Longest common subsequence: Difference between revisions
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An ordered pair (i, j) will be called a match if x[i] == y[j], where 0 <= i < m and 0 <= j < n. |
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. |
Define a strict Cartesian product-order (<) over these ordered pairs, such that (i1, j1) < (i2, j2) iff i1 < j1 and i2 < j2. |
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Given such a product-order |
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. |
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Finding |
Finding an '''LCS''' can then be viewed as the problem of finding a chain of maximum cardinality over the set of matches '''M'''. |
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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. |
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. |
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Any chain '''C''' can then be visualized as a monotonically increasing curve through match bits which are set to True. |
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For example, the sequences "1234" and "1224533324" have an LCS of "1234": |
For example, the sequences "1234" and "1224533324" have an LCS of "1234": |
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