Towards Optimal Range Medians

TL;DR

This paper introduces an efficient algorithm for computing medians in multiple subarray intervals, improving upon previous methods and extending naturally to higher-dimensional problems through range counting queries.

Contribution

The paper presents a simple, space-efficient algorithm that improves query time for range median problems and generalizes to higher dimensions by reducing to range counting queries.

Findings

Achieves $O(n ext{ log }k + k ext{ log }n)$ query time, improving previous algorithms.

Matches the lower bound for $k=O(n)$, indicating optimality.

Generalizes to higher-dimensional problems via range counting queries.

Abstract

We consider the following problem: given an unsorted array of $n$ elements, and a sequence of intervals in the array, compute the median in each of the subarrays defined by the intervals. We describe a simple algorithm which uses O(n) space and needs $O(n\log k + k\log n)$ time to answer the first $k$ queries. This improves previous algorithms by a logarithmic factor and matches a lower bound for $k=O(n)$. Since the algorithm decomposes the range of element values rather than the array, it has natural generalizations to higher dimensional problems -- it reduces a range median query to a logarithmic number of range counting queries.