Efficient Labeling for Reachability in Digraphs

TL;DR

This paper explores efficient labeling schemes for reachability queries in directed graphs, establishing theoretical bounds and presenting a new scheme with significantly reduced label size.

Contribution

It introduces a novel reachability labeling scheme with labels of size n/3+o(n), improving upon previous methods and extending classical approaches.

Findings

Labels of at least n/4 bits are necessary due to information theory.

Existing schemes can achieve labels of about n/2+O(log n) bits.

The new scheme reduces label size to n/3+o(n) bits.

Abstract

We consider labeling nodes of a directed graph for reachability queries. A reachability labeling scheme for such a graph assigns a binary string, called a label, to each node. Then, given the labels of nodes $u$ and $v$ and no other information about the underlying graph, it should be possible to determine whether there exists a directed path from $u$ to $v$. By a simple information theoretical argument and invoking the bound on the number of partial orders, in any scheme some labels need to consist of at least $n/4$ bits, where $n$ is the number of nodes. On the other hand, it is not hard to design a scheme with labels consisting of $n/2+O(\log n)$ bits. In the classical centralised setting, Munro and Nicholson designed a data structure for reachability queries consisting of $n^2/4+o(n^2)$ bits (which is optimal, up to the lower order term). We extend their approach to obtain a scheme with labels consisting of $n/3+o(n)$ bits.