Apply Dynamic Programming

• Optimal Substructure
• Overlapping Subproblems

Note similarity to matrix multiplication:

$$m_{i,j} \;=\; \left\{ \begin{array}{lll} 0 & {\rm if} & i=j \... ...;+\; p_{i-1}p_kp_j\} & {\rm if} & i \neq j \end{array} \right.$$

In fact, our algorithm for computing D involves ``multiplying'' the adjacency matrix by itself n-1 times.