What is the time complexity of Coin Change II?

The optimal solution has a time complexity of O(n*m).

What data structures are used to solve Coin Change II?

This problem uses Array, Dynamic Programming. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Coin Change II?

Insight 1: This is a classic dynamic programming problem. We need to store the number of combinations for each amount from 0 up to the target amount. Insight 2: We can use a 1D array `dp` where `dp[i]` represents the number of combinations to make up amount `i`. The base case is `dp[0] = 1` since there's one way to make an amount of 0 (using no coins).

Which companies ask LeetCode 518. Coin Change II in interviews?

This problem is frequently asked at Morgan Stanley.

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Coin Change II - LeetCode 518 Solution

Coin Change II - Complete Solution Guide

Coin Change II is LeetCode problem 518, a Medium level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

You are given an integer array coins representing coins of different denominations and an integer amount representing a total amount of money. Return the number of combinations that make up that amount . If that amount of money cannot be made up by any combination of the coins, return 0 . You may assume that you have an infinite number of each kind of coin. The answer is guaranteed to fit into a signed 32-bit integer. Example 1: Input: amount = 5, coins = [1,2,5] Output: 4 Explanation: there are

Detailed Explanation

The problem asks us to find the number of distinct combinations of coins that add up to a given target amount. We are given an array of coin denominations and a target amount. We can assume an unlimited supply of each coin denomination. The key is to count the number of combinations, not permutations, meaning the order of the coins doesn't matter (e.g., 1+2 and 2+1 are considered the same combination). If no combination sums to the target amount, we should return 0.

Solution Approach

The solution uses dynamic programming to build a table `dp` where `dp[i]` stores the number of ways to make amount `i` using the given coins. We iterate through each coin and for each coin, we iterate through the amounts from the coin's value up to the target amount. For each amount `i`, we add the number of ways to make `i - coin` to `dp[i]`. This is because if we use the current coin, the number of combinations to reach amount 'i' are increased by the number of combinations to reach amount `i-coin`.

Step-by-Step Algorithm

Step 1: Initialize a `dp` array of size `amount + 1` with all elements set to 0. This array will store the number of combinations for each amount from 0 to `amount`.
Step 2: Set `dp[0] = 1`. This is the base case because there is one way to make an amount of 0 (using no coins).
Step 3: Iterate through each coin in the `coins` array.
Step 4: For each coin, iterate through the amounts from `coin` to `amount` (inclusive).
Step 5: For each amount `i`, update `dp[i]` by adding `dp[i - coin]` to it. This represents adding the current coin to existing combinations that sum to `i - coin`.
Step 6: After iterating through all coins and amounts, `dp[amount]` will contain the total number of combinations to make up the target amount. Return `dp[amount]`.

Key Insights

Insight 1: This is a classic dynamic programming problem. We need to store the number of combinations for each amount from 0 up to the target amount.
Insight 2: We can use a 1D array `dp` where `dp[i]` represents the number of combinations to make up amount `i`. The base case is `dp[0] = 1` since there's one way to make an amount of 0 (using no coins).
Insight 3: The order in which we iterate through the coins is crucial. We iterate through the coins first (outer loop) and then the amounts (inner loop) to ensure that we are counting combinations (order doesn't matter). If we iterate through the amounts first, we would be counting permutations.

Complexity Analysis

Time Complexity: O(n*m)

Space Complexity: O(m)

Topics

This problem involves: Array, Dynamic Programming.

Companies

Asked at: Morgan Stanley.

Coin Change II | LeetCode 518

Medium

Array Dynamic Programming

Time: O(n*m)Space: O(m)

Solution Code

Available in:

class Solution {
    public int change(int amount, int[] coins) {
        int[] dp = new int[amount + 1];
        dp[0] = 1;

        for (int coin : coins) {
            for (int i = coin; i <= amount; i++) {
                dp[i] += dp[i - coin];
            }
        }
        
        return dp[amount];
    }
}

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Problem Info

Description

Problem 518: Coin Change II - Detailed explanation and solution approaches.

Topics

Array Dynamic Programming

Companies

Morgan Stanley

Stats

Difficulty:Medium

Languages:4

Companies:1

Problem #:518

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

6 steps

1
Step 1: Initialize a `dp` array of size `amount + 1` with all elements set to 0. This array will store the number of combinations for each amount from 0 to `amount`.
2
Step 2: Set `dp[0] = 1`. This is the base case because there is one way to make an amount of 0 (using no coins).
3
Step 3: Iterate through each coin in the `coins` array.
4
Step 4: For each coin, iterate through the amounts from `coin` to `amount` (inclusive).
5
Step 5: For each amount `i`, update `dp[i]` by adding `dp[i - coin]` to it. This represents adding the current coin to existing combinations that sum to `i - coin`.
6
Step 6: After iterating through all coins and amounts, `dp[amount]` will contain the total number of combinations to make up the target amount. Return `dp[amount]`.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(n*m)

💾

Space Complexity

O(m)

📝

Why?

The time complexity arises from the nested loops that iterate through each coin and each amount up to the target. The space complexity stems from the `dp` array, which stores the results for each amount from 0 up to the target.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Recursive Approach with Memoization

A recursive solution can be implemented with memoization to avoid redundant calculations. This approach involves defining a recursive function that explores the possibilities of using or not using a coin and storing the results in a memoization table.

Time: O(n*m), where n is the number of coins and m is the target amount.Space: O(n*m), where n is the number of coins and m is the target amount due to the memoization table and recursion stack.

Trade-offs: While conceptually similar, the iterative DP approach is often preferred because it avoids the overhead of recursion and can be slightly more efficient in practice. The recursive approach uses more memory because of the call stack and the memoization table.

✨

Key Insights

1Insight 1: This is a classic dynamic programming problem. We need to store the number of combinations for each amount from 0 up to the target amount.
2Insight 2: We can use a 1D array `dp` where `dp[i]` represents the number of combinations to make up amount `i`. The base case is `dp[0] = 1` since there's one way to make an amount of 0 (using no coins).
3Insight 3: The order in which we iterate through the coins is crucial. We iterate through the coins first (outer loop) and then the amounts (inner loop) to ensure that we are counting combinations (order doesn't matter). If we iterate through the amounts first, we would be counting permutations.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Dynamic Programming

Breaking complex problems into smaller subproblems and storing results.

Edge Cases

Edge case 1: `amount = 0`. The solution correctly handles this because `dp[0]` is initialized to 1, indicating one way to make an amount of 0 (using no coins).
Edge case 2: `coins` array is empty or contains only coins larger than the `amount`. In such scenarios, the code initializes `dp[0]=1` and never changes `dp[amount]` which is initialized to zero. Thus, it returns 0, which is the correct output.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Counting permutations instead of combinations. This happens if the loops are nested in the wrong order (iterating through amounts before coins). The code needs to iterate through the coins first to avoid double-counting combinations.

✗Common mistake 2: Not initializing `dp[0]` to 1. This is the base case and is essential for the dynamic programming approach to work correctly. Without it, the number of combinations will always be 0.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Inventory management. You might want to calculate the number of ways to fulfill an order given different package sizes of a product.

🏦Real-world scenario 2: Financial planning. Determine the number of ways to reach a savings goal given different investment options and their returns.

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

Consider multiple approaches before coding. Mentioning alternatives shows breadth of knowledge.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

👨‍💻

About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: November 6, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

Try solving this without using extra space first. It's a great exercise for understanding the underlying pattern.

@CodeNinjaJul 14

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 7

❓

Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

How do I identify if a problem needs dynamic programming?

DP problems typically have: 1) Overlapping subproblems (same calculation repeated), 2) Optimal substructure (optimal solution uses optimal solutions to subproblems). Look for keywords like "minimum/maximum," "count ways," "is it possible," or "longest/shortest."

What techniques are commonly tested in medium-difficulty problems?

Medium problems often require combining multiple techniques: two-pointer with hash maps, BFS/DFS with state tracking, dynamic programming with optimization, or divide-and-conquer approaches. They test your ability to recognize patterns and choose the right combination of techniques.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Coin Change II"?

For this medium-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether dynamic programming techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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