Recursive Solution

Consider a recursive solution:

Let p = <p₀, p₁, .., p_n> be the sequence of dimensions.

Recursive-Matrix-Chain(p,i,j)
$\;\;\;\;\;$if i = j
$\;\;\;\;\;$then return 0
$\;\;\;\;\;$m[i,j] = $\infty$
$\;\;\;\;\;$for k = i to j - 1
$\;\;\;\;\;$ $\;\;\;\;\;$q = Recursive-Matrix-Chain(p,i,k) +
$\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$Recursive-Matrix-Chain(p,k+1,j) +
$\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$P[i-1]P[k]P[j]
$\;\;\;\;\;$ $\;\;\;\;\;$if q < m[i,j]
$\;\;\;\;\;$ $\;\;\;\;\;$then m[i,j] = q
$\;\;\;\;\;$return m[i,j]

Analysis:

$$T(n) = \left\{ \begin{array}{ll} \Theta(1) & n = 1 \\ \The... ...^{n-1} (T(k) + T(n-k) + \Theta(1)) & n > 1 \end{array} \right.$$

$$\begin{eqnarray*}T(n) & = & \Theta(1) + \sum_{k=1}^{n-1} (T(k) + T(n-k) + \Theta... ..._{k=1}^{n-1} T(k) \\ & = & \Theta(n) + 2 \sum_{k=1}^{n-1} T(k) \end{eqnarray*}$$