Step 2: Define recursive solution

Let , where A_i has dimensions P[i-1] x P[i]. P is an array of dimensions.

For now, the subproblems will be finding the minimum number of scalar multiplications m[i,j] for computing A_{i .. j}(1 $\leq$ i $\leq$ j $\leq$ n).

Define m[i,j].

• If i = j, m[i,j] = 0 (single matrix).

• If i < j, assume an optimal split between A_k and A_k+1(i $\leq$ k < j).
  m[i,j] = cost of computing A_{i .. k} + cost of computing A_{k+1 .. j} + cost of computing A_{i .. k} A_{k+1 .. j}
  = m[i,k] + m[k+1,j] + P[i-1]P[k]P[j]
  However, we do not know the value of k, so we have to try all j-i possibilities.

$$m[i,j] = \left\{ \begin{array}{ll} 0 & {\rm if} \; i=j \\ m... ...,j] + P[i-1]P[k]P[j] & {\rm if} \; i<j \\ \end{array} \right.$$

Note that a recursive algorithm based on this definition would still require exponential time.