Dynamic Programming 1 Neil Tang 4/20/2010 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 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 Time Complexity: O(n) 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 Time Complexity: O(n2) 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. A simple solution: run Dijsktra’s algorithms |V| times, which leads to a time complexity of O(|E||V|log|V|). CS223 Advanced Data Structures and Algorithms

All-Pairs Shortest Path Recursive expression for any 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