Dynamic Programming 2 Neil Tang 4/22/2010 CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Course Survey Please complete the course survey at: http://www.cs.montana.edu/survey/ CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Class Overview Basic idea Matrix multiplication Ordering matrix multiplication CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Basic Idea Mathematically express the problem in the recursive form. Solve it by a non-recursive algorithm that systematically records the answers to the subproblems in a table. CS223 Advanced Data Structures and Algorithms

Matrix Multiplication C=AB. If the size of A is wy, then the size of B must be yz, i.e., the number of columns of A must be equal to the number of rows of B. The size of C is wz and CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms An Example Compute ABCD with |A|=5010, |B| = 1040, |C|=4030 and |D|=305. (((AB)C)D) involves 20,000+60,000+7,500 =87,500 multiplications. (A((BC)D)) involves 12,000+1,500+2,500 =16,000 multiplications. (A(B(CD))) involves 6,000+2,000+2,500 =10,500 multiplications. CS223 Advanced Data Structures and Algorithms

Ordering Matrix Multiplications Consider AleftAleft+1…Ai…Aright mleft,right = min leftiright {mleft,i + mi+1,right+cleft-1cicright} CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Another Example right left right left CS223 Advanced Data Structures and Algorithms

Ordering Matrix Multiplications The time complexity is O(n3) CS223 Advanced Data Structures and Algorithms