Dynamic Programming (Part #2)
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1 Dynamic Programming (Part #) Introduction to Algorithms MIT Press (Chapter 5)
2 Matrix-Chain Multiplication Problem: given a sequence A, A,, A n, compute the product: A A A n Matrix compatibility: C = A B C=A A A i A i+ A n col A = row B col i = row i+ row C = row A col C = col B row C = row A col C = col An
3 MATRIX-MULTIPLY(A, B) if columns[a] rows[b] then error incompatible dimensions else for i to rows[a] do for j to columns[b] k do C[i, j] = for k to columns[a] do C[i, j] C[i, j] + A[i, k] B[k, j] j cols[b] rows[a] cols[a] cols[b] multiplications j cols[b] i * k = i rows[a] A B rows[a] C 3
4 Matrix-Chain Multiplication In what order should we multiply the matrices? A A A n Parenthesize the product to get the order in which matrices are multiplied E.g.: A A A 3 = ((A A ) A 3 ) = (A (A A 3 )) Which one of these orderings should we choose? The order in which we multiply the matrices has a significant impact on the cost of evaluating the product 4
5 Example A A A 3 A : x A : x 5 A 3 : 5 x 5. ((A A ) A 3 ): A A = x x 5 = 5, ( x 5) ((A A ) A 3 ) = x 5 x 5 =,5 Total: 7,5 scalar multiplications. (A (A A 3 )): A A 3 = x 5 x 5 = 5, ( x 5) (A (A A 3 )) = x x 5 = 5, Total: 75, scalar multiplications one order of magnitude difference!! 5
6 Matrix-Chain Multiplication: Problem Statement Given a chain of matrices A, A,, A n, where A i has dimensions p i- x p i, fully parenthesize the product A A A n in a way that minimizes the number of scalar multiplications. A A A i A i+ A n p x p p x p p i- x p i p i x p i+ p n- x p n 6
7 What is the number of possible parenthesizations? Exhaustively checking all possible parenthesizations is not efficient! It can be shown that the number of parenthesizations grows as Ω(4 n /n 3/ ) (see page 333 in your textbook) 7
8 . The Structure of an Optimal Parenthesization Notation: A i j = A i A i+ A j, i j Suppose that an optimal parenthesization of A i j splits the product between A k and A k+, where i k < j A i j = A i A i+ A j = A i A i+ A k A k+ A j = A i k A k+ j 8
9 Optimal Substructure A i j = A i k A k+ j The parenthesization of the prefix A i k must be an optimal parenthesization If there were a less costly way to parenthesize A i k, we could substitute that one in the parenthesization of A i j and produce a parenthesization with a lower cost than the optimum contradiction! An optimal solution to an instance of the matrix-chain multiplication contains within it optimal solutions to subproblems 9
10 . A Recursive Solution Subproblem: determine the minimum cost of parenthesizing A i j = A i A i+ A j for i j n Let m[i, j] = the minimum number of multiplications needed to compute A i j full problem (A..n ): m[, n] i = j: A i i = A i m[i, i] =, for i =,,, n
11 . A Recursive Solution Consider the subproblem of parenthesizing A i j = A i A i+ A j for i j n = A i k A k+ j for i k < j m[i, k] p i- p k p j m[k+,j] Assume that the optimal parenthesization splits the product A i A i+ A j at k (i k < j) m[i, j] = m[i, k] + m[k+, j] + p i- p k p j min # of multiplications to compute A i k min # of multiplications to compute A k+ j # of multiplications to compute A i k A k j
12 . A Recursive Solution (cont.) m[i, j] = m[i, k] + m[k+, j] + p i- p k p j We do not know the value of k There are j i possible values for k: k = i, i+,, j- Minimizing the cost of parenthesizing the product A i A i+ A j becomes: if i = j m[i, j] = min {m[i, k] + m[k+, j] + p i- p k p j } ik<j if i < j
13 3. Computing the Optimal Costs if i = j m[i, j] = min {m[i, k] + m[k+, j] + p i- p k p j } ik<j if i < j Computing the optimal solution recursively takes exponential time! How many subproblems? (n ) n 3 n Parenthesize A i j for i j n One problem for each choice of i and j 3 i 3 j
14 3. Computing the Optimal Costs (cont.) if i = j m[i, j] = min {m[i, k] + m[k+, j] + p i- p k p j } ik<j How do we fill in the tables m[..n,..n]? if i < j Determine which entries of the table are used in computing m[i, j] A i j = A i k A k+ j Subproblems size is one less than the original size Idea: fill in m such that it corresponds to solving problems of increasing length 4
15 3. Computing the Optimal Costs (cont.) if i = j m[i, j] = min {m[i, k] + m[k+, j] + p i- p k p j } ik<j Length = : i = j, i =,,, n Length = : j = i +, i =,,, n- m[, n] gives the optimal solution to the problem n if i < j 3 n Compute rows from bottom to top and from left to right 3 j i 5
16 Example: min {m[i, k] + m[k+, j] + p i- p k p j } m[, 5] = min m[, ] + m[3, 5] + p p p 5 m[, 3] + m[4, 5] + p p 3 p 5 m[, 4] + m[5, 5] + p p 4 p 5 k = k = 3 k = i j Values m[i, j] depend only on values that have been previously computed 6
17 Example min {m[i, k] + m[k+, j] + p i- p k p j } Compute A A A 3 A : x (p x p ) A : x 5 (p x p ) A 3 : 5 x 5 (p x p 3 ) m[i, i] = for i =,, 3 m[, ] = m[, ] + m[, ] + p p p (A A ) = + + ** 5 = 5, m[, 3] = m[, ] + m[3, 3] + p p p 3 (A A 3 ) = + + * 5 * 5 = 5, m[, 3] = min m[, ] + m[, 3] + p p p 3 = 75, (A (A A 3 )) 3 m[, ] + m[3, 3] + p p p 3 = 7,5 ((A A )A 3 )
18 Matrix-Chain-Order(p) O(N 3 ) 8
19 4. Construct the Optimal Solution In a similar matrix s we keep the optimal values of k s[i, j] = a value of k such that an optimal parenthesization of A i..j splits the product between A k and A k+ n 3 3 n k j 9
20 4. Construct the Optimal Solution s[, n] is associated with the entire product A..n The final matrix multiplication will be split at k = s[, n] n 3 n A..n = A..s[, n] A s[, n]+..n For each subproduct recursively find the corresponding value of k that results in an optimal parenthesization 3 j
21 4. Construct the Optimal Solution s[i, j] = value of k such that the optimal parenthesization of A i A i+ A j splits the product between A k and A k i j s[, n] = 3 A..6 = A..3 A 4..6 s[, 3] = A..3 = A.. A..3 s[4, 6] = 5 A 4..6 = A 4..5 A 6..6
22 4. Construct the Optimal Solution (cont.) PRINT-OPT-PARENS(s, i, j) if i = j then print A i else print ( PRINT-OPT-PARENS(s, i, s[i, j]) PRINT-OPT-PARENS(s, s[i, j] +, j) print ) i j
23 Example: A A 6 ( ( A ( A A 3 ) ) ( ( A 4 A 5 ) A 6 ) ) PRINT-OPT-PARENS(s, i, j) if i = j then print A i else print ( PRINT-OPT-PARENS(s, i, s[i, j]) PRINT-OPT-PARENS(s, s[i, j] +, j) print ) P-O-P(s,, 6) s[, 6] = 3 i =, j = 6 ( P-O-P (s,, 3) s[, 3] = i =, j = 3 ( P-O-P(s,, ) A ) s[..6,..6] P-O-P(s,, 3) s[, 3] = i =, j = 3 ( P-O-P (s,, ) A ) i P-O-P (s, 3, 3) A 3 3 j
24 Longest Common Subsequence Given two sequences X = x, x,, x m Y = y, y,, y n find a maximum length common subsequence (LCS) of X and Y E.g.: X = A, B, C, B, D, A, B Subsequences of X: A subset of elements in the sequence taken in order A, B, D, B, C, D, B, etc. 4
25 Example X = A, B, C, B, D, A, B X = A, B, C, B, D, A, B Y = B, D, C, A, B, A Y = B, D, C, A, B, A B, C, B, A and B, D, A, B are longest common subsequences of X and Y (length = 4) B, C, A, however is not a LCS of X and Y 5
26 Brute-Force Solution For every subsequence of X, check whether it s a subsequence of Y There are m subsequences of X to check Each subsequence takes (n) time to check scan Y for first letter, from there scan for second, and so on Running time: (n m ) 6
27 Making the choice X = A, B, D, E Y = Z, B, E Choice: include one element into the common sequence (E) and solve the resulting subproblem X = A, B, D, G Y = Z, B, D Choice: exclude an element from a string and solve the resulting subproblem 7
28 Notations Given a sequence X = x, x,, x m we define the i-th prefix of X, for i =,,,, m X i = x, x,, x i c[i, j] = the length of a LCS of the sequences X i = x, x,, x i and Y j = y, y,, y j 8
29 A Recursive Solution Case : x i = y j e.g.: X i = A, B, D, E Y j = Z, B, E c[i, j] = c[i -, j - ] + Append x i = y j to the LCS of X i- and Y j- Must find a LCS of X i- and Y j- optimal solution to a problem includes optimal solutions to subproblems 9
30 A Recursive Solution Case : x i y j e.g.: X i = A, B, D, G Y j = Z, B, D c[i, j] = max { c[i -, j], c[i, j-] } Must solve two problems find a LCS of X i- and Y j : X i- = A, B, D and Y j = Z, B, D find a LCS of X i and Y j- : X i = A, B, D, G and Y j = Z, B Optimal solution to a problem includes optimal solutions to subproblems 3
31 Overlapping Subproblems To find a LCS of X and Y we may need to find the LCS between X and Y n- and that of X m- and Y Both the above subproblems has the subproblem of finding the LCS of X m- and Y n- Subproblems share subsubproblems 3
32 3. Computing the Length of the LCS if i = or j = c[i, j] = c[i-, j-] + if x i = y j max(c[i, j-], c[i-, j]) if x i y j n y j: y y y n x i x x first second i m x m j 3
33 Additional Information if i,j = c[i, j] = c[i-, j-] + if x i = y j max(c[i, j-], c[i-, j]) if x i y j b & c: x i A B 3 C m D 3 n A C D F y j: c[i-,j] i c[i,j-] j A matrix b[i, j]: For a subproblem [i, j] it tells us what choice was made to obtain the optimal value If x i = y j b[i, j] = Else, if c[i -, j] c[i, j-] else b[i, j] = b[i, j] = 33
34 LCS-LENGTH(X, Y, m, n). for i to m. do c[i, ] 3. for j to n 4. do c[, j] 5. for i to m 6. do for j to n 7. do if x i = y j The length of the LCS if one of the sequences is empty is zero 8. then c[i, j] c[i -, j - ] + 9. b[i, j ]. else if c[i -, j] c[i, j - ]. then c[i, j] c[i -, j]. b[i, j] 3. else c[i, j] c[i, j - ] 4. b[i, j] 5.return c and b Case : x i = y j Case : x i y j Running time: (mn) 34
35 Example X = A, B, C, B, D, A Y = B, D, C, A, B, A If x i = y j b[i, j] = Else if c[i -, j] c[i, j-] else b[i, j] = b[i, j] = if i = or j = c[i, j] = c[i-, j-] x i A B C B D A B if x i = y j max(c[i, j-], c[i-, j]) if x i y j y j B D C A B A
36 4. Constructing a LCS Start at b[m, n] and follow the arrows When we encounter a in b[i, j] x i = y j is an element of the LCS x i A B C B D A B y j B D C A B A
37 PRINT-LCS(b, X, i, j). if i = or j =. then return 3. if b[i, j] = 4. then PRINT-LCS(b, X, i -, j - ) 5. print x i 6. elseif b[i, j] = Running time: (m + n) 7. then PRINT-LCS(b, X, i -, j) 8. else PRINT-LCS(b, X, i, j - ) Initial call: PRINT-LCS(b, X, length[x], length[y]) 37
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