Optimized Dynamic Programming Solution

The previous dynamic programming solution we discussed solves the coin change problem by exploring all possible combinations. However, it has a time complexity of O(N * amount), where N is the number of coins and amount is the desired change. This can be inefficient for large values of amount or a large number of coins.

Fortunately, we can optimize the dynamic programming solution by using a different approach.

The key idea is to use one-dimensional array dp instead of a two-dimensional array to store the minimum number of coins needed to make each value from 0 to amount. This reduces the space complexity from O(N * amount) to O(amount).

Here's the implementation of the optimized dynamic programming solution in C#:

1{{code}}

In this solution, we create an array dp of size amount + 1 and initialize all its values to amount + 1, except for dp[0] which is set to 0 since we don't need any coins to make zero change.

We then iterate through the coins array and for each coin coin, we iterate from coin to amount. For each value i, we update dp[i] to be the minimum between its current value and dp[i - coin] + 1. This represents the minimum number of coins needed to make the amount i using the current coin and any previously calculated solutions.

Finally, we return dp[amount], which will be the minimum number of coins needed to make the given amount of change. If dp[amount] is greater than amount, it means that it was not possible to make the amount using the given coins, so we return -1.

The optimized dynamic programming solution improves the time complexity to O(N * amount), where N is the number of coins and amount is the desired change, and the space complexity to O(amount). This makes it more efficient than the previous solution and suitable for larger inputs.

Let's run the code and see the output:

1Minimum number of coins needed: 3