Sum of Imbalance Numbers of All Subarrays

Sum of Imbalance Numbers of All Subarrays

What is this problem about?

The "imbalance number" of an array is the number of integers x such that x and x + 1 both exist in the array, but they are not "adjacent" in terms of sorted value (meaning x+1 is present but x's contribution to a contiguous sorted range is missing something). More specifically, in a sorted version of the array, it's the count of gaps larger than 1 between consecutive elements. The task is to find the sum of imbalance numbers of all possible subarrays of a given array.

Why is this asked in interviews?

This Hard-level Google question evaluates a candidate's ability to optimize a problem involving every possible subarray (O(N²)). It tests whether they can update a complex property (imbalance) incrementally as they expand a subarray, rather than re-calculating it from scratch. It also tests knowledge of data structures like Hash Sets or sorted containers for tracking neighbors.

Algorithmic pattern used

The pattern is "Incremental Subarray Enumeration with Hash Set."

1. Iterate through all possible starting positions i.
2. For each i, expand the subarray to the right one element at a time (j = i to n-1).
3. Maintain a Hash Set of elements in the current subarray [i...j].
4. As you add a new element nums[j], update the imbalance count. Adding a number might "fill a gap" between existing numbers (decreasing imbalance) or "create a new number" that doesn't have neighbors (increasing imbalance). By doing this, the total complexity is O(N²), which is acceptable for the typical constraints of this problem.

Example explanation

Subarray: [1, 3]. Sorted: [1, 3]. Gap between 1 and 3 is > 1. Imbalance = 1. If we add 2 to get [1, 3, 2]:

• Sorted: [1, 2, 3].
• Gap between 1 and 2 is 1. Gap between 2 and 3 is 1.
• Imbalance = 0. The logic involves checking if val-1 and val+1 are already in the set when val is added.

Common mistakes candidates make

1. Sorting every subarray: Sorting each of the O(N²) subarrays takes O(N³ log N) time, which is much too slow.
2. Wrong Imbalance definition: Misunderstanding the specific rules for what constitutes an "imbalance" (it's essentially counting how many elements x have x+1 in the set).
3. Handling Duplicate Values: Not accounting for the fact that adding the same number twice shouldn't change the imbalance count.

Interview preparation tip

Practice "Incremental DP" or "Incremental property tracking" for subarrays. If you can update the answer for [i...j] to get the answer for [i...j+1] in O(1) or O(log N), you can solve most subarray problems efficiently.