Longest common subsequence: Difference between revisions
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Replaced use of the word Sigma with the actual Greek Letter. Used bold font for emphasis.
(Clarified statement of the Longest Common Subsequence Problem) |
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The '''Longest Common Subsequence''' (or [http://en.wikipedia.org/wiki/Longest_common_subsequence_problem '''LCS''']) is a subsequence of maximum length common to two (or more) strings.
Let x = x[0]... x[m-1] and y = y[0]... y[n-1], m <= n be strings drawn from an alphabet
An ordered pair (i, j) will be called a match if x[i] == y[j], where 0 <= i < m and 0 <= j < n.
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Define a strict Cartesian product-order over these ordered pairs, such that (i1, j1) < (i2, j2) iff i1 < j1 and i2 < j2.
Given such a product-order (<) over a set of matches M, a chain '''C''' is any subset of M where either p1 < p2 or p2 < p1, for distinct pairs p1 and p2 in C.
Finding the LCS can then be viewed as the problem of finding a chain of maximum cardinality over the set of matches '''M'''.
This set of matches can be represented as an m*n bit matrix, where each bit '''M[i, j]''' is True iff there is a match at the corresponding positions of strings x and y.
A chain can then be visualized as any monotonically increasing curve through those match bits which are set to True.
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