#2386

Hard

LeetCode Problem Workspace

Find the K-Sum of an Array

Find the K-Sum of an Array requires computing the kth largest subsequence sum in an array using sorting and heap techniques.

array sorting heap priority queue

Problem Statement

You are given an integer array nums and a positive integer k. Your task is to compute the kth largest subsequence sum of the array, where a subsequence is any subset of elements, not necessarily contiguous. The sum is computed by adding up the elements of the subsequence.

Return the K-Sum, the kth largest sum that can be formed from any subsequence of nums. Note that the subsequences can contain repeated sums, and the order in which sums are calculated does not affect the result. The values should be sorted in decreasing order to find the kth largest sum.

Examples

Example 1

Input: nums = [2,4,-2], k = 5

Output: 2

All the possible subsequence sums that we can obtain are the following sorted in decreasing order:

6, 4, 4, 2, 2, 0, 0, -2. The 5-Sum of the array is 2.

Example 2

Input: nums = [1,-2,3,4,-10,12], k = 16

Output: 10

The 16-Sum of the array is 10.

Constraints

n == nums.length
1 <= n <= 105
-109 <= nums[i] <= 109
1 <= k <= min(2000, 2n)

Solution Approach

Generate Possible Sums

The first step is to generate all possible subsequence sums. This can be achieved by considering the sum of each subsequence formed by the elements in the array. The sum of every possible subsequence is stored.

Sort the Sums

After calculating the sums, sort them in decreasing order to prepare for selecting the kth largest sum. Sorting ensures that we can access the kth largest sum directly.

Use a Heap for Efficiency

To optimize space and time complexity, a max heap can be used. We keep track of the largest sums in the heap and pop the smallest sums to efficiently reach the kth largest.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity of this problem depends on the method of generating the subsequence sums and the use of the sorting or heap algorithm. If all possible sums are generated and sorted, the time complexity is O(2^n log(2^n)), where n is the number of elements in the array. If a heap is used to manage the sums, the time complexity can be reduced to O(n log(k)). Space complexity is also affected by the method used, with a heap-based solution requiring O(k) space.

What Interviewers Usually Probe

The candidate uses a sorting-based approach to generate and retrieve subsequence sums.
The candidate optimizes the problem with the use of heaps or other data structures.
The candidate demonstrates knowledge of handling large input sizes by considering time complexity reductions.

Common Pitfalls or Variants

Common pitfalls

Generating all subsequences without considering the constraints of space and time complexity.
Overlooking the need for efficient sum retrieval and sorting or heap management.
Ignoring the requirement for handling repeated sums and selecting the correct kth value.

Follow-up variants

Handling arrays with both negative and positive values in a way that affects sum calculations.
Optimizing the approach for very large arrays and values of k.
Expanding the problem to include constraints on the subsequence size or type.

FAQ

What is the main algorithmic pattern used in this problem?

This problem primarily involves the combination of array manipulation, sorting, and heap usage to efficiently find the kth largest subsequence sum.

How do I efficiently find the kth largest subsequence sum?

You can use a heap to store the largest subsequence sums and pop smaller sums as you compute them, allowing you to quickly access the kth largest.

What is the time complexity of the brute force solution?

The brute force solution involves generating all subsequences and sorting their sums, leading to a time complexity of O(2^n log(2^n)).

How do negative numbers affect this problem?

Negative numbers can influence the sums of subsequences, but the approach still applies, as we are concerned with sorting the sums and selecting the kth largest value.

Can this problem be solved without sorting?

Sorting is the most straightforward method, but alternatives, such as using a max heap, can be employed to reduce time complexity when k is much smaller than the number of subsequences.

terminal

Solution

Solution 1: Priority Queue (Min-Heap)

First, we find the maximum subarray sum $mx$, which is the sum of all positive numbers.

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class Solution:
    def kSum(self, nums: List[int], k: int) -> int:
        mx = 0
        for i, x in enumerate(nums):
            if x > 0:
                mx += x
            else:
                nums[i] = -x
        nums.sort()
        h = [(0, 0)]
        for _ in range(k - 1):
            s, i = heappop(h)
            if i < len(nums):
                heappush(h, (s + nums[i], i + 1))
                if i:
                    heappush(h, (s + nums[i] - nums[i - 1], i + 1))
        return mx - h[0][0]

Continue Practicing

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