Question

I have the following recursive function:

typedef unsigned long long ull;

ull calc(ull b, ull e)
{
  if (!b) return e;
  if (!e) return b;
  return calc(b - 1, e - 1) + calc(b - 1, e) - calc(b, e - 1);
}

I want to implement it with dynamic programming (i.e. using storage). I have tried to use a map<pair<ull, ull>, ull> but it is too slow also. I couldn’t implement it using arrays O(1) too.

I want to find a solution so that this function solves quickly for large b, es.

Answer 1

If a bottom up representation is what you want then this would do fine.

Fill up the table as MBo has shown

This can be done as:

for e from 0 to n:
  DP[0][e] = e
for b from 0 to n:
  DP[b][0] = b
for i from 1 to n:
   for j from 1 to n:
      DP[i][j] = DP[i-1][j-1] + DP[i-1][j] - DP[i][j-1]

now your answer for any b,e is simply DP[b][e]