Orientability of Random Hypergraphs and the Power of Multiple Choices

  • Nikolaos Fountoulakis
  • Konstantinos Panagiotou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)


A hypergraph H = (V, E) is called s-orientable, if there is an assignment of each edge e ∈ E to one of its vertices v ∈ e such that no vertex is assigned more than s edges. Let H n,m,k be a hypergraph, drawn uniformly at random from the set of all k-uniform hypergraphs with n vertices and m edges. In this paper we establish the threshold for the 1-orientability of H n,m,k for all k ≥ 3, i.e., we determine a critical quantity \(c_k^*\) such that with probability 1 − o(1) the graph H n,cn,k has a 1-orientation if \(c < c_k^*\), but fails doing so if \(c > c_k^*\).

We present two applications of this result that involve the paradigm of multiple choices. First, we show how it implies sharp load thresholds for cuckoo hash tables, where each element chooses k out of n locations. Particularly, for each k ≥ 3 we prove that with probability 1 − o(1) the maximum number of elements that can be hashed is \((1 - o(1))c_k^* n\), and more items prevent the successful allocation. Second, we study random graph processes, where in each step we have the choice among any edge connecting k random vertices. Here we show the existence of a phase transition for avoiding a giant connected component, and quantify precisely the dependence on k.


Random Graph Multiple Choice Degree Sequence Giant Component Poisson Random Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bohman, T., Frieze, A.: Avoiding a Giant Component. Random Structures & Algorithms 19, 75–85 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bohman, T., Frieze, A., Krivelevich, M., Loh, P., Sudakov, B.: Ramsey Games with Giants (Submitted)Google Scholar
  3. 3.
    Bohman, T., Frieze, A., Wormald, N.: Avoiding a Giant Component in Half the Edge Set of a Random Graph. Random Structures & Algorithms 25, 432–449 (2004)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bohman, T., Kim, J.H.: A Phase Transition for Avoiding a Giant Component. Random Structures & Algorithms 28(2), 195–214 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cain, J.A., Sanders, P., Wormald, N.: The Random Graph Threshold for k-orientiability and a Fast Algorithm for Optimal Multiple-choice Allocation. In: Bansal, N., Pruhs, K., Stein, C. (eds.) SODA 2007, pp. 469–476. SIAM, Philadelphia (2007)Google Scholar
  6. 6.
    Cooper, C.: The Cores of Random Hypergraphs with a Given Degree Sequence. Random Structures & Algorithms 25(4), 353–375 (2004)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Devroye, L., Morin, P.: Cuckoo Hashing: Further Analysis. Information Processing Letters 86(4), 215–219 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dietzfelbinger, M., Goerdt, A., Mitzenmacher, M., Montanari, A., Pagh, R., Rink, M.: Tight Thresholds for Cuckoo Hashing via XORSAT. In: Proceedings of ICALP (to appear, 2010)Google Scholar
  9. 9.
    Dietzfelbinger, M., Schellbach, U.: On Risks of Using Cuckoo Hashing with Simple Universal Hash Classes. In: Mathieu, C. (ed.) SODA 2009, pp. 795–804. SIAM, Philadelphia (2009)Google Scholar
  10. 10.
    Dietzfelbinger, M., Weidling, C.: Balanced Allocation and Dictionaries with Tightly Packed Constant Size Bins. Theoretical Computer Science 380(1-2), 47–68 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Drmota, M., Kutzelnigg, R.: A Precise Analysis of Cuckoo Hashing (submitted)Google Scholar
  12. 12.
    Erdős, P., Rényi, A.: On the Evolution of Random Graphs. Publication of the Mathematical Institute of the Hungarian Academy of Sciences 5, 17–61 (1960)Google Scholar
  13. 13.
    Fernholz, D., Ramachandran, V.: The k-orientability Thresholds for G n, p. In: Bansal, N., Pruhs, K., Stein, C. (eds.) SODA 2007, pp. 459–468. SIAM, Philadelphia (2007)Google Scholar
  14. 14.
    Fotakis, D., Pagh, R., Sanders, P., Spirakis, P.: Space efficient hash tables with worst case constant access time. In: Habib, M., Alt, H. (eds.) STACS 2003. LNCS, vol. 2607, pp. 271–282. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Frieze, A., Melsted, P.: Maximum Matchings in Random Bipartite Graphs and the Space Utilization of Cuckoo Hashtables (2009) (manuscript),
  16. 16.
    Gao, P., Wormald, N.: Load Balancing and Orientability Thresholds for Random Hypergraphs. In: Proceedings of STOC (to appear, 2010)Google Scholar
  17. 17.
    Janson, S., Łuczak, T., Ruciński, A.: Random Graphs. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York (2000)zbMATHGoogle Scholar
  18. 18.
    Kim, J.H.: Poisson Cloning Model for Random Graphs (2006) (manuscript)Google Scholar
  19. 19.
    Mitzenmacher, M., Vadhan, S.: Why Simple Hash Functions Work: Exploiting the Entropy in a Data Stream. In: Teng, S.-H. (ed.) SODA 2008, pp. 746–755. SIAM, Philadelphia (2008)Google Scholar
  20. 20.
    Molloy, M.: Cores in Random Hypergraphs and Boolean Formulas. Random Structures & Algorithms 27(1), 124–135 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Pagh, R., Rodler, F.F.: Cuckoo Hashing. In: auf der Heide, M. (ed.) ESA 2001. LNCS, vol. 2161, pp. 121–133. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  22. 22.
    Spencer, J., Wormald, N.: Birth Control for Giants. Combinatorica 27(5), 587–628 (2007)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Nikolaos Fountoulakis
    • 1
  • Konstantinos Panagiotou
    • 1
  1. 1.Max-Planck-Institute for InformaticsSaarbrückenGermany

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