Step-by-Step Tutorial: Solving the Change Making Problem Using Dynamic Programming

Overview:

In this tutorial, we will learn how to solve the change making problem using dynamic programming. The goal is to find the fewest number of coins needed to make change for a given amount using a set of coin denominations.

Step 1: Understanding the Problem

• The change making problem involves finding the minimum number of coins needed to make change for a given amount using specific coin denominations.
• We are given an amount and a set of coin denominations.
• The objective is to determine the least amount of coins required to make change for the given amount.

Step 2: Approach Overview

• We will use dynamic programming to solve the change making problem.
• There are two approaches we will explore: top-down and bottom-up.

Step 3: Top-Down Approach

1. Start with the given amount and consider using each coin denomination as the last coin to make change.
2. Recursively explore all possible paths by subtracting the chosen coin from the amount and calculating the minimum number of coins needed for the remaining amount.
3. Cache the results of subproblems to avoid redundant calculations.
4. Choose the path with the fewest coins at each step.
5. The final result at the top level will give the minimum number of coins needed to make change for the initial amount.

Step 4: Bottom-Up Approach

1. Create a table to store the minimum number of coins needed for each amount from 0 to the given amount.
2. Initialize the table with an arbitrarily large value for all amounts except 0, which is initialized to 0.
3. Iterate through each amount and each coin denomination, updating the table with the minimum number of coins required for each amount.
4. Calculate the minimum number of coins needed for the given amount based on the subproblems solved earlier.
5. The final result will be the minimum number of coins needed to make change for the given amount.

Step 5: Time and Space Complexity Analysis

• The time complexity for both approaches is O(amount * numCoins), where amount is the given amount and numCoins is the number of coin denominations.
• The space complexity for the top-down approach includes the call stack depth, which can be up to the amount, and caching results for subproblems.
• The space complexity for the bottom-up approach involves storing results for all subproblems in the table.

Conclusion:

By following these steps, you can effectively solve the change making problem using dynamic programming. Understanding both the top-down and bottom-up approaches will help you tackle similar problems efficiently. Remember to analyze the time and space complexity to optimize your solution.