F(0)=0 F(1)=1 F(n)=F(n-1)+F(n-2)

Transcription

1 Algorithms Dana Shapira Lesson #4: Dynamic programming Fibonacci Series F()= F()= F(n)=F(n-)+F(n-) Write a Divide and Conquer Algorithm! What is its running time? Binomial Coefficients n! n = ( )! n! Recursive equation: n n n = + Write a Divide and Conquer Algorithm! What is its running time? Dynamic Programming Approach to Optimization Problems. Characterize structure of an optimal solution.. Recursively define value of an optimal solution.. Compute value of an optimal solution in bottom-up fashion. 4. Construct an optimal solution from computed information. 4

2 World Series Odds Two teams A and B play to see who is the first to win n games. In world series games n=4. Assumption: A and B are equally competent, each has a 5% chance to win any particular game. World Series Odds P(,j)= j> P(i,)= i> P(i,j)=½ P(i-,j)+½P(i,j-) i> and j> P(i,j) - the probability that A needs i extra games to win and B needs j extra games to win. 5 6 Recursive Solution P(i,j){ if i= return elseif j= return else return ½ P(i-,j)+½P(i,j-) What is its running time? 7 Dynamic Programming P(n,m){ for(i=; i n; i++) T[,i] = for(j=; j m; j++) T[j,] = for(i=; i n; i++) for(j=; j m; j++) return T[n,m] T[i,j]= ½T[i-,j]+ ½T[i,j-]; 4 / /4 /8 /6 / What is its running time? 4 /4 7/8 5/6 8

3 -Problem: Given n matrices M,M,,M n, compute the product M M M M n, where M i has dimension d i- x d i, for i =,,n. -objective compute M,M,,M n with the minimum number of scalar multiplications. -Problem: Parenthesize the product M M M n in a way to minimize the number of scalar multiplications. -Example: M --- x 5 M x M --- x 4 -Given matrices A with dimension p x q and B with dimension q x r, multiplication AB taes pqr scalar multiplications M (M M ) = multiplications (M M )M = 54 multiplications 9 Let m(i,j) be the number of multiplications performed using optimal parenthesis of M i M i+ M j- M j. i = j m( i, j ) = min{m(i,) + m(+,j)+di- d d j i < j i <j MUL(i,j){ if (i= =j) return ; else{ x for =i to j- y MUL(i,)+MUL(+,j)+d i- d d j if y<x { x y S(i,j) What is its running time?

4 Recursive Running Time n ( ) = ( ( ) + ( ) + ) T n T T n = n ( ) = ( ) + ( ) T n T n = n ( ) = ( ) + ( ) T n T n = ( ) ( ) = ( ) + ( ) = ( ) + > ( ) > ( ) > > ( ) T n T n T n T n T n T n T n T n Dynamic Programming -Example: M --- x 5 M x M --- x 4 i = j m( i, j ) = min{m(i,) + m(+,j)+di- d d j i < j i <j m 54 6 S 4 MUL(n){ for (i=;i n;i++) m[i,i]=; for (diff=;diff n-;diff++){ for (i=;i n-diff;i++){ j i+diff x for (=i; j-;++) y m[i,]+m[+,j]+d i- d d j if y<x { x y S(i,j) m[i,j]=x; Longest Common Subsequence Let x= x x x n y= y y y m A common subsequence of X and Y of length exists if there are indices i i... i and j j j such that for every l x = y - objective Find the longest Common Subsequence of x and y (LCS) il jl

5 LCS(X,Y){ LCS for (i=;i m;i++) C[i,]=; for (j=;j n;i++) C[,j]=; for (i=;i m;i++){ for (j=;j n;i++){ if (x i =y j ) C[i,j] C[i-,j-]+ else C[i,j] max(c[i,j-],c[i-,j]) return C[m,n] 7 Optimal Polygon Given a convex polygon P=< v,v,,v n- >, a chord v i v j divides P into two polygons <v i,v i+,,v j > and <v j,v j+,,v i > (assume v n =v or more generally, v =v mod n). v v 8 Optimal Polygon Given a convex polygon P=< v,v,,v n- >, a chord v i v j divides P into two polygons <v i,v i+,,v j > and <v j,v j+,,v i > (assume v n =v or more generally, v =v mod n). Given a convex polygon P=< v,v,,v n- >, it can always be divided into n- non-overlapping triangles using n- chords. v v v 9 v

6 Given a convex polygon P=< v,v,,v n- >, it can always be divided into n- non-overlapping triangles using n- chords. Given a convex polygon P=< v,v,,v n- >, there could be a lot of triangulations (in fact, an exponential number of them). v v v v Given a convex polygon P=< v,v,,v n- >, we want to compute an optimal triangulation. Given a convex polygon P=< v,v,,v n- >, we want to compute an optimal triangulation whose weight is minimized. The weight of a triangulation is the weight of all its triangles and the weight of a triangle is the sum of its edge lengths. Optimal on what? v v v v 4

7 Given a convex polygon P=< v,v,,v n- >, we want to compute an optimal triangulation whose weight is minimized. The weight of a triangulation is the weight of all its triangles and the weight of a triangle is the sum of its edge lengths. The weight of is + + Dynamic Solution: Let t[i,j] be the weight of an optimal triangulation of polygon <v i-,v i,,v j >. v j v + v v v i- v i v 5 6 Dynamic Solution: Let t[i,j] be the weight of an optimal triangulation of polygon <v i-,v i,,v j >. Dynamic Solution: Let t[i,j] be the weight of an optimal triangulation of polygon <v i-,v i,,v j >. t[i,j] = min { t[i,] + t[+,j] + w( v i- v v j ), i<j v j t[i,j] = min { t[i,] + t[+,j] + w( v i- v v j ), i<j v j t[i,i] = t[i,i] = v + v Almost identical to the matrix chain multiplication problem! v + v v i- v i v i- v i 7 8