Succinct Data Structure for Path Graphs

TL;DR

This paper introduces two space-efficient data structures for path graphs that enable fast adjacency, neighborhood, and degree queries, improving query times while maintaining succinct space usage.

Contribution

The paper presents novel succinct data structures for path graphs supporting efficient queries, utilizing heavy path decomposition and wavelet trees, with space and time improvements over previous methods.

Findings

First data structure supports adjacency in O(log n) time.

Second data structure supports adjacency and degree in O(1) time.

Both structures use advanced decomposition and range search techniques.

Abstract

We consider the problem of designing a succinct data structure for {\it path graphs} (which are a proper subclass of chordal graphs and a proper superclass of interval graphs) on $n$ vertices while supporting degree, adjacency, and neighborhood queries efficiently. We provide the following two solutions for this problem: - an $n \log n+o(n \log n)$-bit succinct data structure that supports adjacency query in $O(\log n)$ time, neighborhood query in $O(d \log n)$ time and finally, degree query in $\min\{O(\log^2 n), O(d \log n)\}$ where $d$ is the degree of the queried vertex. - an $O(n \log^2 n)$-bit space-efficient data structure that supports adjacency and degree queries in $O(1)$ time, and the neighborhood query in $O(d)$ time where $d$ is the degree of the queried vertex. Central to our data structures is the usage of the classical heavy path decomposition by Sleator and Tarjan~\cite{ST}, followed by a careful bookkeeping using an orthogonal range search data structure using wavelet trees~\cite{Makinen2007} among others, which maybe of independent interest for designing succinct data structures for other graph classes.