Succinct Permutation Graphs

TL;DR

This paper introduces new succinct data structures for permutation and circular permutation graphs that support various queries efficiently, including the first exact distance oracle for these classes.

Contribution

It presents the first succinct exact distance oracle for permutation graphs and extends support for multiple queries, including neighborhood, degree, and shortest paths, with optimal or near-optimal space and time complexity.

Findings

Supports distance, adjacency, and neighborhood queries in optimal time.

Introduces a succinct data structure for bipartite permutation graphs.

Develops semi-distributed representations that bypass lower bounds on distance labeling.

Abstract

We present a succinct data structure for permutation graphs, and their superclass of circular permutation graphs, i.e., data structures using optimal space up to lower order terms. Unlike concurrent work on circle graphs (Acan et al. 2022), our data structure also supports distance and shortest-path queries, as well as adjacency and neighborhood queries, all in optimal time. We present in particular the first succinct exact distance oracle for (circular) permutation graphs. A second succinct data structure also supports degree queries in time independent of the neighborhood's size at the expense of an $O(\log n/\log \log n)$-factor overhead in all running times. Furthermore, we develop a succinct data structure for the class of bipartite permutation graphs. We demonstrate how to run algorithms directly over our succinct representations for several problems on permutation graphs: Clique, Coloring, Independent Set, Hamiltonian Cycle, All-Pair Shortest Paths, and others. Finally, we initiate the study of semi-distributed graph representations; a concept that smoothly interpolates between distributed (labeling schemes) and centralized (standard data structures). We show how to turn some of our data structures into semi-distributed representations by storing only $O(n)$ bits of additional global information, circumventing the lower bound on distance labeling schemes for permutation graphs.