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Lecture 8. Paradigm #6 Dynamic Programming Popularized by Richard Bellman ("Dynamic Programming", Princeton University Press, 1957; call number QA 264.B36). Chapter 15 of CLRS. Typically, dynamic programming reduces the complexity of a problem from 2n to O(n3) or O(n2) or even O(n). It does so by keeping track of already computed results in a bottom-up fashion, hence avoiding enumerating all possibilities. Typically applies to optimization problems. Example 1. Efficient multiplication of matrices (Section 15.2 of CLRS.) Suppose we are given the following 3 matrices: M1 10 x 100 M2 100 x 5 M3 5 x 50 There are two ways to compute M1*M2*M3: M1 (M2 M3) or (M1 M2) M3 Since the cost of multiplying a p x q matrix by a q x r matrix is pqr multiplications, the cost of M1 (M2 M3) is 100 x 5 x 50 + 10 x 100 x 50 = 75,000 multiplications, while the cost of (M1 M2) M3 is 10 x 100 x 5 + 10 x 5 x 50 = 7,500 multiplications: a difference of a factor of 10. Naïve approach We could enumerate all possibilities, and then take the minimum. How many possibilities are there? The LAST multiplication performed is either M1*(M2 ... Mn), or (M1 M2)*(M3 ... Mn), or ... (M1 M2 ...)(Mn). Therefore, W(n), the number of ways to compute M1 M2 ... Mn, satisfies the following recurrence: W(n) = Σ1 ≤ k < n W(k)W(n-k) --- Catalan number Now it can be proved by induction that W(n) = (2n-2 choose n1)/n. Using Stirling's approximation, which says that n! = √(2πn) nn e-n (1 + o(1)), we have (2n choose n) ~ 22n/√(π n), We conclude that W(n) ~ 4n n-3/2, which means our naive approach will simply take too long (about 1010 steps when n = 20). Dynamic Programming approach Let’s avoid all the re-computation of the recursive approach. Observe: Suppose the optimal method to compute M1 M2 ... Mn were to first compute M1 M2 ... Mk (in some order), then compute Mk+1 ... Mn (in some order), and then multiply these together. Then the method used for M1 M2 ... Mk must be optimal, for otherwise we could substitute a superior method and improve the optimal method. Similarly, the method used to compute Mk+1 ... Mn must also be optimal. The only thing left to do is to find the best possible k, and there are only n choices for that. Letting m[i,j] represent the optimal cost for computing the product Mi ... Mj, we see that m[i,j] = min { m[i,k] + m[k+1,j] + p[i-1]p[k]p[j] }, i ≤ k < j k represents the optimal place to break the product Mi ... Mj into two pieces. Here p is an array such that M1 is of dimension p[0] × p[1], M2 is of dimension p[1] × p[2], ... etc. Implementing it --- O(n3) time Like the Fibonacci number example, we cannot implement this by recursion. It will be exponential time. MATRIX-MULT-ORDER(p) /* p[0..n] is an array holding the dimensions of the matrices; matrix i has dimension p[i-1] x p[i] */ for i := 1 to n do m[i,i] := 0 for d := 1 to n-1 do // d is the size of the sub-problem. for i := 1 to n-d do j := i+d m[i,j] := infinity; for k := i to j-1 do q := m[i,k] + m[k+1,j] + p[i-1]*p[k]*p[j] if q < m[i,j] then m[i,j] := q s[i,j] := k // optimal position for breaking m[i,j] return(m,s) Actually multiply the matrices We have stored the break points k’s in the array s. s[i,j] represents the optimal place to break the product Mi ... Mj. We can use s now to multiply the matrices: MATRIX-MULT(M, s, i, j) /* Given the matrix s calculated by MATRIX-MULT-ORDER. The list of matrices M = [M1, M2, ... , Mn]. Starting and finishing indices i and j. This routine computes the product Mi ... Mj using the optimal method */ if j > i then X := MATRIX-MULT(M, s, i, s[i,j]); Y := MATRIX-MULT(M, s, s[i,j]+1, j); return(X*Y); else return(Mi) Longest Common Subsequence (LCS) Application: comparison of two DNA strings Ex: X= {A B C B D A B }, Y= {B D C A B A} Longest Common Subsequence: X= AB C BDAB Y= BDCAB A Brute force algorithm would compare each subsequence of X with the symbols in Y LCS Algorithm if |X| = m, |Y| = n, then there are 2m subsequences of x; we must compare each with Y (n comparisons) So the running time of the brute-force algorithm is O(n 2m) Notice that the LCS problem has optimal substructure: solutions of subproblems are parts of the final solution – often, this is when you can use dynamic programming. Subproblems: “find LCS of pairs of prefixes of X and Y” LCS Algorithm First we’ll find the length of LCS. Later we’ll modify the algorithm to find LCS itself. Let Xi, Yj be the prefixes of X and Y of length i and j respectively Let c[i,j] be the length of LCS of Xi and Yj Then the length of LCS of X and Y will be c[m,n] if x[i] y[ j ], c[i 1, j 1] 1 c[i, j ] max(c[i, j 1], c[i 1, j ]) otherwise LCS recursive solution if x[i] y[ j ], c[i 1, j 1] 1 c[i, j ] max(c[i, j 1], c[i 1, j ]) otherwise We start with i = j = 0 (empty substrings of x and y) Since X0 and Y0 are empty strings, their LCS is always empty (i.e. c[0,0] = 0) LCS of empty string and any other string is empty, so for every i and j: c[0, j] = c[i,0] = 0 LCS recursive solution if x[i] y[ j ], c[i 1, j 1] 1 c[i, j ] max(c[i, j 1], c[i 1, j ]) otherwise When we calculate c[i,j], we consider two cases: First case: x[i]=y[j]: one more symbol in strings X and Y matches, so the length of LCS Xi and Yj equals to the length of LCS of smaller strings Xi-1 and Yi-1 , plus 1 LCS recursive solution if x[i] y[ j ], c[i 1, j 1] 1 c[i, j ] max(c[i, j 1], c[i 1, j ]) otherwise Second case: x[i] != y[j] As symbols don’t match, our solution is not improved, and the length of LCS(Xi , Yj) is the same as before, we take the maximum of LCS(Xi, Yj-1) and LCS(Xi-1,Yj) Think: Why can’t we just take the length of LCS(Xi-1, Yj-1)12? 2/13/2015 LCS Length Algorithm LCS-Length(X, Y) 1. m = length(X) // get the # of symbols in X 2. n = length(Y) // get the # of symbols in Y 3. for i = 1 to m c[i,0] = 0 // special case: Y0 4. for j = 1 to n c[0,j] = 0 // special case: X0 5. for i = 1 to m // for all Xi 6. for j = 1 to n // for all Yj 7. if ( Xi == Yj ) 8. c[i,j] = c[i-1,j-1] + 1 9. else c[i,j] = max( c[i-1,j], c[i,j-1] ) 10. return c LCS Example We’ll see how LCS algorithm works on the following example: X = ABCB Y = BDCAB LCS(X, Y) = BCB X=AB C B Y= BD CAB LCS Example (0) j i 0 1 0 1 Yj B 2 3 4 5 D C A B Xi A 2 B 3 C 4 B X = ABCB; m = |X| = 4 Y = BDCAB; n = |Y| = 5 Allocate array c[5,4] ABCB BDCAB LCS Example (1) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 2 B 3 C 0 4 B 0 0 for i = 1 to m for j = 1 to n 2/13/2015 ABCB BDCAB c[i,0] = 0 c[0,j] = 0 16 LCS Example (2) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 2 B 3 C 0 4 B 0 ABCB BDCAB 0 if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 17 LCS Example (3) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 2 B 3 C 0 4 B 0 ABCB BDCAB 0 if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 18 LCS Example (4) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 2 B 3 C 0 4 B 0 ABCB BDCAB 0 if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 19 LCS Example (5) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 1 2 B 3 C 0 4 B 0 ABCB BDCAB 0 if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 20 LCS Example (6) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 1 0 1 2 B 3 C 0 4 B 0 ABCB BDCAB if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 21 LCS Example (7) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 2 B 3 C 0 4 B 0 ABCB BDCAB if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 22 LCS Example (8) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 2 2 B 3 C 0 4 B 0 ABCB BDCAB if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 23 LCS Example (10) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 2 1 1 2 B 3 C 0 4 B 0 ABCB BDCAB if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 24 LCS Example (11) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 2 1 1 2 2 B 3 C 0 4 B 0 ABCB BDCAB if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 25 LCS Example (12) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 2 1 1 2 2 2 2 B 3 C 0 4 B 0 ABCB BDCAB if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 26 LCS Example (13) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 2 1 2 2 2 2 B 3 C 0 1 4 B 0 1 ABCB BDCAB if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 27 LCS Example (14) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 2 2 2 B 3 C 0 1 1 2 2 4 B 0 1 1 2 2 ABCB BDCAB if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 28 LCS Example (15) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 2 2 B 3 C 0 1 1 2 2 2 4 B 0 1 1 2 2 3 ABCB BDCAB if ( Xi == Yj ) c[i,j] = c[i-1,j-1] + 1 else c[i,j] = max( c[i-1,j], c[i,j-1] ) 2/13/2015 29 LCS Algorithm Running Time LCS algorithm calculates the values of each entry of the array c[m,n] So what is the running time? O(m*n) since each c[i,j] is calculated in constant time, and there are m*n elements in the array 2/13/2015 30 How to find actual LCS So far, we have just found the length of LCS, but not LCS itself. We want to modify this algorithm to make it output Longest Common Subsequence of X and Y Each c[i,j] depends on c[i-1,j] and c[i,j-1] or c[i-1, j-1] For each c[i,j] we can say how it was acquired: 2 2 2 3 2/13/2015 For example, here c[i,j] = c[i-1,j-1] +1 = 2+1=3 31 How to find actual LCS - continued Remember that if x[i] y[ j ], c[i 1, j 1] 1 c[i, j ] max(c[i, j 1], c[i 1, j ]) otherwise So we can start from c[m,n] and go backwards Whenever c[i,j] = c[i-1, j-1]+1, remember x[i] (because x[i] is a part of LCS) When i=0 or j=0 (i.e. we reached the beginning), output remembered letters in reverse order 2/13/2015 32 Finding LCS j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 2 2 B 3 C 0 1 1 2 2 2 4 B 0 1 1 2 2 3 2/13/2015 33 Finding LCS (2) j i 0 1 Xi A 0 1 2 3 4 5 Yj B D C A B 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 2 2 B 3 C 0 1 1 2 2 2 4 B 0 1 1 2 2 3 LCS (reversed order): B C B B C B LCS (straight order): (this string turned out to be a palindrome) 2/13/2015 34 If we have time, we will do some exercises in class: Edit distance: Given two text strings A of length n and B of length m, you want to transform A into B with a minimum number of operations of the following types: delete a character from A, insert a character into A, or change some character in A into a new character. The minimal number of such operations required to transform A into B is called the edit distance between A and B. Balanced Partition: Given a set of n integers each in the range 0 ... K. Partition these integers into two subsets such that you minimize |S1 - S2|, where S1 and S2 denote the sums of the elements in each of the two subsets.