Coin Change Leetcode Problem 322 [Python Solution]

In this blog post, we're going to dive into the Coin Change problem, which is a classic example of a dynamic programming challenge.

This problem is categorized as a 1-D Dynamic Programming problem and is featured on LeetCode under Problem 322. We'll provide a Python solution to tackle this problem efficiently.

Problem Overview

The problem statement is as follows: You are given an integer array coins, representing coins of different denominations, and an integer amount, representing a total amount of money.

Your task is to find the fewest number of coins needed to make up that amount.

If it's impossible to reach the target amount with the given coins, return -1. You can assume that you have an infinite number of each kind of coin.

Let's break down the problem step by step.

Example 1:

Input:
coins = [1,2,5], amount = 11

Output:
3

Explanation: To make 11 with coins of values 1, 2, and 5, we can use two 5 coins and one 1 coin, which totals 11. This represents the fewest number of coins required.

Example 2:

Input:
coins = [2], amount = 3

Output:
-1

Explanation: In this case, there's no combination of the single coin with value 2 that can sum up to 3. So, we return -1.

Example 3:

Input:
coins = [1], amount = 0

Output:
0

Explanation: If the amount is 0, we don't need any coins, so the answer is 0.

Constraints:
– 1 <= coins.length <= 12
– 1 <= coins[i] <= 231 – 1
– 0 <= amount <= 104

Now that we understand the problem, let's explore how to approach it and find an efficient Python solution.

Understanding the Problem Constraints

Before we dive into the solution, let's understand the problem constraints.

The constraints indicate that the number of coins can be relatively small (up to 12), and the values of the coins and the target amount are within reasonable limits.

This suggests that a dynamic programming approach is feasible.

The problem asks for the minimum number of coins needed to reach a specific amount.

To solve this, we'll use a dynamic programming table (DP table) to keep track of the minimum number of coins required to make up different amounts.

We'll initialize the table with values representing unattainable amounts and update it iteratively.

Coin Change LeetCode Problem Solution

Let's provide a Python solution to the Coin Change problem using dynamic programming.

We'll first explore a brute-force approach to understand the problem's complexities and then optimize it with a dynamic programming approach.

1. Bruteforce Approach

The brute-force approach involves recursively trying out all possible combinations of coins to reach the target amount and finding the one that requires the fewest coins.

While this approach works, it can be extremely inefficient and time-consuming, especially for larger inputs.

Here's a Python function that implements the brute-force approach:

def coinChangeBruteforce(coins, amount):
    def dfs(remaining_amount):
        if remaining_amount == 0:
            return 0
        if remaining_amount < 0:
            return -1

        min_coins = float('inf')
        for coin in coins:
            result = dfs(remaining_amount - coin)
            if result >= 0:
                min_coins = min(min_coins, result + 1)

        return -1 if min_coins == float('inf') else min_coins

    result = dfs(amount)
    return result if result != float('inf') else -1

While this code accurately solves the problem, it's not efficient.

It has an exponential time complexity due to the recursive approach, making it impractical for larger inputs.

We can do better with dynamic programming.

2. An Efficient Approach with Dynamic Programming

Dynamic programming allows us to store and reuse intermediate results, significantly reducing redundant calculations.

We'll use a 1-D DP array to keep track of the minimum number of coins needed to make up each amount from 0 to the target amount.

Here's a Python solution that utilizes dynamic programming:

View on GitHub ➹

def coinChange(coins, amount):
    # Initialize a DP array to store the minimum number of coins for each amount.

dp = [amount + 1] * (amount + 1)
    dp[0] = 0

    # Iterate through all possible amounts from 1 to the target amount.

for a in range(1, amount + 1):
        for c in coins:
            if a - c &gt;= 0:
                dp[a] = min(dp[a], 1 + dp[a - c])

    return dp[amount] if dp[amount] != amount + 1 else -1

Let's break down the dynamic programming solution step by step:

We initialize a DP array called dp, where each element represents the minimum number of coins needed to make up the corresponding amount.

We initialize all elements to amount + 1 as a placeholder value.

We set dp[0] to 0 because it takes zero coins to make an amount of 0.
We iterate through all possible amounts from 1 to the target amount (amount).
For each amount a, we iterate through the available coin denominations (coins).
If subtracting the coin's value (c) from the current amount (a - c) results in a non-negative value, it means we can consider using that coin to form the current amount.
We update dp[a] by taking the minimum of its current value and 1 + dp[a - c].

This represents our dynamic programming recurrence relation.

Finally, we return dp[amount] if it's not equal to amount + 1, indicating that we found a valid solution.

Otherwise, we return -1, indicating that it's impossible to make up the target amount with the given coins.

This dynamic programming approach is highly efficient, with a time complexity of O(amount * number of coin denominations) and a space complexity of O(amount).

It avoids the redundancy of the brute-force approach and provides a much faster solution, even for larger inputs.

Time and Space Complexity

Now, let's analyze the time and space complexity of our efficient dynamic programming solution:

Time Complexity: O(amount * number of coin denominations)
– The outer loop iterates through the range of 1 to the target amount, which has a time complexity of O(amount).
– The inner loop iterates through the coin denominations, and in the worst case, it iterates through all coin denominations (number of coin denominations), which has a time complexity of O(number of coin denominations).
– Combining these, we get O(amount * number of coin denominations) as the overall time complexity.

Space Complexity: O(amount)
– We use a DP array of size amount + 1 to

store the minimum number of coins for each amount, leading to a space complexity of O(amount).

With this efficient dynamic programming solution, we can solve the Coin Change problem for larger inputs in a reasonable amount of time and with minimal memory usage.

Reasoning Behind Our Approach

The dynamic programming approach we've used is based on the principle of optimally solving subproblems and building up the solution for larger problems.

We start with the simplest subproblem (amount = 0) and iteratively compute solutions for more complex problems.

The core idea is to minimize the number of coins needed to reach each amount by considering all possible coin denominations.

This approach effectively avoids redundant calculations by storing intermediate results in the DP array.

Whenever we encounter a previously computed amount, we reuse its result instead of recomputing it.

By iteratively solving subproblems and combining their solutions, we arrive at the optimal solution for the main problem (the target amount).

The time and space complexity analysis demonstrates the efficiency of our approach, making it suitable for solving real-world instances of the Coin Change problem.

Conclusion

In this blog post, we tackled the Coin Change problem, a classic dynamic programming challenge featured on LeetCode.

We discussed the problem statement, constraints, and provided both a brute-force approach and an efficient dynamic programming solution in Python.

The brute-force approach, while conceptually simple, suffers from exponential time complexity, making it impractical for larger inputs.

In contrast, the dynamic programming solution optimally solves subproblems and builds the solution for the main problem, resulting in a much faster and efficient algorithm.

Dynamic programming is a powerful technique for solving optimization problems by breaking them down into smaller, overlapping subproblems and reusing computed results.

In this case, it allowed us to find the fewest number of coins required to make up a given amount while considering all possible coin denominations.

If you found this post helpful, please consider liking and subscribing to support our our platform.

Feel free to comment, ask questions, make suggestions, or share the content.

We hope this guide has been beneficial, especially for beginners, and that you're now better equipped to tackle dynamic programming challenges like the Coin Change problem.

Question Link

Happy coding!