Common Subsequence

A subsequence of a given sequence is obtained by deleting some elements (possibly none) from the original sequence. Given a sequence X = \langle x_1, x_2, \ldots, x_m \rangle, another sequence Z = \langle z_1, z_2, \ldots, z_k \rangle is a subsequence of $X$ if there is a strictly increasing sequence of indices \langle i_1, i_2, \ldots, i_k \rangle such that for all j = 1, 2, \ldots, k, x_{i_j} = z_j. For example, Z = \langle a, b, f, c \rangle is a subsequence of X = \langle a, b, c, f, b, c \rangle with indices \langle 1, 2, 4, 6 \rangle.

Given two sequences $X$ and $Y$, find the length of the longest common subsequence of $X$ and $Y$.