Sum of Subsequence Widths Solution in Python

Problem Description

Welcome to the whimsical world of subsequences! Imagine you have a bag of marbles, each with a different size. Now, if you were to pick some marbles and measure the width of the subsequence you created, you might wonder: “What’s the total width of all possible subsequences?” Well, that’s exactly what this problem is about!

In the Sum of Subsequence Widths problem, you are given an array of integers, and your task is to calculate the sum of the widths of all possible subsequences. The width of a subsequence is defined as the difference between the maximum and minimum values in that subsequence.

So, if you think about it, it’s like trying to figure out how much space your marbles take up when you pick them in different combinations. Spoiler alert: it’s a lot of math and sorting!

Code Solution


class Solution:
    def sumSubseqWidths(self, nums: list[int]) -> int:
        kMod = 1_000_000_007
        n = len(nums)
        ans = 0
        exp = 1

        nums.sort()

        for i in range(n):
            ans += (nums[i] - nums[n - 1 - i]) * exp
            ans %= kMod
            exp = exp * 2 % kMod

        return ans

Approach

The approach taken in this solution is quite clever! First, we sort the array of numbers. This allows us to easily identify the maximum and minimum values in any subsequence. The key insight is that for each number in the sorted array, we calculate its contribution to the total width based on its position.

For each number, we determine how many times it will be the maximum and how many times it will be the minimum in all possible subsequences. The exp variable helps us keep track of the number of subsequences that can be formed with the current number as the maximum.

By iterating through the sorted array and calculating the contributions, we can efficiently compute the total sum of widths.

Time and Space Complexity

Complexity Type	Complexity
Time Complexity	O(n log n)
Space Complexity	O(1)

Real-World Example

Imagine you are organizing a race with different runners, each having a different speed. The width of the subsequence could represent the difference in speed between the fastest and slowest runners in any group you form. If you want to know how much faster the fastest runner is compared to the slowest in all possible groups, you would be calculating the sum of subsequence widths!