Dynamic Planar Point Location with Sub-Logarithmic Local Updates

TL;DR

This paper presents a data structure for planar point location that supports efficient local updates, insertions, deletions, and queries, with sub-logarithmic time for local updates, improving dynamic geometric data structures.

Contribution

It introduces a linear size data structure enabling sub-logarithmic local updates in planar point location, a significant improvement over previous methods.

Findings

Supports insertions, deletions, and queries in logarithmic time

Allows local updates in sub-logarithmic time on pointer machines

Achieves linear size data structure for dynamic planar point location

Abstract

We study planar point location in a collection of disjoint fat regions, and investigate the complexity of \emph {local updates}: replacing any region by a different region that is "similar" to the original region. (i.e., the size differs by at most a constant factor, and distance between the two regions is a constant times that size). We show that it is possible to create a linear size data structure that allows for insertions, deletions, and queries in logarithmic time, and allows for local updates in sub-logarithmic time on a pointer machine.