UVa 348 - Optimal Array Multiplication Sequence

348 - Optimal Array Multiplication Sequence

 * http://acm.uva.es/p/v3/348.html

Summary
Compute the optimal ordering of multiplications for all sequences of matrices of lengths 1 through n, in that order.

Explanation
Suppose for matrices $$A_1 A_2 ... A_{n-1} A_n$$ we know the optimal number of multiplications needed to multiply any consecutive sequence of less than n of the matrices. Then the optimal number of multiplications needed to multiply the n matrices is $$a(1,n)=min[a(1,i)+a(i,n)+rows(A_1)*rows(A_i)*columns(B_n), 1<=i<=n]$$. Straightforward dynamic programming: first calculate all $$a(i,i)$$ as 0, then calculate all $$a(i,j)$$ with $$j-i=1$$, then with $$j-i=2$$, $$j-i=3$$, etc. Since we want the parenthesized expression rather than the optimal number of multiplications, we can simply store a partial solution at each stage, i.e., store a string "(A1 x (A2 x A3))" as b(1,3) in case 3 of the examples.

Input
3 1 5 5 20 20 1 3 5 10 10 20 20 35 6 30 35 35 15 15 5 5 10 10 20 20 25 0

Output
Case 1: (A1 x (A2 x A3)) Case 2: ((A1 x A2) x A3) Case 3: ((A1 x (A2 x A3)) x ((A4 x A5) x A6))