Planar Reachability in Linear Space and Constant Time

TL;DR

This paper introduces a linear-space data structure for planar digraphs that allows constant-time reachability queries, achieving optimal efficiency in both construction and query performance.

Contribution

It presents a novel linear-space, constant-time reachability data structure for planar digraphs, improving upon previous solutions with higher space complexity.

Findings

Achieves constant-time reachability queries in planar digraphs.

Uses linear space and construction time, optimal in theory.

Outperforms previous solutions with higher space requirements.

Abstract

We show how to represent a planar digraph in linear space so that distance queries can be answered in constant time. The data structure can be constructed in linear time. This representation of reachability is thus optimal in both time and space, and has optimal construction time. The previous best solution used $O(n\log n)$ space for constant query time [Thorup FOCS'01].