What is the time complexity of Subarray Sum Equals K?

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

What data structures are used to solve Subarray Sum Equals K?

This problem uses Array, Hash Table, Prefix Sum. Understanding these concepts is essential for solving this Medium problem efficiently.

What are the key insights for solving Subarray Sum Equals K?

Insight 1: Using prefix sums can efficiently calculate the sum of any subarray in O(1) time. The sum of subarray nums[i:j] is prefix_sum[j] - prefix_sum[i-1]. Insight 2: Employing a hash table to store the counts of prefix sums allows us to quickly determine how many times a particular prefix sum has appeared before. This lets us find subarray sums that equal `k`.

Which companies ask LeetCode 560. Subarray Sum Equals K in interviews?

This problem is frequently asked at AMD, Apollo.io, ByteDance, Capgemini, Cisco and 20 more companies.

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Subarray Sum Equals K - LeetCode 560 Solution

Q: How to solve Subarray Sum Equals K?

The provided solutions use a prefix sum approach combined with a hash table to efficiently count the number of subarrays summing to `k`. The algorithm iterates through the `nums` array, maintaining a running `current_sum`. For each element, it checks if there exists a previous prefix sum such that the difference between the `current_sum` and `k` is equal to that previous prefix sum. This indicates that the subarray between those two prefix sum indices has a sum equal to `k`. The hash table stores the counts of each prefix sum encountered so far, allowing for O(1) lookup of previous prefix sums.

Subarray Sum Equals K - Complete Solution Guide

Subarray Sum Equals K is LeetCode problem 560, 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

Given an array of integers nums and an integer k , return the total number of subarrays whose sum equals to k . A subarray is a contiguous non-empty sequence of elements within an array. Example 1: Input: nums = [1,1,1], k = 2 Output: 2 Example 2: Input: nums = [1,2,3], k = 3 Output: 2 Constraints: 1 <= nums.length <= 2 * 10 4 -1000 <= nums[i] <= 1000 -10 7 <= k <= 10 7

Detailed Explanation

The problem asks us to find the number of subarrays within a given array `nums` whose elements sum up to a target value `k`. A subarray is a contiguous, non-empty sequence of elements from the original array. We need to count how many such subarrays exist.

Solution Approach

The provided solutions use a prefix sum approach combined with a hash table to efficiently count the number of subarrays summing to `k`. The algorithm iterates through the `nums` array, maintaining a running `current_sum`. For each element, it checks if there exists a previous prefix sum such that the difference between the `current_sum` and `k` is equal to that previous prefix sum. This indicates that the subarray between those two prefix sum indices has a sum equal to `k`. The hash table stores the counts of each prefix sum encountered so far, allowing for O(1) lookup of previous prefix sums.

Step-by-Step Algorithm

Step 1: Initialize a `count` to 0 to store the number of subarrays with sum `k`.
Step 2: Initialize a `current_sum` to 0 to keep track of the running sum as we iterate through the array.
Step 3: Create a hash table `prefix_sum_counts` to store the counts of each prefix sum encountered. Initialize `prefix_sum_counts[0] = 1` to handle cases where a subarray starting from the beginning sums to `k`.
Step 4: Iterate through the `nums` array.
Step 5: In each iteration, update `current_sum` by adding the current element `num`.
Step 6: Calculate `diff = current_sum - k`. If `diff` exists as a key in `prefix_sum_counts`, it means there is a previous prefix sum such that the subarray between that prefix sum and the current index has a sum of `k`. Increment `count` by the value associated with `diff` in `prefix_sum_counts`.
Step 7: Update the count of `current_sum` in `prefix_sum_counts`. If `current_sum` already exists, increment its count; otherwise, add it to the hash table with a count of 1.
Step 8: After iterating through the entire array, return `count`.

Key Insights

Insight 1: Using prefix sums can efficiently calculate the sum of any subarray in O(1) time. The sum of subarray nums[i:j] is prefix_sum[j] - prefix_sum[i-1].
Insight 2: Employing a hash table to store the counts of prefix sums allows us to quickly determine how many times a particular prefix sum has appeared before. This lets us find subarray sums that equal `k`.
Insight 3: Initializing the hash table with a prefix sum of 0 occurring once handles the case where a subarray starting from the beginning of the array sums to `k`.

Complexity Analysis

Time Complexity: O(n)

Space Complexity: O(n)

Topics

This problem involves: Array, Hash Table, Prefix Sum.

Companies

Asked at: AMD, Apollo.io, ByteDance, Capgemini, Cisco, Disney, Flipkart, Goldman Sachs, IBM, Infosys, J.P. Morgan, Morgan Stanley, Nagarro, PayPal, Quora, Ripple, Scale AI, ServiceNow, Snap, Swiggy, Tesla, Tinkoff, Walmart Labs, Yandex, tcs.

Subarray Sum Equals K | LeetCode 560

Medium

Array Hash Table Prefix Sum

Time: O(n)Space: O(n)

Solution Code

Available in:

import java.util.HashMap;
import java.util.Map;

class Solution {
    public int subarraySum(int[] nums, int k) {
        int count = 0;
        int current_sum = 0;
        Map<Integer, Integer> prefix_sum_counts = new HashMap<>();
        prefix_sum_counts.put(0, 1);

        for (int num : nums) {
            current_sum += num;

            int diff = current_sum - k;
            if (prefix_sum_counts.containsKey(diff)) {
                count += prefix_sum_counts.get(diff);
            }

            prefix_sum_counts.put(current_sum, prefix_sum_counts.getOrDefault(current_sum, 0) + 1);
        }

        return count;
    }
}

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Problem Info

Description

Problem 560: Subarray Sum Equals K - Detailed explanation and solution approaches.

Topics

Array Hash Table Prefix Sum

Companies

AMDApollo.ioByteDanceCapgeminiCiscoDisneyFlipkartGoldman Sachs

Stats

Difficulty:Medium

Languages:4

Companies:25

Problem #:560

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

8 steps

1
Step 1: Initialize a `count` to 0 to store the number of subarrays with sum `k`.
2
Step 2: Initialize a `current_sum` to 0 to keep track of the running sum as we iterate through the array.
3
Step 3: Create a hash table `prefix_sum_counts` to store the counts of each prefix sum encountered. Initialize `prefix_sum_counts[0] = 1` to handle cases where a subarray starting from the beginning sums to `k`.
4
Step 4: Iterate through the `nums` array.
5
Step 5: In each iteration, update `current_sum` by adding the current element `num`.
6
Step 6: Calculate `diff = current_sum - k`. If `diff` exists as a key in `prefix_sum_counts`, it means there is a previous prefix sum such that the subarray between that prefix sum and the current index has a sum of `k`. Increment `count` by the value associated with `diff` in `prefix_sum_counts`.
7
Step 7: Update the count of `current_sum` in `prefix_sum_counts`. If `current_sum` already exists, increment its count; otherwise, add it to the hash table with a count of 1.
8
Step 8: After iterating through the entire array, return `count`.

💡

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)

👍 Linear - Good! Visits each element once.

💾

Space Complexity

O(n)

📦 Linear space - Memory grows with input size.

📝

Why?

The time complexity is O(n) because we iterate through the array only once. The hash table operations (get, put) take O(1) on average. The space complexity is O(n) because in the worst case, we store all n prefix sums in the hash table. This can happen if the numbers in nums are all distinct and non-zero.

📖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

Brute Force

Iterate through all possible subarrays and calculate their sum. Compare the sum to `k` and increment a counter.

Time: O(n^2)Space: O(1)

Trade-offs: This approach is simple to understand but inefficient for large arrays due to the nested loops. It avoids using extra space like the hash table approach but has significantly worse time complexity. Not recommended for larger input sizes.

✨

Key Insights

1Insight 1: Using prefix sums can efficiently calculate the sum of any subarray in O(1) time. The sum of subarray nums[i:j] is prefix_sum[j] - prefix_sum[i-1].
2Insight 2: Employing a hash table to store the counts of prefix sums allows us to quickly determine how many times a particular prefix sum has appeared before. This lets us find subarray sums that equal `k`.
3Insight 3: Initializing the hash table with a prefix sum of 0 occurring once handles the case where a subarray starting from the beginning of the array sums to `k`.

📚

Key Terms Explained

For Students

Array

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

Hash Table

A data structure that maps keys to values for quick lookup. Like a dictionary!

Edge Cases

Edge case 1: Empty input array. The code correctly handles this by returning 0, as the loop won't execute.
Edge case 2: All elements are 0 and k = 0. The hash table correctly counts all subarrays as summing to 0.
Edge case 3: Negative numbers in the input array. The algorithm correctly handles negative numbers by maintaining the running sum and checking for differences.
Edge case 4: k = 0, but there are no zeros in the array. Correctly returns 0 as no subarray sums to zero.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Forgetting to initialize the hash table with `prefix_sum_counts[0] = 1`. This is crucial for handling cases where the subarray starting from the beginning sums to `k`.

✗Common mistake 2: Not considering negative numbers in the input array. The prefix sum approach works correctly with negative numbers as long as the running sum is maintained accurately.

✗Common mistake 3: Assuming the subarray must start from index 0. The problem asks for *any* subarray, so all possible contiguous subarrays must be considered.

✅

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: Analyzing financial data to find periods where the cumulative profit or loss reaches a specific target value.

🏦Real-world scenario 2: Signal processing, where finding sections of a signal that have a specific energy level (sum of squares) is required.

🎮Real-world scenario 3: In bioinformatics, identifying regions in a DNA sequence with a specific GC content (sum of G and C bases).

💡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.

👨‍💻 From Experience

This problem pattern appears frequently in real codebases. I've used similar logic when building production systems.

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: June 21, 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

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 13

The key insight here is recognizing the pattern. Once you see it, the solution becomes straightforward.

@AlgoMasterJul 6

❓

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.

What are the trade-offs of using a hash table?

Hash tables provide O(1) average-case lookup, insertion, and deletion, making them ideal for frequency counting and duplicate detection. However, they use O(n) extra space and may degrade to O(n) time in worst-case scenarios with hash collisions. Always consider if the space trade-off is acceptable.

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 "Subarray Sum Equals K"?

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 hash table techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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