coin change top-down

The coin change problem can be solved using a top-down approach in C++. This approach involves using recursion and memoization to find the minimum number of coins needed to make a certain amount of change. Here are the steps to solve the problem:

1. Define a recursive function to solve the coin change problem. This function should take the amount of change to make and an array of coin denominations as parameters.

2. Inside the function, check if the base case is met. The base case occurs when the amount of change to make is 0. In this case, return 0 as no coins are needed.

3. If the base case is not met, initialize a variable to store the minimum number of coins needed. Set its initial value to a large number.

4. Iterate through the available coin denominations. For each denomination, check if it is less than or equal to the amount of change to make.

5. If the current denomination is valid, make a recursive call to the function with the updated amount of change (original amount minus the current denomination) and add 1 to the result (since we are using one coin of the current denomination).

6. Compare the result of the recursive call with the current minimum number of coins needed. If the result is smaller, update the minimum number of coins.

7. Store the result in a memoization array to avoid redundant calculations for the same amount of change.

8. Return the minimum number of coins needed.

By following these steps, you can implement the top-down approach to solve the coin change problem in C++. This method efficiently handles overlapping subproblems and can significantly improve the performance of the solution.