The Minimum Coin Change Problem is a classic algorithmic problem in computer science. The problem statement is as follows: given a target amount of money and a list of coin denominations, find the minimum number of coins needed to make up the target amount.
Here's an example to illustrate the problem: suppose the target amount is $11, and the available coin denominations are $1, $2, and $5. To make up the target amount of $11, you could use one $5 coin, one $2 coin, and two $1 coins, for a total of four coins. However, you could also use three $2 coins and one $1 coin, which would be a total of four coins as well. The goal is to find the minimum number of coins required.
The problem can be solved using a dynamic programming approach. Here's a step-by-step algorithm to solve the Minimum Coin Change Problem -
Create an array, let's call it "dp," with a length of target amount + 1, initialized with a value greater than the target amount (e.g., infinity).
Set dp[0] to 0 since we need zero coins to make up a target amount of zero.
Iterate over the array dp from 1 to target amount:
For each index i, iterate over the available coin denominations:
If the current coin denomination is less than or equal to i and dp[i - coin denomination] + 1 is less than dp[i], update dp[i] with dp[i - coin denomination] + 1.
After the iterations, dp[target amount] will contain the minimum number of coins required to make up the target amount.
If dp[target amount] is still greater than the target amount, it means it is not possible to make up the target amount with the given coin denominations.
Here's an example Python code implementation of the Minimum Coin Change Problem:
def min_coin_change(coins, target_amount):
dp = [float('inf')] * (target_amount + 1)
dp[0] = 0
for i in range(1, target_amount + 1):
for coin in coins:
if coin <= i and dp[i - coin] + 1 < dp[i]:
dp[i] = dp[i - coin] + 1
if dp[target_amount] > target_amount:
return -1 # Not possible to make up the target amount
else:
return dp[target_amount]
You can use this function by passing the list of coin denominations and the target amount as arguments. For example -
coins = [1, 2, 5] target_amount = 11 result = min_coin_change(coins, target_amount) print(result) # Output: 3
In this example, it will return 3, indicating that the minimum number of coins required to make up the target amount of 11 is 3.
Certainly! The Minimum Coin Change Problem has some additional variations and considerations that can be explored.
Finding the actual coin combinations: In addition to finding the minimum number of coins required, you can also modify the algorithm to determine the actual coin combinations that make up the target amount. This can be achieved by keeping track of the chosen coin at each step while building the dp array.
Unbounded vs. bounded coin change: The problem can be further categorized into unbounded and bounded coin change problems. In the unbounded version, you can use an unlimited number of coins of each denomination, while in the bounded version, you have a finite number of coins for each denomination. The algorithm can be adjusted accordingly.
Different optimization goals: Instead of minimizing the number of coins, you can modify the problem to optimize for other goals, such as minimizing the total value of coins used or finding the maximum number of ways to make change using the minimum number of coins.
Time and space complexity: The dynamic programming solution to the Minimum Coin Change Problem has a time complexity of O(n*m), where n is the target amount and m is the number of coin denominations. The space complexity is O(n), as we only need to store the dp array.
Greedy algorithm approach: While the dynamic programming solution guarantees an optimal result, there are cases where a greedy algorithm can provide a feasible solution but not necessarily the minimum. For example, if the available coin denominations are 1, 3, and 4, and the target amount is 6, the greedy algorithm would choose two 3 coins. However, the minimum number of coins required is actually two 2 coins.
Alternate algorithms: Besides dynamic programming, there are other algorithms that can solve the Minimum Coin Change Problem, such as recursive backtracking with memoization or using a matrix-based approach.
These are some additional aspects and variations related to the Minimum Coin Change Problem. By exploring these variations and considering different scenarios, you can deepen your understanding of the problem and its solutions.