Dynamic Programming 1 Neil Tang 4/15/2008 CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Class Overview Basic Idea Fibonacci numbers Recursive equation evaluation All-pairs shortest paths 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

CS223 Advanced Data Structures and Algorithms Fibonacci Numbers fib(N) = fib(N-1) + fib(N-2) CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Fibonacci Numbers CS223 Advanced Data Structures and Algorithms

CS223 Advanced Data Structures and Algorithms Fibonacci Numbers A dynamic programming based algorithm CS223 Advanced Data Structures and Algorithms

Recursive Equation Evaluation CS223 Advanced Data Structures and Algorithms

Recursive Equation Evaluation CS223 Advanced Data Structures and Algorithms

Recursive Equation Evaluation A dynamic programming based algorithm CS223 Advanced Data Structures and Algorithms

All-pairs Shortest Path The all-pairs shortest path problem: Given a weighted graph G, find the shortest (minimum cost) path between every pair of nodes in G. CS223 Advanced Data Structures and Algorithms

All-Pairs Shortest Path Recursive expression for node pair (i, j) Dk,i,j = min{Dk-1,i,j, Dk-1,i,k+Dk-1,k,j} CS223 Advanced Data Structures and Algorithms

All-Pairs Shortest Path Time Complexity: O(|V|3) CS223 Advanced Data Structures and Algorithms