Number of Subsequences That Satisfy the Given Sum Condition

Problem Description

You are given an array of integers nums and an integer target. Return the number of non-empty subsequences of nums such that the sum of the minimum and maximum element on it is less or equal to target. Since the answer may be too large, return it modulo 10^9 + 7.

Key Insights

• A subsequence is formed from the original array by deleting some or no elements without changing the order of the remaining elements.
• The condition to check is that the sum of the minimum and maximum elements of each subsequence must be less than or equal to the target.
• Sorting the array can help efficiently find valid pairs of minimum and maximum using a two-pointer technique.
• Utilizing combinatorial counting can help efficiently calculate the number of valid subsequences.

Space and Time Complexity

Time Complexity: O(n log n) for sorting the array, and O(n) for the two-pointer traversal, resulting in a total of O(n log n) due to sorting. Space Complexity: O(1) for the pointers, O(n) for the storage of the sorted array.

Solution

The solution involves first sorting the array to easily identify minimum and maximum elements. By using a two-pointer approach, we can efficiently find the valid pairs of minimum and maximum that satisfy the condition. For each valid pair, we can calculate the number of non-empty subsequences that can be formed using combinatorial mathematics.

1. Sort the array.
2. Use two pointers: one starting from the beginning (left) and the other from the end (right) of the array.
3. For each pair of elements pointed by left and right, check if their sum is less than or equal to the target.
4. If valid, calculate the number of subsequences that can be formed with elements between the left and right pointers.
5. Move the pointers accordingly, and continue until all pairs are checked.

Code Solutions