What is the optimal time complexity of solving Coin Change?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Coin Change?

The space complexity is O(1) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Coin Change?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Coin Change?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Coin Change

Learn how to solve the 'Coin Change' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Medium

You are given an integer array coins representing coins of different denominations and an integer amount representing a total amount of money.

Return the fewest number of coins that you need to make up that amount. If that amount of money cannot be made up by any combination of the coins, return -1.

You may assume that you have an infinite number of each kind of coin.

Write a function coinChange(coins: List[int], amount: int) -> int.

Constraints

•1 <= len(coins) <= 12
•1 <= coins[i] <= 2^31 - 1
•0 <= amount <= 10^4

Examples

Example 1

Input

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

Output

Explanation

11 = 5 + 5 + 1

Example 2

Input

coins = [2], amount = 3

Output

-1

Explanation

3 cannot be formed using only coins of denomination 2.

Example 3

Input

coins = [1], amount = 0

Output

Explanation

No coins are needed to make amount 0.

Need a Hint?

Define subproblem states, establish the recurrence relation, and use memoization (top-down) or tabulation (bottom-up).

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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