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Longest Common Subsequence (LCS)

Problem: Given two sequences X = $\langle x_1, .., x_m \rangle$ and Y = $\langle y_1, .., y_n \rangle$ , find the longest subsequence Z = $\langle z_1, .., z_k \rangle$ that is common to x and y.

A subsequence is a subset of elements from the sequence with strictly increasing order (not necessarily contiguous).

For example, if X = $\langle$ A,B,C,B,D,A,B $\rangle$ and Y = $\langle$ B,D,C,A,B,A $\rangle$ , then some common subsequences are:

$\langle$ A $\rangle$
$\langle$ B $\rangle$
$\langle$ C $\rangle$
$\langle$ D $\rangle$
$\langle$ A,A $\rangle$
$\langle$ B,B $\rangle$
$\langle$ B,C,A $\rangle$
$\langle$ B,C,B,A $\rangle$ This is one of the longest common subsequences.
$\langle$ B,D,A,B $\rangle$ This is one of the longest common subsequences.

Brute Force: Check all 2^m subsequences of X for an occurrence in Y.

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