Succinct Posets

• Published in: Algorithmica Vol. 76; no. 2; pp. 445 - 473
• Main Authors: Munro, J. Ian, Nicholson, Patrick K.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2016; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Computer Science; Computer Systems Organization and Communication Networks; Data Structures and Information Theory; Graph theory; Graphical representations; Mathematics of Computing; Set theory; Theory of Computation; Partial orders; Graph compression; Succinct data structure; Rank and select; Data compression; Balanced bipartite graphs; Graph representations; Zarankiewicz problem
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-015-0047-1

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. Furthermore, a slight extension to this data structure yields a succinct oracle for reachability in arbitrary directed graphs. Thus, using no more than a quarter of the space required to represent an arbitrary directed graph, reachability queries can be supported in constant time. We also consider the operation of listing all the successors or predecessors of a given element, and show how to do this in constant time per element reported using a slightly modified version of our succinct data structure.