Succinct Posets

TL;DR

This paper introduces a space-efficient data structure for representing posets and directed graphs, enabling constant-time reachability queries while occupying near-theoretical minimum space.

Contribution

It presents a novel compression algorithm and a succinct data structure for posets that supports constant-time precedence queries, reducing space to about half of traditional methods.

Findings

Space usage is approximately half of an adjacency matrix.

Supports constant-time precedence and reachability queries.

Achieves near-information-theoretic lower bounds in worst case.

Abstract

We describe an algorithm for compressing a partially ordered set, or \emph{poset}, so that it occupies space matching the information theory lower bound (to within lower order terms), in the worst case. Using this algorithm, we design a succinct data structure for representing a poset that, given two elements, can report whether one precedes the other in constant time. This is equivalent to succinctly representing the transitive closure graph of the poset, and we note that the same method can also be used to succinctly represent the transitive reduction graph. For an $n$ element poset, the data structure occupies $n^2/4 + o(n^2)$ bits, in the worst case, which is roughly half the space occupied by an upper triangular matrix. Furthermore, a slight extension to this data structure yields a succinct oracle for reachability in arbitrary directed graphs. Thus, using roughly a quarter of the space required to represent an arbitrary directed graph, reachability queries can be supported in constant time.