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Dynamic Programming

1. Optimal Substructure.

Define: Given X = $\langle x_1, .., x_m \rangle$ , the ith prefix of X, i = 0, .., m, is $X_i \;=\; \langle x_1, .., x_i \rangle$ . X₀ is empty.

Theorem 16.1

Let X = $\langle x_1, .., x_m \rangle$ and Y = $\langle y_1, .., y_n \rangle$ be sequences, and Z = $\langle z_1, .., z_k \rangle$ be any LCS of X and Y.

1.

If x_m = y_n, then z_k = x_m = y_n and Z_k-1 is an LCS of X_m-1 and Y_n-1.

2.

If $x_m \neq y_n$ , then $z_k \neq x_m$ implies that Z is an LCS of X_m-1 and Y.

3.

If $x_m \neq y_n$ , then $z_k \neq y_n$ implies that Z is an LCS of X and Y_n-1.

Thus the LCS problem has optimal substructure.

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