Coin Change Problem: Mastering Dynamic Programming in JavaScript

12.2.4 Coin Change Problem

The Coin Change problem is a classic algorithmic challenge that exemplifies the power of dynamic programming. It is a quintessential problem for understanding how to break down complex problems into manageable subproblems, optimize solutions, and implement efficient algorithms in JavaScript.

Problem Description

The Coin Change problem can be described as follows:

• Objective: Given a set of coin denominations and a target amount, determine the minimum number of coins required to make up the target amount.
• Constraints: Assume an infinite supply of each coin denomination.

This problem is not only a staple in computer science education but also a practical problem-solving scenario in financial applications, vending machines, and resource allocation systems.

Understanding the Problem

The Coin Change problem is an optimization problem with overlapping subproblems, making it a perfect candidate for dynamic programming. The goal is to find the minimum number of coins needed to achieve a given amount using the available denominations.

Example

Consider the coin denominations [1, 2, 5] and a target amount of 11. The minimum number of coins required to make 11 is 3 (using two 5 coins and one 1 coin).

Dynamic Programming Approach

Dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is particularly useful for problems with overlapping subproblems and optimal substructure properties.

Formulating the DP Solution

1. Define the State:

   • Let dp[amount] represent the minimum number of coins needed to make up the amount.

2. Base Case:

   • dp[0] = 0: Zero coins are needed to make up the amount 0.

3. Recurrence Relation:

   • For each coin in the set of denominations, update the dp array:

     for (let coin of coins) {
       if (coin <= amount) {
         dp[amount] = Math.min(dp[amount], dp[amount - coin] + 1);
       }
     }
     

4. Final Solution:

   • The solution for the target amount will be stored in dp[amount]. If dp[amount] is still Infinity, it means the amount cannot be made up with the given coins.

Implementing the Solution in JavaScript

Below is the JavaScript implementation of the Coin Change problem using dynamic programming:

function coinChange(coins, amount) {
  const dp = Array(amount + 1).fill(Infinity);
  dp[0] = 0;

  for (let i = 1; i <= amount; i++) {
    for (let coin of coins) {
      if (coin <= i) {
        dp[i] = Math.min(dp[i], dp[i - coin] + 1);
      }
    }
  }
  return dp[amount] === Infinity ? -1 : dp[amount];
}

// Example usage:
const coins = [1, 2, 5];
const amount = 11;
console.log(coinChange(coins, amount)); // Output: 3

Explanation of the Code

• Initialization: The dp array is initialized with Infinity to signify that initially, no amount can be made up. dp[0] is set to 0 because no coins are needed to make the amount 0.
• Nested Loops: The outer loop iterates over each amount from 1 to amount. The inner loop iterates over each coin denomination.
• Updating dp: For each coin, if it can contribute to the current amount (i), update dp[i] to the minimum of its current value or dp[i - coin] + 1.
• Result: If dp[amount] is Infinity, return -1 indicating the amount cannot be made up. Otherwise, return dp[amount].

Edge Cases and Considerations

• Unreachable Amounts: If the target amount cannot be made up with the given denominations, the function returns -1.
• Zero Amount: If the target amount is 0, the function should return 0 as no coins are needed.
• Negative Amounts: The problem typically assumes non-negative amounts, but handling negative inputs gracefully is good practice.

Practice Exercises

To solidify your understanding, try solving these practice problems:

1. Test with Different Coin Sets:

   • Use the function with various sets of coins and amounts to see how it performs.

2. Output the Actual Coins Used:

   • Extend the problem to not only return the minimum number of coins but also the specific coins used to make up the amount.

3. Handle Large Inputs:

   • Test the function with large amounts and coin sets to evaluate its performance and efficiency.

Visualization

To better understand the dynamic programming approach, consider the following flowchart that illustrates the process of filling the dp array:

    graph TD;
	    A[Start] --> B[Initialize dp array with Infinity];
	    B --> C[Set dp[0] = 0];
	    C --> D[Iterate over amounts from 1 to target amount];
	    D --> E[For each coin, check if it can contribute to the current amount];
	    E --> F[Update dp[i] = min(dp[i], dp[i - coin] + 1)];
	    F --> G[Continue until all amounts are processed];
	    G --> H{dp[target amount] == Infinity?};
	    H -->|Yes| I[Return -1];
	    H -->|No| J[Return dp[target amount]];
	    I --> K[End];
	    J --> K;

Optimization Tips

• Space Optimization: The current implementation uses an array of size amount + 1. Consider using a rolling array to reduce space complexity.
• Early Termination: If the exact amount is found early, consider breaking out of the loop to save computation time.

Common Pitfalls

• Incorrect Initialization: Ensure the dp array is correctly initialized with Infinity and dp[0] is set to 0.
• Boundary Conditions: Pay attention to the conditions in the loops, especially when checking if a coin can contribute to the current amount.

Real-World Applications

The Coin Change problem has practical applications in various fields:

• Finance: Calculating the minimum number of currency notes or coins needed for a transaction.
• Resource Allocation: Optimizing the distribution of resources in logistics and supply chain management.
• Vending Machines: Determining the optimal way to dispense change.

Further Reading and Resources

For more in-depth understanding and practice, consider exploring the following resources:

• Dynamic Programming on GeeksforGeeks
• Coin Change Problem on LeetCode
• Introduction to Algorithms by Cormen et al.

Quiz Time!