Memoization

Top-Down recursive solution that remembers intermediate results.

For example, intermediate results found in m[2,4] are useful in determining the value of m[1,3].

Memoized-Matrix-Chain(p)
1 $\;\;\;\;\;$n = length(p) - 1
2 $\;\;\;\;\;$for i = 1 to n
3 $\;\;\;\;\;$ $\;\;\;\;\;$for j = i to n
4 $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$m[i,j] = $\infty$
5 $\;\;\;\;\;$return Lookup-Chain(p, 1, n)

Lookup-Chain(p, i, j)
1 $\;\;\;\;\;$if m[i,j] <$\infty$
2 $\;\;\;\;\;$then return m[i,j]
3 $\;\;\;\;\;$if i = j
4 $\;\;\;\;\;$then m[i,j] = 0
5 $\;\;\;\;\;$else for k = i to j-1
6 $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$q = Lookup-Chain(p, i, k) +
$\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$Lookup-Chain(p, k+1, j) + P[i-1]P[k]P[j]
7 $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$if q < m[i,j]
8 $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$then m[i,j] = q
9 $\;\;\;\;\;$return m[i,j]

In this algorithm each of $\Theta(n^2)$ entries is initialized once (line 4) and is filled in by one call to Lookup-Chain.

Each of calls to Lookup-Chain takes n steps ignoring recursion, so the total time required is .

The algorithm requires memory.