Dynamic Range Selection in Linear Space

TL;DR

This paper introduces a new linear space data structure for dynamic range selection queries in the plane, achieving improved space efficiency while maintaining fast query and update times.

Contribution

It presents a novel linear space data structure for dynamic range selection with improved space complexity and competitive query and update times.

Findings

Supports range selection queries in O((log n / log log n)^2) time.

Supports insertions and deletions in O((log n / log log n)^2) amortized time.

Reduces space usage by a factor of Θ(log n / log log n) compared to previous solutions.

Abstract

Given a set $S$ of $n$ points in the plane, we consider the problem of answering range selection queries on $S$: that is, given an arbitrary $x$-range $Q$ and an integer $k > 0$, return the $k$-th smallest $y$-coordinate from the set of points that have $x$-coordinates in $Q$. We present a linear space data structure that maintains a dynamic set of $n$ points in the plane with real coordinates, and supports range selection queries in $O((\lg n / \lg \lg n)^2)$ time, as well as insertions and deletions in $O((\lg n / \lg \lg n)^2)$ amortized time. The space usage of this data structure is an $\Theta(\lg n / \lg \lg n)$ factor improvement over the previous best result, while maintaining asymptotically matching query and update times. We also present a succinct data structure that supports range selection queries on a dynamic array of $n$ values drawn from a bounded universe.