Dynamic Programming

1. Dynamic Programming Heejin Park College of Information and Communications Hanyang University

2. Content • Introduction • Assembly-line scheduling • Matrix-chain multiplication • Elements of dynamic programming • Longest common subsequence

3. Matrix-chain multiplication • Matrix-chain multiplication • Given a chain of n matrices <A1, A2, ..., An>, compute the product A1A2…An. • The number of ways to compute the product A1A2 …An. = the number of ways to fully parenthesize the product. • The number of ways to compute the product A1A2A3 is 2. • (A1A2) A3 • A1(A2 A3)

4. Matrix-chain multiplication • The order of multiplications • The order of multiplications does not change the value of the product because matrix multiplication is associative. • For example, whether the left multiplication is done first or the right multiplication is done first does not matter. ( A1·A2) ·A3 = A1· (A2·A3) • However, the order of multiplication affects the number of scalar multiplications needed to compute the product.

5. (A2x3) (B 3x2) (C2x2) Matrix-chain multiplication • Multiplying two matrices A and B • We can multiply them if they are compatible: the number of columns of A must equal the number of rows of B. • If A is a p×qmatrix and B is aq×rmatrix, the resulting matrix is a p×r matrix. x =

6. Matrix-chain multiplication • The number of scalar multiplications to multiply A and B. • It is pqr because we compute pr elements and computing each element needs qscalar multiplications.

7. Matrix-chain multiplication • The order of multiplications affects the number of scalar multiplications. • Computing A1A2A3 where A1: 10×100 A2: 100×5 A3 : 5×50 • (A1A2) A3 • (A1A2) = 10*100*5 = 5000, (10 ×5) A3 = 10*5*50 = 2500 =>5000 + 2500 = 7,500 • A1 (A2A3) • (A2A3) = 100*5*50 =25000, A1(100 × 50) = 10 *100*50 = 50000 =>25000 + 50000 = 75,000 • Computing (A1A2) A3 is 10 times faster.

8. Matrix-chain multiplication • Matrix-chain multiplication problem • Given a chain A1, A2, ..., An of n matrices, where matrix Ai has dimension pi-1×pi, find the order of matrix multiplications minimizing the scalar multiplications to compute the product. • That is, to fully parenthesize the product of matrices minimizing scalar multiplications. • For example, for the product A1A2A3A4, a fully parenthesization is ((A1A2) A3) A4.

9. Matrix-chain multiplication • Solutions of the matrix-chain multiplication problem • Brute-force approach • Enumerate all possible parenthesizations. • Compute the number of scalar multiplications of each parenthesization. • Select the parenthesization needing the least number of scalar multiplications.

10. Matrix-chain multiplication • The Brute-force approach is inefficient. • The number of enumerated parenthesizations is Ω(4n/n3/2). • The number of parenthesizations of a product of n matrices, denoted by P(n), is as follows. if n=1 if n≥2

11. Matrix-chain multiplication • The product A1A2A3A4 can be fully parenthesized in five distinct ways. A1(A2(A3A4)), A1((A2A3) A4), (A1A2)(A3A4), (A1(A2A3))A4, ((A1A2) A3) A4.

12. Matrix-chain multiplication • P(n): The number of alternative parenthesizations of a sequence of n matrices. • The split between the two subproducts may occur between the kth and (k + 1)st matrices for any k = 1, 2, ..., n – 1. if n=1 if n≥2

13. if i=j if i<j Matrix-chain multiplication • Dynamic programming • Optimal substructure • m[i,j]: The minimum number of scalar multiplications for computing AiAi+1 …Aj. • matrix Ai : pi-1× pi • computing Ai..k Ak+1..jtakes pi-1pk pjscalar multiplications. • s[i, j] stores the optimal kfor tracing the optimal solution.

14. Matrix-chain multiplication 15750 7875 9375 11875 15125 1 1 3 3 3 3 2625 4375 7125 10500 2 3 3 750 5375 3 3 3 2500 1000 3500 4 5 5 5000 s m +2500 +35·15·20 0 =13000, 2625 +1000 +35·5·20 =7125, 4375 +0 +35·10·20 =11375 i=2, j=5, i≤k<j

15. Matrix-chain multiplication • Running time • O(n3) time in total • Θ(n2) subproblems • O(n) time for each subproblem • Space consumption • Θ(n2) space to store the m and s tables.

16. Content • Introduction • Assembly-line scheduling • Matrix-chain multiplication • Elements of dynamic programming • Longest common subsequence

17. Elements of dynamic programming • Elements of dynamic programming • Optimal substructure • Overlapping subproblems

18. Element of dynamic programming • Subtleties • Unweighted longest simple path problem • Does it have optimal substructure? p1 p2 u w v p

19. Elements of dynamic programming q r s t

20. Elements of dynamic programming • Overlapping subproblems • When a recursive algorithm revisits the same problem over and over again, that is the optimization problem has overlapping subproblems.

21. Elements of dynamic programming 1...4 Matrix chain multiplication: top-down vs. bottom-up 1...1 2...4 1...2 3...4 1...3 4...4 2...2 3...4 2...3 4...4 1...1 2...2 3...3 4...4 1...1 2...3 1...2 3...3 3...3 4...4 2...2 3...3 2...2 3...3 1...1 2...2

22. Elements of dynamic programming • Memoization • Recursive solution but solve each subproblem only once. • Fills the table in recursive way. • In most case, it is slower than dynamic programming. • It is useful when only a part of subproblems is solved.

23. Elements of dynamic programming • The running time of a dynamic-programming algorithm depends on the product of two factors. • The number of subproblems overall. • How many choiceseach subproblem has. • Assembly line scheduling • Θ(n) subproblems · 2 choices = Θ(n) • Matrix chain multiplication • Θ(n2) subproblems · (n-1) choices = O(n3)