Introduction to Algorithms, Second Edition

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The University of Texas at Austin Lecture 8 Department of Computer Sciences Professor Vijaya Ramachandran Dynamic Programming CS357: ALGORITHMS, Spring 2006 1 Dynamic Programming In this lecture we will study an algorithmic technique called ‘dynamic programming’. Dynamic programming and the greedy strategy that we will study later solve ‘optimization’ problems. An optimization problem is one for which an input has a collection of feasible solutions, each with an associated cost, and we need to find a feasible solution that optimizes the cost; here ‘optimizes’ would mean either minimizes or maximizes , depending on the nature of the problem. Such a solution is called an optimal solution. We will now study a dynamic programming algorithm for the Longest Common Subsequence (LCS) problem. 1.1 Longest Common Subsequence Problem A sequence Z = < z 1 , · · · , z k > is a subsequence of another sequence X = < x 1 , · · · , x m > if there exists a strictly increasing sequence of indices 1 i 1 < i 2 · · · < i k m such that x i j = z j , 1 j k .
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