, Volume 76, Issue 2, pp 445–473 | Cite as

Succinct Posets



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.


Succinct data structure Graph representations Partial orders Rank and select Zarankiewicz problem Balanced bipartite graphs Graph compression Data compression 


  1. 1.
    Agrawal, R., Borgida, A., Jagadish, H.V.: Efficient management of transitive relationships in large data and knowledge bases. In: Proceedings of the ACM International Conference on Management of Data (SIGMOD), pp. 253–262. ACM Press (1989)Google Scholar
  2. 2.
    Barbay, J., Claude, F., Navarro, G.: Compact binary relation representations with rich functionality. Inf. Comput. 232, 19–37 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Barbay, J., He, M., Munro, J.I., Rao, S.S.: Succinct indexes for strings, binary relations and multilabeled trees. ACM Trans. Algorithms (TALG) 7(4), 52 (2011)MathSciNetMATHGoogle Scholar
  4. 4.
    Bender, M.A., Farach-Colton, M., Pemmasani, G., Skiena, S., Sumazin, P.: Lowest common ancestors in trees and directed acyclic graphs. J. Algorithms 57(2), 75–94 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brightwell, G.: The average number of linear extensions of a partial order. J. Comb. Theory Ser. A 73(2), 193–206 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)MATHGoogle Scholar
  7. 7.
    Daskalakis, C., Karp, R.M., Mossel, E., Riesenfeld, S., Verbin, E.: Sorting and selection in posets. SIAM J. Comput. 40(3), 597–622 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    De Loof, K., De Meyer, H., De Baets, B.: Exploiting the lattice of ideals representation of a poset. Fundam. Inf. 71(2, 3), 309–321 (2006)MathSciNetMATHGoogle Scholar
  9. 9.
    Farzan, A.: Succinct representation of trees and graphs. Ph.D. thesis, University of Waterloo (2009)Google Scholar
  10. 10.
    Farzan, A., Fischer, J.: Compact representation of posets. In: Proceedings of the 22nd International Symposium on Algorithms and Computation (ISAAC), LNCS, vol. 7074, pp. 302–311. Springer (2011)Google Scholar
  11. 11.
    Farzan, A., Munro, J.I.: Succinct representations of arbitrary graphs. Theoret. Comput. Sci. 513, 38–52 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Feder, T., Motwani, R.: Clique partitions, graph compression and speeding-up algorithms. J. Comput. Syst. Sci. 51(2), 261–272 (1995)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ferragina, P., Nitto, I., Venturini, R.: Succinct Oracles for Exact Distances in Undirected Unweighted Graphs. Technical Report TR-07-11, Università di Pisa (2007)Google Scholar
  14. 14.
    Gambosi, G., Nešetřil, J., Talamo, M.: Efficient representation of taxonomies. In: Proceedings of the International Joint Conference on Theory and Practice of Software Development (TAPSOFT), LNCS, vol. 249, pp. 232–240. Springer (1987)Google Scholar
  15. 15.
    Gambosi, G., Nešetřil, J., Talamo, M.: On locally presented posets. Theoret. Comput. Sci. 70(2), 251–260 (1990)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gambosi, G., Nešetřil, J., Talamo, M.: Posets, boolean representations and quick path searching. In: Proceedings of the 14th International Colloquium on Automata, Languages and Programming (ICALP), LNCS, vol. 267, pp. 404–424. Springer, Berlin (1987)Google Scholar
  17. 17.
    Gambosi, G., Protasi, M., Talamo, M.: An efficient implicit data structure for relation testing and searching in partially ordered sets. BIT Numer. Math. 33(1), 29–45 (1993)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Garg, V.K., Skawratananond, C.: String realizers of posets with applications to distributed computing. In: Proceedings of the 20th Annual ACM symposium on Principles of Distributed Computing (PODC), pp. 72–80. ACM (2001)Google Scholar
  19. 19.
    Grossi, R., Gupta, A., Vitter, J.S.: High-order entropy-compressed text indexes. In: Proceedings of the 14th Symposium on Discrete Algorithms (SODA), pp. 841–850 (2003)Google Scholar
  20. 20.
    Habib, M., Huchard, M., Nourine, L.: Embedding partially ordered sets into chain-products. In: Proceedings of Knowledge Retrieval, Use and Storage for Efficiency (KRUSE), pp. 147–161 (1995)Google Scholar
  21. 21.
    Habib, M., Nourine, L.: Bit-vector encoding for partially ordered sets. In: International Workshop on Orders, Algorithms, and Applications (ORDAL), LNCS, vol. 831, pp. 1–12. Springer (1994)Google Scholar
  22. 22.
    Habib, M., Nourine, L.: Tree structure for distributive lattices and its applications. Theoret. Comput. Sci. 165(2), 391–405 (1996)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    He, M.: Succinct indexes. Ph.D. thesis, University of Waterloo (2007)Google Scholar
  24. 24.
    Hegde, R., Jain, K.: The hardness of approximating poset dimension. Electron. Notes Discret. Math. 29, 435–443 (2007)CrossRefMATHGoogle Scholar
  25. 25.
    Jacobson, G.: Space-efficient static trees and graphs. In Proceedings of 30th Annual IEEE Symposium on Foundations of Computer Science pp. 549–554 (1989)Google Scholar
  26. 26.
    Jin, R., Xiang, Y., Ruan, N., Fuhry, D.: 3-HOP: a high-compression indexing scheme for reachability query. In: Proceedings of the ACM International Conference on Management of Data (SIGMOD), pp. 813–826. SIGMOD ’09, ACM (2009)Google Scholar
  27. 27.
    Kleitman, D.J., Rothschild, B.L.: The number of finite topologies. Proc. Am. Math. Soc. 25, 276 (1970)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kleitman, D.J., Rothschild, B.L.: Asymptotic enumeration of partial orders on a finite set. Trans. Am. Math. Soc. 205, 205–220 (1975)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Kővári, T., Sós, V.T., Turán, P.: On a problem of Zarankiewicz. Colloq. Math. 3(1954), 50–57 (1954)MathSciNetMATHGoogle Scholar
  30. 30.
    Mittal, N., Garg, V.K.: Rectangles are Better than Chains for Encoding Partially Ordered Sets. Technical Report, University of Texas at Austin, Department of Electrical and Computer Engineering, Austin, TX (2004)Google Scholar
  31. 31.
    Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)CrossRefMATHGoogle Scholar
  32. 32.
    Mubayi, D., Turán, G.: Finding bipartite subgraphs efficiently. Inf. Process. Lett. 110(5), 174–177 (2010)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Mucha, M., Sankowski, P.: Maximum matchings via Gaussian elimination. In: Proceedings of 45th Symposium on Foundations of Computer Science (FOCS), pp. 248–255. IEEE Computer Society (2004)Google Scholar
  34. 34.
    Pǎtraşcu, M.: Succincter. In: Proceedings of 49th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 305–313. IEEE Computer Society (2008)Google Scholar
  35. 35.
    Raman, R., Raman, V., Rao, S.S.: Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets. ACM Trans. Algorithms 3(4) (2007). doi:10.1145/1290672.1290680
  36. 36.
    Raymond, D.R.: Partial-order databases. Ph.D. thesis, University of Waterloo (1996)Google Scholar
  37. 37.
    Raynaud, O., Thierry, E.: The complexity of embedding orders into small products of chains. Order 27(3), 365–381 (2010)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Talamo, M., Vocca, P.: Representing graphs implicitly using almost optimal space. Discret. Appl. Math. 108(1–2), 193–210 (2001)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Talamo, M., Vocca, P.: Fast lattice browsing on sparse representation. In: International Workshop on Orders, Algorithms, and Applications (ORDAL), LNCS, vol. 831, pp. 186–204. Springer (1994)Google Scholar
  40. 40.
    Talamo, M., Vocca, P.: An efficient data structure for lattice operations. SIAM J. Comput. 28(5), 1783–1805 (1999)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Taraz, A.R.: Phase transitions in the evolution of partially ordered sets. Ph.D. thesis, Humboldt-Universität zu Berlin (1999)Google Scholar
  42. 42.
    Thorup, M.: Compact oracles for reachability and approximate distances in planar digraphs. J. ACM 51(6), 993–1024 (2004)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 1–24 (2005)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1992)Google Scholar
  45. 45.
    Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Algebr. Discret. Methods 3(3), 351–358 (1982)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.David R. Cherition School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Max Planck Institut für InformatikSaarbrückenGermany

Personalised recommendations