Algorithm to bracket an expression in order to maximize its value

Question

I found this while looking up problems on dynamic programming. You are given an unparanthesized expression of the form V0 O0 V1 O1 .... Vn-1

We have to put brackets

Answer 1

It is easier to analyze calculation of A[i,j] elements from shorter ranges to longer ranges. Algorithm for that looks like:

# Initialization of single values
for i in 0, ..., n-1:
  A[i,i] = V[i]

# Iterate through number of operation
for d in 1, ..., n-1:
  # Range start
  for i in 0, ..., n-1-d:
    j = i + d
    A[i,j] = max( A[i,x] O_x A[x+1,j] for x in i, ..., j-1)

print 'Result', A[0,n-1]

Since A[] can be implemented with constant time access (array) than algorithm is O(n^3).