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Succinct Posets

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Abstract

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.

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Notes

  1. In this paper we use \(\lg x\) to denote \(\log _2 x\).

  2. All posets we discuss are finite.

  3. We note that the height is sometimes defined as the number of vertices in its maximum chain minus one, but we find our definition slightly more convenient for our purposes.

  4. Note that a forbidden polygon is different from a directed cycle: a forbidden polygon is acyclic.

  5. A typical poset on \(n\) vertices has order dimension about \(n/4\) [44, p. 181].

  6. Further explanation: In their enumeration argument (see Lemma 3) they count the set \(B_{n+1}\), which contains partial orders that have a source vertex \(s_1\) that is below every vertex in some set Q, of size \(\sqrt{n}\), where Q has the property that at least \(n/2\) vertices in the poset are below some vertex in Q. This provides an efficient way to encode the relations with \(s_1\), as any vertex below a vertex in Q cannot be above \(s_1\), as it would form a forbidden polygon; in this case a triangle. Thus, we only need to store the set Q, as well as a bit for each vertex not covered by a vertex in Q, of which there are at most \(n/ 2 - \sqrt{n}\). However, creating a data structure using this property seems difficult, as the data is implicit in terms of the neighbours of Q, which are not explicitly stored.

  7. The construction time is not explicitly stated by Raman, Raman, and Rao, but each of the (constant number of) indices that comprise this data structure can be constructed in \(O(U)\) time, so the observation follows.

  8. We are specifically using the term “ordered” to avoid overloading the term “rank”.

  9. We note that we can use the same bit string to record these edges as the one used in the index of Lemma 6, when we eventually combine everything.

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Acknowledgments

The authors wish to thank the anonymous referees for their detailed and valuable comments.

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Authors and Affiliations

  1. David R. Cherition School of Computer Science, University of Waterloo, 200 University Avenue West, Waterloo, N2L 3G1, Canada

    J. Ian Munro

  2. Max Planck Institut für Informatik, Campus E1 4, 66123, Saarbrücken, Germany

    Patrick K. Nicholson

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  1. J. Ian Munro
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Correspondence to Patrick K. Nicholson.

Additional information

A preliminary version of this paper appeared in the the proceedings of the 20th annual European Symposium on Algorithms (ESA 2012). This research was funded in part by NSERC of Canada, the Canada Research Chairs program, an NSERC PGS-D Scholarship, the Derick Wood Graduate Scholarship Award, and a David R. Cheriton Scholarship.

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Munro, J.I., Nicholson, P.K. Succinct Posets. Algorithmica 76, 445–473 (2016). https://doi.org/10.1007/s00453-015-0047-1

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