Longest common subsequence problem

http://en.wikipedia.org/wiki/Longest_common_subsequence_problem

 

LCS function defined

Let two sequences be defined as follows: X = (x1, x2...xm) and Y = (y1, y2...yn). The prefixes of X are X1, 2,...m; the prefixes of Y are Y1, 2,...n. Let LCS(Xi, Yj) represent the set of longest common subsequence of prefixes Xi and Yj. This set of sequences is given by the following.

LCS\left(X_{i},Y_{j}\right) = \begin{cases} \empty & \mbox{ if }\ i = 0 \mbox{ or } j = 0 \\ \textrm{ ( } LCS\left(X_{i-1},Y_{j-1}\right), x_i) & \mbox{ if } x_i = y_j \\ \mbox{longest}\left(LCS\left(X_{i},Y_{j-1}\right),LCS\left(X_{i-1},Y_{j}\right)\right) & \mbox{ if } x_i \ne y_j \\ \end{cases}

To find the longest subsequences common to Xi and Yj, compare the elements xi and yi. If they are equal, then the sequence LCS(Xi-1, Yj-1) is extended by that element, xi. If they are not equal, then the longer of the two sequences, LCS(Xi, Yj-1), and LCS(Xi-1, Yj), is retained. (If they are both the same length, but not identical, then both are retained.) Notice that the subscripts are reduced by 1 in these formulas. That can result in a subscript of 0. Since the sequence elements are defined to start at 1, it was necessary to add the requirement that the LCS is empty when a subscript is zero.

 

http://www.csie.ntnu.edu.tw/~u91029/LongestCommonSubsequence.html 

posted on 2010-05-21 20:42  cutepig  阅读(522)  评论(0)  编辑  收藏  举报

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