Sum of Imbalance Numbers of All Subarrays

Problem Description

The imbalance number of a 0-indexed integer array arr of length n is defined as the number of indices in sarr = sorted(arr) such that:

• 0 <= i < n - 1, and
• sarr[i+1] - sarr[i] > 1

Given a 0-indexed integer array nums, return the sum of imbalance numbers of all its subarrays.

Key Insights

• A subarray is a contiguous non-empty sequence of elements within an array.
• The imbalance number is calculated based on the sorted version of a subarray.
• We need to efficiently compute the sum of imbalance numbers for all possible subarrays.

Space and Time Complexity

Time Complexity: O(n^3 log n) in the naive approach, but can be optimized. Space Complexity: O(n) for storing sorted subarrays.

Solution

To solve the problem, we can use a two-pointer approach to generate all possible subarrays. For each subarray, we will sort it and then count the imbalance numbers by iterating through the sorted array and checking the difference between consecutive elements. This approach ensures that we capture the required imbalance conditions efficiently.

1. Use two nested loops to generate all subarrays of the input array.
2. For each subarray, sort the elements.
3. Count the number of indices where the difference between consecutive sorted elements is greater than 1.
4. Accumulate the counts to get the final sum of imbalance numbers.

Code Solutions