Coin Change in c#

General
1

The Coin Change problem involves determining the minimum number of coins needed to make a certain amount of money using a given set of coin denominations. You can solve this problem using dynamic programming for optimal efficiency.

Algorithm Explanation

  1. Initialization: Create an array to hold the minimum number of coins needed for each amount from 0 to the target amount. Initialize this array with a value larger than any possible number of coins (like amount + 1), except for the 0 amount, which should be 0 coins.

  2. Dynamic Programming Approach: Iterate through each coin denomination and for each coin, update the array for all amounts from that coin’s value up to the target amount. For each amount, determine if using the coin leads to a smaller number of coins than previously recorded.

  3. Final Result: After processing all coins, the last element in the array will either contain the minimum number of coins needed for the target amount or indicate that it’s not possible to form that amount.

C# Implementation

Here’s a sample implementation of the Coin Change problem:

using System;

class Program
{
    static void Main()
    {
        int[] coins = { 1, 2, 5 }; // Coin denominations
        int amount = 11; // Target amount
        int result = CoinChange(coins, amount);
        
        Console.WriteLine(result == -1 ? "Not possible to make that amount" : $"Minimum coins needed: {result}");
    }

    static int CoinChange(int[] coins, int amount)
    {
        // Initialize the DP array
        int[] dp = new int[amount + 1];
        for (int i = 1; i <= amount; i++)
        {
            dp[i] = amount + 1; // Set to a value larger than any possible number of coins
        }

        dp[0] = 0; // Base case: zero coins are needed to make the amount of 0

        // Fill the DP array
        foreach (var coin in coins)
        {
            for (int i = coin; i <= amount; i++)
            {
                dp[i] = Math.Min(dp[i], dp[i - coin] + 1);
            }
        }

        // Check if we can make the target amount
        return dp[amount] > amount ? -1 : dp[amount];
    }
}

Explanation of Code

  1. Input: The program defines an array of coin denominations (coins) and a target amount (amount).
  2. CoinChange Function:
    • It initializes a dynamic programming (DP) array dp of size amount + 1. Each index represents the minimum number of coins required to make that amount.
    • The array is initialized such that dp[i] is set to amount + 1 for all amounts greater than 0. The base case dp[0] is set to 0 since no coins are needed to make the amount 0.
  3. Filling the DP Array:
    • For each coin, iterate through all amounts from that coin’s value up to the target amount.
    • Update the DP array at each index i by considering the minimum coins needed between the current value and the value of using the current coin (dp[i - coin] + 1).
  4. Return the Result: After filling the DP array, check if dp[amount] is still greater than amount. If it is, return -1 (indicating it’s not possible to form that amount); otherwise, return dp[amount].

Complexity

  • Time Complexity: O(n * m), where n is the target amount and m is the number of coin denominations. This is because we have a nested loop over coins and amounts.
  • Space Complexity: O(n), for the DP array that holds the minimum coins for each amount up to the target.

This dynamic programming approach provides an efficient way to solve the Coin Change problem, making it manageable even for larger amounts and coin sets.