Problem Statement

Given an integer array nums and a positive integer k, return any subsequence of nums of length k, which has the largest sum.

A subsequence maintains the original order of array elements but may exclude some, without reordering the remaining ones.

Examples

• Example 1:

  • Input: nums = [5, -2, 3, 8], k = 2
  • Expected Output: [5, 8]
  • Justification: The subsequence [5, 8] gives the largest sum of 13, which is the maximum sum we can obtain from any subsequence of length 2.

• Example 2:

  • Input: nums = [-1, -2, -3, -4], k = 3
  • Expected Output: [-1, -2, -3]
  • Justification: Although all numbers are negative, the subsequence [-1, -2, -3] has the least negative sum, making it the largest sum (-6) for a length of 3.

• Example 3:

  • Input: nums = [4, 3, 1, 2, 5], k = 4
  • Expected Output: [4, 3, 2, 5]
  • Justification: The subsequence [4, 3, 2, 5] sums up to 14, which is the highest possible sum for any subsequence of length 4 in the given array.

Solution

To solve this problem, we will adopt a two-pronged approach that ensures efficiency and accuracy. Initially, we identify and sort the array elements by their value, retaining their original indexes. This step is crucial for ensuring that we can always trace back to the original sequence of elements after sorting. Subsequently, we select the top k elements with the highest values from this sorted array. The key to this strategy is not just sorting but maintaining a relationship between the sorted elements and their original positions in the array.

This approach works effectively because sorting helps us quickly identify the elements that contribute most to the sum. By selecting the top k elements, we guarantee that the sum is maximized. However, simply choosing these elements without considering their original order could disrupt the sequence, violating the subsequence rule. Hence, after picking these top elements, we refer back to their original indexes to reconstruct the subsequence while preserving the original order of elements. This method is preferred for its blend of computational efficiency and its respect for the sequence integrity of the subsequence.

Step-by-Step Algorithm

1. Pair each element with its index: Create a list of pairs where each pair consists of an element from the input array nums and its corresponding index. This step ensures we can track the original position of each element after sorting.

2. Sort the pairs by their values: Sort the created list of pairs in descending order based on the values of the elements. This sorting step is crucial because it allows us to easily find the k elements with the largest values, which are needed to form the subsequence with the largest sum.

3. Select the top k elements: After sorting, the first k elements in the list will be the ones contributing to the largest sum. However, these elements are currently ordered by their values, not their original sequence.

4. Sort the selected elements by their original indices: To restore the original order of the selected k elements, sort them based on their indices. This step is necessary because a subsequence must maintain the original ordering of elements.

5. Extract the sorted subsequence: Finally, create a new array (or list) to hold the subsequence. Iterate through the sorted list of k elements and extract the values to form the final subsequence.

6. Return the subsequence: The resulting array from the previous step is the subsequence of length k with the largest sum, maintaining the original order from nums. This subsequence is returned as the output.

Algorithm Walkthrough

Let's walk through the algorithm using the input array nums = [4, 3, 1, 2, 5] and k = 4.

1. Pair each element with its index:
   Original pairs: [(4, 0), (3, 1), (1, 2), (2, 3), (5, 4)]

2. Sort the pairs by their values:
   Sorted by values (descending): [(5, 4), (4, 0), (3, 1), (2, 3), (1, 2)]

3. Select the top k elements:
   Since k = 4, we select the first four pairs: [(5, 4), (4, 0), (3, 1), (2, 3)]

4. Sort the selected elements by their original indices:
   Sorted by original indices: [(4, 0), (3, 1), (2, 3), (5, 4)]
   Note: The element (1, 2) is excluded because we are only selecting the top k elements for k = 4.

5. Extract the sorted subsequence:
   Extracting the values from the sorted pairs: [4, 3, 2, 5]

6. Return the subsequence:
   The final subsequence returned is [4, 3, 2, 5].

Code

Complexity Analysis

Time Complexity

1. Sorting the array based on values: This step has a time complexity of O(N \log N), where (N) is the number of elements in the input array. Sorting is the most time-consuming part of the algorithm.
2. Sorting the top k elements based on their original indices: The worst-case time complexity for this step is also O(N \log N). However, since this sorting is applied to only the top k elements, it could be considered O(k \log k) in practice. But in the worst case, where (k = N), it becomes O(N \log N).

Given these considerations, the overall time complexity of the algorithm is O(N \log N), dominated by the sorting operations.

Space Complexity

1. Storing pairs of elements and their original indices: This requires O(N) space.
2. Additional space for sorting and storing the result: Sorting algorithms may require additional space, but for the sake of complexity analysis, the in-place sort is considered, leading to a constant space complexity O(1) for the sorting operation itself. However, storing the result requires O(k) space.

Thus, the total space complexity of the algorithm is O(N) + O(k) = O(N), as (N) is the dominant term and encompasses any additional space needed for the output and intermediate steps.