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 kth 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 a sum of 0.
Example 1:
Input: nums = [2,4,-2], k = 5 Output: 2 Explanation: 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 Explanation: The 16-Sum of the array is 10.
Constraints:
n == nums.length1 <= n <= 105-109 <= nums[i] <= 1091 <= k <= min(2000, 2n)This approach involves generating all possible subsequence sums, sorting them, and then performing a binary search to find the k-th largest sum.
Given the constraints, this approach may not be feasible for the upper limits of n due to its time complexity. However, it illustrates a straightforward way to conceptualize the problem.
This solution generates all possible subsequences by using combinations from itertools, calculates their sums, and stores these in a list. The list is then sorted in descending order, and the k-th largest sum is returned by accessing the list at the k-1 index.
JavaScript
Time Complexity: O(2^n * n) for generating subsequences and computing sums, and O(2^n log(2^n)) for sorting.
Space Complexity: O(2^n) for storing the sums.
This approach uses a max-heap (priority queue) to efficiently find the k-th largest subsequence sum. This method is more effective and feasible when handling large arrays.
The idea is to add sums to the heap and always maintain only the k largest sums, popping from the heap when necessary to ensure it does not grow beyond size k.
This Python solution maintains a max-heap of sums. We start with a heap initialized with 0, signifying the empty subsequence sum. For each number, we calculate potential new sums by adding the number to each current sum in the heap, expanding the list of sums. We then keep only the largest k elements in the heap. This ensures we eventually have the k-th largest sum at the k-1 index.
Java
Time Complexity: O(n * k log k) due to maintaining a heap of size k across n iterations.
Space Complexity: O(k) for the heap storage.
| Approach | Complexity |
|---|---|
| Approach 1: Sorting and Binary Search | Time Complexity: O(2^n * n) for generating subsequences and computing sums, and O(2^n log(2^n)) for sorting. |
| Approach 2: Max-Heap and Iterative Summation | Time Complexity: O(n * k log k) due to maintaining a heap of size k across n iterations. |
Maximum Subarray - Amazon Coding Interview Question - Leetcode 53 - Python • NeetCode • 605,143 views views
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