Fast and Compact Planar Embeddings

TL;DR

This paper enhances Turán's planar graph representation by adding minimal bits to enable fast navigation, resulting in a practical, compact, and efficiently constructible embedding suitable for large datasets.

Contribution

It introduces a method to augment Turán's representation with few bits for quick navigation, and provides the first efficient parallel construction of a compact planar graph embedding.

Findings

Uses about 6 bits per edge in practice

Supports navigation operations within a few microseconds

Builds the encoding sequentially at below 1 microsecond per edge

Abstract

There are many representations of planar graphs, but few are as elegant as Tur\'an's (1984): it is simple and practical, uses only 4 bits per edge, can handle self-loops and multi-edges, and can store any specified embedding. Its main disadvantage has been that "it does not allow efficient searching" (Jacobson, 1989). In this paper we show how to add a sublinear number of bits to Tur\'an's representation such that it supports fast navigation while retaining simplicity. As a consequence of the inherited simplicity, we offer the first efficient parallel construction of a compact encoding of a planar graph embedding. Our experimental results show that the resulting representation uses about 6 bits per edge in practice, supports basic navigation operations within a few microseconds, and can be built sequentially at a rate below 1 microsecond per edge, featuring a linear speedup with a parallel efficiency around 50\% for large datasets.