Navigating Planar Topologies in Near-Optimal Space and Time

TL;DR

This paper presents a near-optimal space encoding for planar graph embeddings that allows efficient topological queries, achieving constant-time operations with linear space and near-information-theoretic bounds.

Contribution

It introduces a succinct encoding of planar graph embeddings that supports fast topological queries in near-optimal space, improving query efficiency and space usage.

Findings

Encoding uses 4m bits, close to the theoretical lower bound.

Supports topological queries in near-constant or constant time.

Achieves linear space with constant-time query performance.

Abstract

We show that any embedding of a planar graph can be encoded succinctly while efficiently answering a number of topological queries near-optimally. More precisely, we build on a succinct representation that encodes an embedding of $m$ edges within $4m$ bits, which is close to the information-theoretic lower bound of about $3.58m$. With $4m+o(m)$ bits of space, we show how to answer a number of topological queries relating nodes, edges, and faces, most of them in any time in $\omega(1)$. Further, we show that with $O(m)$ bits of space we can solve all those operations in $O(1)$ time.