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The Multiple-Orientability Thresholds for Random Hypergraphs

  • NIKOLAOS FOUNTOULAKIS (a1), MEGHA KHOSLA (a2) and KONSTANTINOS PANAGIOTOU (a3)
Abstract

A k-uniform hypergraph H = (V, E) is called ℓ-orientable if there is an assignment of each edge eE to one of its vertices ve such that no vertex is assigned more than ℓ edges. Let Hn,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 ℓ-orientability of Hn,m,k for all k ⩾ 3 and ℓ ⩾ 2, that is, we determine a critical quantity c*k,ℓ such that with probability 1 − o(1) the graph Hn,cn,k has an ℓ-orientation if c < c*k,ℓ , but fails to do so if c > c*k,ℓ .

Our result has various applications, including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays.

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An extended abstract of this work appeared in the Proceedings of the 22nd ACM–SIAM Symposium on Discrete Algorithms: SODA'11.

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[1] Bender, E. A. and Canfield, E. R. (1978) The asymptotic number of labelled graphs with given degree sequence. J. Combin. Theory Ser. A 24 296307.
[2] Bollobás, B. (1980) A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Europ. J. Combin. 1 311316.
[3] Cain, J. A., Sanders, P. and Wormald, N. (2007) The random graph threshold for k-orientability and a fast algorithm for optimal multiple-choice allocation. In Proc. 18th Annual ACM–SIAM Symposium on Discrete Algorithms: SODA 2007, pp. 469–476.
[4] Cooper, C. (2004) The cores of random hypergraphs with a given degree sequence. Random Struct. Alg. 25 353375.
[5] Dietzfelbinger, M., Goerdt, A., Mitzenmacher, M., Montanari, A., Pagh, R. and Rink, M. (2010) Tight thresholds for cuckoo hashing via XORSAT. In Proc. 37th International Colloquium on Automata, Languages and Programming: ICALP 2010, Vol. 6198 of Lecture Notes in Computer Science, Springer, pp. 213225.
[6] Ellis, R. (2006) Entropy, Large Deviations, and Statistical Mechanics, Classics in Mathematics. Springer.
[7] Fernholz, D. and Ramachandran, V. (2007) The k-orientability thresholds for Gn,p . In Proc. 18th Annual ACM–SIAM Symposium on Discrete Algorithms: SODA 2007, pp. 459–468.
[8] Fotakis, D., Pagh, R., Sanders, P. and Spirakis, P. (2003) Space efficient hash tables with worst case constant access time. In Proc. 20th Annual Symposium on Theoretical Aspects of Computer Science: STACS 2003, Vol. 2607 of Lecture Notes in Computer Science, Springer, pp. 271282.
[9] Fountoulakis, N. and Panagiotou, K. (2010) Orientability of random hypergraphs and the power of multiple choices. In Proc. 37th International Colloquium on Automata, Languages and Programming: ICALP 2010, Vol. 6198 of Lecture Notes in Computer Science, Springer, pp. 348359.
[10] Fountoulakis, N. and Panagiotou, K. (2012) Sharp load thresholds for cuckoo hashing. Random Struct. Alg. 41 306333.
[11] Frieze, A. and Melsted, P. (2012) Maximum matchings in random bipartite graphs and the space utilization of cuckoo hash tables. Random Struct. Alg. 41 334364.
[12] Gao, P. and Wormald, N. C. (2010) Load balancing and orientability thresholds for random hypergraphs. In Proc. 42nd ACM Symposium on Theory of Computing: STOC 2010, pp. 97–104.
[13] Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience.
[14] Kim, J. H. (2004) Poisson cloning model for random graphs. Manuscript.
[15] Leconte, M., Lelarge, M. and Massoulié, L. (2013) Convergence of multivariate belief propagation, with applications to cuckoo hashing and load balancing. In Proc. 24th ACM–SIAM Symposium on Discrete Algorithms: SODA 2013, pp. 35–46.
[16] Lelarge, M. (2012) A new approach to the orientation of random hypergraphs. In Proc. 23th ACM–SIAM Symposium on Discrete Algorithms: SODA 2012, pp. 251–264.
[17] Molloy, M. (2005) Cores in random hypergraphs and boolean formulas. Random Struct. Alg. 27 124135.
[18] Pagh, R. and Rodler, F. F. (2001) Cuckoo hashing. In Proc. 9th Annual European Symposium on Algorithms: ESA 2001, pp. 121–133.
[19] Sanders, P., Egner, S. and Korst, J. (1999) Fast concurrent access to parallel disks. In Proc. 11th Annual ACM–SIAM Symposium on Discrete Algorithms: SODA 1999, pp. 849–858.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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