Coin Change Problem: Dynamic Programming Solution
Find minimum coins to make an amount and the number of possible combinations using dynamic programming. Interactive visualizer with step-by-step examples.
Visualize the coin change problem dynamic programming solution step by step.
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The coin change problem is a classic dynamic programming problem. Given coin denominations and a target amount, find the minimum coins to make the amount or the number of ways coin change combinations possible. This interactive coin change dynamic programming visualizer shows both solutions step by step.
📌 The Coin Change Recurrence Relation:
dp[i] = min(dp[i], dp[i – coin] + 1) for minimum coins
dp[i] += dp[i – coin] for number of combinations
Coin Change Visualizer
Find minimum coins to make amount or number of ways coin change using dynamic programming.
Coin Change Example: Minimum Coins for Amount 11
Walk through a complete coin change example step by step.
Problem: Coins = [1, 2, 5], Amount = 11
Step 1: Initialize dp[0] = 0, all other dp[i] = infinity
Step 2: For each coin, update dp[i] = min(dp[i], dp[i-coin] + 1)
Step 3: After processing all coins, dp[11] = minimum coins needed
Step 4: Backtrack to find which coins were used
Answer: Minimum coins = 3 (5 + 5 + 1)
For Number of Ways: dp[0] = 1, dp[i] += dp[i-coin] → dp[11] = 4 ways
Coin Change Algorithm Explained
Understanding the coin change algorithm and dynamic programming approach.
Define the DP Array
Create an array dp[] where dp[i] represents the minimum coins needed to make amount i (or number of ways). Initialize dp[0] = 0, others to infinity (minimum coins) or 0 (ways).
Apply the Recurrence Relation
For each coin, update dp[i] = min(dp[i], dp[i-coin] + 1) for minimum coins. For combinations, dp[i] += dp[i-coin]. This is the core of the coin change recurrence relation.
Trace Back Solution
For minimum coins, trace back through the dp array to find which coins were used to achieve the optimal solution.
Frequently Asked Questions
Common questions about the coin change problem and dynamic programming.
What is the coin change problem?
The coin change problem asks: given coin denominations and a target amount, what is the minimum number of coins needed to make that amount (or how many possible combinations exist)? It’s a classic dynamic programming problem.
How do you solve the coin change problem?
The coin change algorithm uses dynamic programming. Create a dp array where dp[i] represents the minimum coins to make amount i. For each coin, update dp[i] = min(dp[i], dp[i-coin] + 1). The answer is dp[amount].
What is the recurrence relation for coin change?
The coin change recurrence relation is: dp[i] = min(dp[i], dp[i – coin] + 1) for minimum coins. For number of ways: dp[i] += dp[i – coin] with dp[0] = 1.
What’s the time complexity of coin change?
Time complexity is O(n × amount), where n is the number of coin denominations. Space complexity is O(amount).
Is this coin change visualizer free?
Yes! This coin change problem solver is completely free. No signup, no ads. Just open the tool and start learning dynamic programming.
Additional Resources
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Free coin change problem visualizer for computer science students learning dynamic programming.