Power Update After K-th Largest Insertion II - Solution & Explanation

Problem Overview: You receive a sequence of insertions and must continuously track the k-th largest element in the structure. After each insertion, a "power" value must be updated based on the current ordering of elements relative to that k-th largest threshold. A naive recomputation quickly becomes too slow as the dataset grows.

Approach 1: Re-sort After Every Insertion (Brute Force) (Time: O(n log n) per update, Space: O(n))

The simplest idea stores all inserted values in a list. After each insertion, sort the entire array and read the element at index n - k to obtain the k-th largest value. Once that value is known, recompute the power metric by iterating through the elements that influence the formula. This approach is easy to reason about and helps confirm correctness for small inputs. The downside is the repeated sort operation, which makes it impractical for large streams of updates.

Approach 2: Min Heap of Size K (Time: O(n log k), Space: O(k))

A more efficient approach maintains a min heap containing only the top k elements seen so far. The smallest element in the heap is always the current k-th largest. When a new value arrives, push it into the heap and remove the smallest element if the heap grows beyond size k. The heap root immediately gives the updated threshold needed to compute the power update. This technique relies on operations such as push and pop, each costing O(log k). Heaps are a common pattern in streaming problems involving order statistics and appear frequently in heap-based interview questions.

Approach 3: Fenwick Tree / Order Statistic Structure (Time: O(n log n), Space: O(n))

When the power update depends on aggregate information about many elements, maintaining only the k largest may not be enough. Instead, compress the value range and store frequencies in a Fenwick Tree (Binary Indexed Tree). Each insertion updates the frequency structure, and a binary search over prefix sums finds the index corresponding to the k-th largest value. Additional Fenwick trees or prefix sums can maintain sums needed for the power calculation. Each update and query runs in O(log n). This pattern is common in problems involving dynamic ranks and appears frequently alongside segment tree and binary search techniques.

Recommended for interviews: The min‑heap solution is usually the expected answer if the task only requires tracking the k-th largest element in a stream. It reduces the complexity from repeated sorting to O(n log k). If the power update depends on aggregated values across the dataset, interviewers often expect a Fenwick Tree or segment tree because it supports fast rank queries and prefix computations. Mentioning the brute force approach first shows baseline reasoning, while presenting the heap or Fenwick solution demonstrates algorithmic optimization skills.