It is rather complicated to consider the advance level of dynamic programming at the beginning, so I have decided to start with a basic example: finding Fibonacci numbers.

Since the concepts of recursion and memorization are often required in dynamic programming, therefore, I have implemented a version of memorization.

- Set base case of 0 and 1 equal to 1
- Check if
`fib(n)`

already exist in the array - If yes, return it directly
- If no, return it as
`fib(n-1)`

+`fib(n-2)`

like ordinary approach

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#include<bits/stdc++.h> using namespace std; int fib(int n, vector<int>&mem){ printf("fib(%d)\n", n); int res = 0; if(n == 0 || n == 1) return 1; if(mem[n] != -1){ printf("Using fib(%d)\n", n); return mem[n]; } else{ mem[n] = fib(n-1, mem) + fib(n-2, mem); //printf("fib(%d) = %d saved \n", n, res); } return mem[n]; } int main(){ int n = 10; vector<int> mem (n + 1, -1); printf("Ans = %d\n", fib(n, mem)); return 0; } |

Console output:

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fib(10) fib(9) fib(8) fib(7) fib(6) fib(5) fib(4) fib(3) fib(2) fib(1) fib(0) fib(1) fib(2) Using fib(2) fib(3) Using fib(3) fib(4) Using fib(4) fib(5) Using fib(5) fib(6) Using fib(6) fib(7) Using fib(7) fib(8) Using fib(8) |

It can be observed that starting from `fib(2)`

, we have obtained the result and store it into the table, which give `O(1)`

future access time.