Coin Change Problem in Algorithms

You can get training on our article to master one of the most fundamental problems in algorithm design: the Coin Change Problem. This problem is a staple in the field of algorithms, particularly in the study of dynamic programming. Whether you're preparing for coding interviews or building applications that involve optimization, understanding the Coin Change Problem will enhance your problem-solving toolkit. In this article, we will explore the problem in detail, delve into recursive and dynamic programming solutions, and analyze the time and space complexities of these approaches.

What is the Coin Change Problem?

The Coin Change Problem is a classic problem in computer science that asks the following:

For example, suppose you have coins with denominations {1, 2, 5} and you need to make a target amount of 11. The optimal solution would use three coins: 5 + 5 + 1 = 11.

The problem can be generalized to two variations:

• Minimum Coin Change Problem: Find the fewest number of coins to make the target amount.
• Number of Ways to Make Change: Find how many ways you can make the target amount using the coins.

The Coin Change Problem is particularly interesting because it can be solved using a variety of algorithmic techniques, from brute force recursion to dynamic programming approaches that optimize for both time and space.

Recursive Solution to Coin Change Problem

The recursive solution to the Coin Change Problem is often the most intuitive but also the least efficient. Here’s the basic idea:

• For every coin in the denominations, subtract its value from the target amount and recursively call the function for the remaining amount.
• If the target amount becomes 0, a solution is found.
• If the target becomes negative, discard that branch as no solution exists.

For instance, let’s say the target is 11 and the denominations are {1, 2, 5}. The recursion tree would explore every possible combination of coins, checking if the sum equals the target amount.

A simple recursive implementation in Python would look like this:

def coin_change_recursive(coins, amount):
    if amount == 0:
        return 0
    if amount < 0:
        return float('inf')
    
    min_coins = float('inf')
    for coin in coins:
        result = coin_change_recursive(coins, amount - coin)
        min_coins = min(min_coins, result + 1)
    
    return min_coins

While this approach works for small inputs, it is highly inefficient due to overlapping subproblems. Each subproblem is solved repeatedly, making the time complexity exponential, O(2^n), where n is the target amount.

Dynamic Programming Approach to Coin Change

Dynamic programming (DP) is a paradigm that optimizes problems by breaking them into smaller subproblems and storing the results of those subproblems to avoid redundant computations. Unlike the recursive approach, DP ensures that each subproblem is solved only once.

The two main techniques in DP are memoization and tabulation, both of which are effective in solving the Coin Change Problem.

Memoization in Coin Change Problem

Memoization is a top-down approach where we solve the problem recursively but store the results of subproblems in a cache. This eliminates the need to recompute solutions for overlapping subproblems.

Here’s how the memoized solution looks in Python:

def coin_change_memo(coins, amount, memo):
    if amount == 0:
        return 0
    if amount < 0:
        return float('inf')
    if amount in memo:
        return memo[amount]
    
    min_coins = float('inf')
    for coin in coins:
        result = coin_change_memo(coins, amount - coin, memo)
        min_coins = min(min_coins, result + 1)
    
    memo[amount] = min_coins
    return memo[amount]

# Example usage
coins = [1, 2, 5]
amount = 11
print(coin_change_memo(coins, amount, {}))  # Output: 3

With memoization, the time complexity is reduced to O(n * m), where n is the target amount and m is the number of coin denominations. The space complexity is also significantly improved.

Tabulation in Coin Change Problem

Tabulation is a bottom-up approach where we iteratively build a table to store the solutions to subproblems. This eliminates recursion and provides even better performance in some cases.

Here’s how the tabulation solution looks in Python:

def coin_change_tabulation(coins, amount):
    dp = [float('inf')] * (amount + 1)
    dp[0] = 0  # Base case: 0 coins are needed to make amount 0
    
    for i in range(1, amount + 1):
        for coin in coins:
            if i - coin >= 0:
                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(coin_change_tabulation(coins, amount))  # Output: 3

Tabulation uses a one-dimensional array dp to store the minimum coins required for each amount from 0 to the target. The time complexity remains O(n * m), but the iterative nature avoids the overhead of recursion.

Time Complexity of Coin Change Algorithms

Understanding the time complexity is critical when comparing different approaches:

• Recursive Approach: Exponential, O(2^n), due to redundant computations.
• Memoization: Polynomial, O(n * m), where n is the target amount and m is the number of coins.
• Tabulation: Also O(n * m), but often faster in practice due to the absence of recursion overhead.

Space Complexity of Coin Change Algorithms

The space complexity varies depending on the approach:

• Recursive Approach: O(n) due to the recursion stack.
• Memoization: O(n) for the recursion stack and the memoization table.
• Tabulation: O(n) for the DP table, making it the most space-efficient approach.

Summary

The Coin Change Problem is a fundamental problem that showcases the power of dynamic programming. While the recursive solution provides an intuitive understanding, it is not feasible for large inputs due to its exponential time complexity. By employing dynamic programming techniques like memoization and tabulation, we can drastically improve efficiency, reducing both time and space complexity.

For developers, mastering the Coin Change Problem is essential, as it is representative of a wide range of optimization problems. By understanding the intricacies of dynamic programming, you can tackle similar challenges with confidence and efficiency. Whether you're preparing for a coding interview or optimizing a real-world application, the lessons from this problem will serve you well.

Last Update: 25 Jan, 2025