Point Location in Constant Time

TL;DR

This paper presents a method to preprocess a planar subdivision in near-linear time, enabling point location queries to be answered in constant time, significantly improving efficiency over previous methods.

Contribution

The authors introduce a preprocessing algorithm that reduces point location query time to constant, improving upon the previous $O(rac{n}{ ext{polylog} n})$ query time with faster preprocessing.

Findings

Preprocessing time improved to $O(n\sqrt{\log n})$

Point location query time reduced to $O(1)$

Significant efficiency gains over previous methods

Abstract

We preprocess the input subdivision with $n$ points on the plane in $O(n\sqrt{\log n})$ time to facilitate point location in constant time. Previously the preprocessing time is $O(n\log n)$ and point location takes $O(\log n)$ time.