What is the time complexity of Find the K-Sum of an Array?

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

What data structures are used to solve Find the K-Sum of an Array?

This problem uses Array, Sorting, Heap (Priority Queue). Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Find the K-Sum of an Array?

Insight 1: The problem can be transformed into finding the k-th smallest sum of subsequences, where the elements are the absolute values of the original array's elements, subtracted from the maximum possible sum (formed by summing all positive numbers). Insight 2: Using a priority queue (min-heap) efficiently finds the k-th smallest sum. The queue stores the candidate subsequence sums and their corresponding indices.

Which companies ask LeetCode 2386. Find the K-Sum of an Array in interviews?

This problem is frequently asked at Hubspot.

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Find the K-Sum of an Array - LeetCode 2386 Solution

Find the K-Sum of an Array - Complete Solution Guide

Find the K-Sum of an Array is LeetCode problem 2386, a Hard 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 nums and a positive integer k . You can choose any subsequence of the array and sum all of its elements together. We define the K-Sum of the array as the k th largest subsequence sum that can be obtained ( not necessarily distinct). Return the K-Sum of the array . A subsequence is an array that can be derived from another array by deleting some or no elements without changing the order of the remaining elements. Note that the empty subsequence is considered to have

Detailed Explanation

The problem asks us to find the k-th largest sum of all possible subsequences of a given array 'nums'. A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements. The empty subsequence has a sum of 0. The k-th largest subsequence sum is the k-th element when all possible subsequence sums are sorted in descending order.

Solution Approach

The solution uses a priority queue to find the k-th smallest subtraction from the maximum possible sum. First, the maximum possible sum is calculated by adding all positive numbers in the input array. Then, the absolute values of all elements in the input array are taken and sorted. A priority queue is used to maintain the candidate subtraction values. The priority queue is initialized with the first element (smallest absolute value). In each iteration (k-1 times), the smallest value is extracted from the priority queue. Two new subtraction values are then added to the queue using the next element and the previous element. The algorithm then returns the maximum possible sum minus the k-th smallest subtraction value found.

Step-by-Step Algorithm

Step 1: Calculate the maximum possible sum by summing all positive numbers in the input array.
Step 2: Calculate the absolute values of all elements in the input array and sort them in ascending order.
Step 3: Initialize a priority queue (min-heap) with the first element (smallest absolute value) from the sorted array and its index (0).
Step 4: Iterate k-1 times: extract the smallest element (sum and index) from the priority queue. Store the current smallest sum as the k-th subtraction value.
Step 5: If the index of the extracted element is less than n-1 (where n is the size of nums), add two new elements to the priority queue:
Step 5a: The first element's sum is the current smallest sum + the absolute value at index i+1, and index is i+1.
Step 5b: The second element's sum is the current smallest sum - the absolute value at index i + the absolute value at index i+1, and index is i+1
Step 6: Finally, return the maximum possible sum minus the k-th subtraction value.

Key Insights

Insight 1: The problem can be transformed into finding the k-th smallest sum of subsequences, where the elements are the absolute values of the original array's elements, subtracted from the maximum possible sum (formed by summing all positive numbers).
Insight 2: Using a priority queue (min-heap) efficiently finds the k-th smallest sum. The queue stores the candidate subsequence sums and their corresponding indices.
Insight 3: The maximum possible sum can be precomputed by summing only positive numbers from the input array. This avoids generating all possible subsequence sums, which would be computationally expensive.

Complexity Analysis

Time Complexity: O(k log k)

Space Complexity: O(k)

Topics

This problem involves: Array, Sorting, Heap (Priority Queue).

Companies

Asked at: Hubspot.

Find the K-Sum of an Array | LeetCode 2386

Hard

Array Sorting Heap (Priority Queue)

Time: O(k log k)Space: O(k)

Solution Code

Available in:

import java.util.Arrays;
import java.util.PriorityQueue;

class Solution {
    public long kSum(int[] nums, int k) {
        long maxSum = 0;
        for (int x : nums) {
            if (x > 0) {
                maxSum += x;
            }
        }

        int[] absNums = new int[nums.length];
        for (int i = 0; i < nums.length; i++) {
            absNums[i] = Math.abs(nums[i]);
        }
        Arrays.sort(absNums);

        int n = nums.length;

        long kthSubtraction = 0;
        if (k == 1) {
            return maxSum;
        }

        PriorityQueue<long[]> pq = new PriorityQueue<>((a, b) -> Long.compare(a[0], b[0]));
        pq.offer(new long[]{absNums[0], 0});

        for (int i = 0; i < k - 1; i++) {
            long[] current = pq.poll();
            kthSubtraction = current[0];
            int index = (int) current[1];

            if (index + 1 < n) {
                pq.offer(new long[]{kthSubtraction + absNums[index + 1], index + 1});
                pq.offer(new long[]{kthSubtraction - absNums[index] + absNums[index + 1], index + 1});
            }
        }

        return maxSum - kthSubtraction;
    }
}

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Problem Info

Description

Problem 2386: Find the K-Sum of an Array - Detailed explanation and solution approaches.

Topics

Array Sorting Heap (Priority Queue)

Companies

Hubspot

Stats

Difficulty:Hard

Languages:4

Companies:1

Problem #:2386

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: Calculate the maximum possible sum by summing all positive numbers in the input array.
2
Step 2: Calculate the absolute values of all elements in the input array and sort them in ascending order.
3
Step 3: Initialize a priority queue (min-heap) with the first element (smallest absolute value) from the sorted array and its index (0).
4
Step 4: Iterate k-1 times: extract the smallest element (sum and index) from the priority queue. Store the current smallest sum as the k-th subtraction value.
5
Step 5: If the index of the extracted element is less than n-1 (where n is the size of nums), add two new elements to the priority queue:
6
Step 5a: The first element's sum is the current smallest sum + the absolute value at index i+1, and index is i+1.
7
Step 5b: The second element's sum is the current smallest sum - the absolute value at index i + the absolute value at index i+1, and index is i+1
8
Step 6: Finally, return the maximum possible sum minus the k-th subtraction value.

💡

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(k log k)

💾

Space Complexity

O(k)

📝

Why?

The priority queue stores at most 2*k elements. Each heap operation (push and pop) takes O(log k) time. Since the loop iterates k-1 times, the overall time complexity is O(k log k). The space complexity is O(k) because the priority queue stores at most 2*k elements.

📖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

Generating all possible subsequence sums

Generate all possible subsequences and calculate their sums. Then, sort the sums and return the k-th largest.

Time: O(2^n log(2^n)) = O(n * 2^n)Space: O(2^n)

Trade-offs: This approach is only feasible for very small input arrays due to the exponential time and space complexity. It's generally not practical for n > 20.

✨

Key Insights

1Insight 1: The problem can be transformed into finding the k-th smallest sum of subsequences, where the elements are the absolute values of the original array's elements, subtracted from the maximum possible sum (formed by summing all positive numbers).
2Insight 2: Using a priority queue (min-heap) efficiently finds the k-th smallest sum. The queue stores the candidate subsequence sums and their corresponding indices.
3Insight 3: The maximum possible sum can be precomputed by summing only positive numbers from the input array. This avoids generating all possible subsequence sums, which would be computationally expensive.

📚

Key Terms Explained

For Students

Array

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

Sorting

Arranging elements in a specific order (ascending/descending).

Edge Cases

Edge case 1: Input array contains only negative numbers. The maximum sum is 0, and the algorithm correctly calculates the k-th smallest subsequence sum.
Edge case 2: k = 1. The algorithm directly returns the maximum sum, avoiding unnecessary priority queue operations.
Edge case 3: Large values in the input array. The solution uses `long long` to prevent integer overflow in calculations of sums.

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Integer overflow. The sums of subsequences can be large, leading to integer overflow if not using `long long` (or similar 64-bit integers) data types.

✗Common mistake 2: Not handling negative numbers correctly. Failing to take the absolute values and sort them leads to incorrect results.

✗Common mistake 3: Using an inefficient priority queue implementation or not using one at all, leading to a time complexity greater than O(k log k).

✅

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: Resource allocation. Given a set of tasks with varying costs (positive or negative representing gains), finding the best combination of tasks within a budget constraint can be modeled as finding a k-th sum of a subset.

🏦Real-world scenario 2: Financial portfolio optimization. Finding the k-th most profitable portfolio given a set of investments with varying returns (positive or negative) is an example where subsequence sums are relevant.

💡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

It's okay to not finish hard problems in interviews. Show your problem-solving process.

⚠️ 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: July 17, 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

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

@AlgoMasterJul 13

I was asked this exact problem at my Google interview. The interviewer wanted me to optimize space complexity.

@InterviewProJul 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.

How do I approach hard algorithm problems?

Hard problems often require: 1) Breaking down into subproblems, 2) Combining multiple advanced techniques, 3) Careful optimization of brute force. Start with a working solution (even O(n²) or worse), then optimize. Draw examples, identify patterns, and think about what makes this problem unique.

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 "Find the K-Sum of an Array"?

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

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