Approach

To effectively answer the interview question on implementing a dynamic programming algorithm for the coin change problem, follow this structured framework:

1. Understand the Problem: Clearly define what the coin change problem is.

2. Identify Inputs and Outputs: Specify the inputs (coin denominations, amount) and the desired output (minimum number of coins).

3. Outline the Dynamic Programming Approach: Discuss how dynamic programming will optimize the solution by storing intermediate results.

4. Develop the Algorithm: Walk through the algorithm's steps in detail.

5. Code Implementation: Provide a code snippet demonstrating the solution.

6. Test Cases and Complexity: Discuss potential test cases and analyze the time and space complexity.

Key Points

• Clarity on the Problem: Understand that the coin change problem involves determining the minimum number of coins needed to make a specific amount from given denominations.

• Dynamic Programming Advantages: Highlight the efficiency of storing results to avoid redundant calculations.

• Code Clarity: Ensure that your code is well-documented and easy to follow.

• Performance Analysis: Be prepared to discuss the algorithm's performance and potential edge cases.

Standard Response

The coin change problem is a classic problem in dynamic programming, where the goal is to determine the minimum number of coins needed to make a specific amount. Here’s how I would approach the problem:

Problem Definition

Given an array of coin denominations and a total amount, we need to find the minimum number of coins that sum up to that amount. If it’s not possible to make the amount from the given denominations, we should return -1.

Inputs and Outputs

• Input:

• coins[]: An array of integers representing the coin denominations.

• amount: An integer representing the total amount we want to make.

• Output:

• An integer representing the minimum number of coins needed, or -1 if it cannot be made.

Dynamic Programming Approach

• Create a DP Array: Initialize an array dp where dp[i] will store the minimum number of coins needed to make amount i.

• Base Case:

• Set dp[0] = 0 (0 coins are needed to make the amount 0).

• Initialize all other entries as Infinity (or a large number), as we want to minimize coin count.

• Fill the DP Array:

• Iterate through each coin.

• For each coin, update the dp array for all amounts from the coin value to the target amount.

• For each amount i, update dp[i] = min(dp[i], dp[i - coin] + 1) if dp[i - coin] is not Infinity.

Code Implementation

Here’s a sample Python implementation:

def coinChange(coins, amount):
 # Initialize dp array with infinity
 dp = [float('inf')] * (amount + 1)
 dp[0] = 0 # Base case

 # Fill the dp array
 for coin in coins:
 for i in range(coin, amount + 1):
 dp[i] = min(dp[i], dp[i - coin] + 1)

 return dp[amount] if dp[amount] != float('inf') else -1

# Example usage
coins = [1, 2, 5]
amount = 11
print(coinChange(coins, amount)) # Output: 3 (11 = 5 + 5 + 1)

Test Cases and Complexity

• Test Cases:

• Input: coins = [1, 2, 5], amount = 11 ➔ Output: 3

• Input: coins = [2], amount = 3 ➔ Output: -1

• Input: coins = [1], amount = 0 ➔ Output: 0

• Time Complexity: O(n * m), where n is the number of coin denominations and m is the target amount.

• Space Complexity: O(m) for the dp array.

Tips & Variations

Common Mistakes to Avoid

• Ignoring Edge Cases: Always consider scenarios where the amount is 0 or where no combination of coins can make the amount.

• Inefficient Algorithms: Avoid recursive solutions without memoization, as they lead to exponential time complexity.

Alternative Ways to Answer

• For a technical role, you might emphasize efficiency and complexity analysis.

• For a managerial role, discuss how this problem relates to resource allocation and optimization in business.

Role-Specific Variations