a elemet i a sorted array: Divide: Check middle elemet. Coquer: Recursively search subarray. Combie: Trivial. Example: Fid

10 Biary search Fid a elemet i a sorted array: Divide: Check middle elemet. Coquer: Recursively search subarray. Combie: Trivial. Example: Fid

11 Biary search Fid a elemet i a sorted array: Divide: Check middle elemet. Coquer: Recursively search subarray. Combie: Trivial. Example: Fid

12 Master theorem (reprise) T() = at(/b) + f () CASE : f () = O( log b a ε ) T() = O( log b a ). CASE 2: f () = Θ( lg b a lg k ) T() = O( log b a lg k+ ). CASE 3: f () = Ω( log b a + ε ) ad a f (/b) c f () T() = Θ( f ()). Merge sort: a = 2, b = 2 log b a = CASE 2 (k = 0) T() = Θ( lg ). 2

13 Recurrece for biary search T() = T(/2) +Θ() # subproblems subproblem size work dividig ad combiig log a b log 2 0 = = = T() = Θ(lg ) CASE 2 (k = 0) 3

14 Powerig a umber Problem: Compute a, where N. Naive algorithm: Θ(). Divide-ad-coquer algorithm: a a a / 2 a / 2 = ( )/ 2 ( )/ 2 a T() = Θ(lg ) a if is eve; if is odd. 4

15 Polyomial multiplicatio Iput: a(x)=a 0 +a x+ +a x, Output: b(x)=b 0 +b x+ +b x, c(x)= a(x)*b(x)=c 0 +c x+ +c 2 x 2 c i =a 0 b i +a b i- + + a i- b +a i b 0 Example: (a 0 +a x) * (b 0 +b x) = a 0 b 0 + (a 0 b +a b 0 )x + a b x 2 = c 0 + c x + c 2 x 2 5

16 Motivatio Essetially equivalet to multiplyig large itegers: 6046*600 = (6* *0 + 0* *0 3 ) * (* *0 + 0* *0 3 ) = a(0) * b(0) = c(0), where c(x)=a(x)*b(x) c(0) = c c c The coefficiets of c form the digits of the product c(0) 6

17 How to multiply two polyomials From the defiitio: Θ( 2 ) time Faster? Use divide ad coquer Divide a(x) = a 0 +a x+ +a x = (a a /2 x /2 ) + x /2 (a /2 x a x /2 ) = p(x) + x /2 q(x) = p + x /2 q I the same way: b(x) = s+x /2 t 7

18 Coquer Observe that a*b = (p+x /2 q) * (s+x /2 t) = p*s + x /2 (p*t+q*s) + x q*t But p,q,s,t have degree /2 ca compute the products recursively! (ad the perform Θ() additios) 8

19 The great momet T() = 4T(/2) +Θ() # subproblems subproblem size work dividig ad combiig log a b log2 4 2 = = CASE T() = Θ( 2 ) No better tha the ordiary algorithm??? 9

20 Need to be more clever Compute: p*s q*t (p+q) * (s+t) = p*s + (p*t + q*s) + q*t (all polyomials have degree /2 ) Ca extract (p*t + q*s) without ay additioal multiplicatios! 20

21 The truly great momet T() = 3T(/2) +Θ() # subproblems subproblem size work dividig ad combiig log a b log = = Much better tha Θ( 2 )! 2

22 22 Matrix multiplicatio Iput: A = [a ij ], B = [b ij ]. Output: C = [c ij ] = A B. } = 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.,,2,, j i K =

23 Stadard algorithm for i to // the i th row do for j to // the j th colum do c ij 0 // the (i th, j th ) elemet for k to do c ij c ij + a ik b kj Ruig time = Θ( 3 ) 23

24 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 s a b = e f t u c d g h C = A B 8 mults of (/2) (/2) submatrices 4 adds of (/2) (/2) submatrices 24

25 Aalysis of D&C algorithm T() = 8T(/2) +Θ( 2 ) # subproblems subproblem size work dividig ad combiig log a b log2 8 3 = = CASE T() = Θ( 3 ) No better tha the ordiary algorithm. 25

26 Strasse s idea(969) Multiply 2 2 matrices with oly 7 recursive mults. P = a ( f h) r = P5 + P4 P2 + P6 P 2 = (a + b) h s = P + P2 P 3 = (c + d) e t = P3 + P4 P 4 = d (g e) u = P5 + P P3 P7 P 5 = (a + d) (e + h) P 6 = (b d) (g + h) P 7 = (a c) (e + f ) 7 mults, 8 adds/subs. Note: No reliace o commutativity of mult! 26

27 Strasse s idea Multiply 2 2 matrices with oly 7 recursive mults. P = a ( f h) r = P5 + P4 P2 + P6 P 2 = (a + b) h = (a + d) (e + h) P 3 = (c + d) e + d (g e) (a + b) h P 4 = d (g e) + (b d) (g + h) P 5 = (a + d) (e + h) = ae + ah + de + dh P 6 = (b d) (g + h) + dg de ah bh P 7 = (a c) (e + f ) + bg+ bh dg dh = ae + bg 27

28 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() = 7T(/2) + Θ( 2 ) 28

29 Aalysis of Strasse loga b log T() = 7T(/2) + Θ( 2 ) = CASE T() = Θ( lg7 ) 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 30 or so. Best to date (of theoretical iterest oly): Θ( ). 29

30 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). Ca lead to more efficiet algorithms 30

31 VLSI layout Problem: Embed a complete biary tree with leaves i a grid usig miimal area. 3

32 H-tree embeddig 32

33 Fiboacci umbers Recursive defiitio: F = 0 F if if = = 0; ; F 2 if 2. Naive recursive algorithm: Ω(φ ) (expoetial time), where φ = ( + 5) / 2 is the golde ratio. 33

34 Computig Fiboacci Numbers 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. Bottom-up: Compute F 0, F, F 2,, F i order, formig each umber by summig the two previous. Ruig time: Θ(). 34

35 35 Recursive squarig Theorem: Algorithm: Recursive squarig. Time = Θ (lg ). Proof of theorem. (Iductio o.) Base ( = ): F F F F = = F F F F

36 36 Recursive squarig Iductive step ( 2): = F F F F F F F F = 0 0 = 0

37 Next week Dyamic programmig Chapter 5 37