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Longest common
Longest Common Subsequence The Longest Common Subsequence (LCS) is the longest sequence of characters that appears in every substring of a given text. It...
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Longest Common Subsequence The Longest Common Subsequence (LCS) is the longest sequence of characters that appears in every substring of a given text. It...
The Longest Common Subsequence (LCS) is the longest sequence of characters that appears in every substring of a given text. It's like the longest shared "road" between all the different parts of the text.
Formal Definition:
Given two strings, x and y, the Longest Common Subsequence Length (LCS) is the length of the longest subsequence in x that is also a subsequence of y.
Example:
Suppose we have the following two strings:
abcba
abcabc
The longest common subsequence for these two strings is "abc". This is because it appears in both strings in the same order and length.
Properties:
The length of the LCS is always greater than or equal to the length of the strings.
If x and y are disjoint (don't overlap), then the LCS length is 0.
The LCS of two strings is always a subsequence of the LCS of their superstrings.
Applications:
Sequence alignment: finding the most similar sequences in a collection of strings.
String searching: finding all occurrences of a substring in a larger string.
Pattern recognition: identifying repeating patterns in a larger dataset.
Algorithm:
The most efficient way to find the LCS is dynamic programming. We can build a table where each cell holds the length of the LCS for a given substring of the two strings. Then, the LCS can be retrieved by traversing the table from the bottom-up.
Time complexity: O(n * m), where n and m are the lengths of the two strings.
Space complexity: O(n * m), as we need to build the table on the fly