Tabulation – the Bottom-Up Approach

The heart of dynamic programming is tabulation, which is the inverse approach to memoization. In fact, though the term dynamic programming is sometimes applied to both memoization and tabulation, its use is generally assumed to refer specifically to the latter.

The standard implementation of tabulation consists of storing the solutions for the base cases and then iteratively filling a table with the solutions for every subproblem, which can then be reused to find the solutions for other subproblems. Tabulated solutions are generally considered to be a bit harder to conceptualize than memoized ones because the state of each subproblem must be represented in a way that can be expressed iteratively.

A tabulated solution to computing the Fibonacci sequence would look like this:

int Fibonacci(int n)

{

        vector<int> DP(n + 1, 0);

        DP[1] = 1;

   ...