A Dynamic Stabbing-Max Data Structure with Sub-Logarithmic Query Time

TL;DR

This paper introduces a dynamic data structure for one-dimensional stabbing-max queries with optimal sub-logarithmic query time, linear space, and efficient updates, extending to multi-dimensional cases.

Contribution

It presents a novel dynamic data structure achieving sub-logarithmic query time for stabbing-max problems in one and multiple dimensions.

Findings

Answers 1D stabbing-max queries in $O(rac{ ext{log} n}{ ext{log} ext{log} n})$ time.

Supports insertions and deletions in $O( ext{log} n)$ and $O(rac{ ext{log} n}{ ext{log} ext{log} n})$ amortized time respectively.

Extends to multi-dimensional stabbing-max queries with polylogarithmic time and space complexity.

Abstract

In this paper we describe a dynamic data structure that answers one-dimensional stabbing-max queries in optimal $O(\log n/\log\log n)$ time. Our data structure uses linear space and supports insertions and deletions in $O(\log n)$ and $O(\log n/\log \log n)$ amortized time respectively. We also describe a $O(n(\log n/\log\log n)^{d-1})$ space data structure that answers $d$-dimensional stabbing-max queries in $O((\log n/\log\log n)^{d})$ time. Insertions and deletions are supported in $O((\log n/\log\log n)^d\log\log n)$ and $O((\log n/\log\log n)^d)$ amortized time respectively.