Range Medians

TL;DR

This paper generalizes median finding to multiple intervals, providing an efficient algorithm with near-optimal comparison bounds for batch median queries in unsorted arrays.

Contribution

It introduces a new algorithm for batch median queries with proven comparison complexity bounds, extending classical median finding to multiple intervals.

Findings

Algorithm uses $O(n ext{log}n + k ext{log}k ext{log}n)$ comparisons.

Lower bound of $oldsymbol{ ext{Omega}(n ext{log}k)}$ comparisons established.

Algorithm is optimal for $k=O(n/ ext{log}n)$.

Abstract

We study a generalization of the classical median finding problem to batched query case: given an array of unsorted $n$ items and $k$ (not necessarily disjoint) intervals in the array, the goal is to determine the median in {\em each} of the intervals in the array. We give an algorithm that uses $O(n\log n + k\log k \log n)$ comparisons and show a lower bound of $\Omega(n\log k)$ comparisons for this problem. This is optimal for $k=O(n/\log n)$.