Minimum Coin Change Problem in Python
The minimum coin change problem is a classical problem in computer science and algorithms, where the goal is to determine the minimum number of coins needed to make a specific amount using a given set of denominations. Here's a step-by-step guide to solving this problem using dynamic programming in Python.
Dynamic Programming Approach
The dynamic programming approach involves building a solution incrementally by solving subproblems and storing their results to avoid redundant calculations.
Function Definition
Below is a Python function that implements this approach:
def min_coins(coins, amount):
# Create a list to store the minimum coins required for each amount
dp = [float('inf')] * (amount + 1)
dp[0] = 0 # Base case: 0 coins are required to make amount 0
# Iterate through each coin in the coins list
for coin in coins:
for x in range(coin, amount + 1):
# Update the dp array by checking the minimum coins required
dp[x] = min(dp[x], dp[x - coin] + 1)
# Check if the amount can be formed; if not, return -1
return dp[amount] if dp[amount] != float('inf') else -1
# Example usage:
coins = [1, 3, 4]
amount = 6
print(min_coins(coins, amount)) # Output: 2 (2 coins of denomination 3)
Explanation of the Code
1. Initialization:
- An array dp is created where dp[i] holds the minimum number of coins needed to make the amount i. Initialize dp with float('inf') to represent that the amount cannot be formed initially. The base case dp[0] is set to 0 since no coins are needed to make amount 0.
2. Filling the DP Array:
- The outer loop iterates over each coin denomination.
- The inner loop iterates through all amounts from the current coin value to the target amount. For each amount, it checks if using the coin would result in a smaller number of coins compared to the current minimum stored in dp.
3. Final Decision:
- Finally, the function checks if it's possible to form the specified amount. If dp[amount] is still float('inf'), it means the amount cannot be formed with the given coins, and the function returns -1. Otherwise, it returns the minimum number of coins needed.
Time Complexity
The time complexity of this algorithm is \( O(n \times m) \), where \( n \) is the number of coin denominations and \( m \) is the target amount. The space complexity is \( O(m) \) due to the usage of the dp array.
Conclusion
This dynamic programming approach efficiently computes the minimum number of coins needed for a specified amount. By utilizing subproblem results, it avoids unnecessary recalculations, making it suitable for larger inputs within reasonable limits. This method highlights the power of dynamic programming in solving optimization problems in an effective manner.