Tight Thresholds for Cuckoo Hashing via XORSAT

(Extended Abstract)
  • Martin Dietzfelbinger
  • Andreas Goerdt
  • Michael Mitzenmacher
  • Andrea Montanari
  • Rasmus Pagh
  • Michael Rink
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)


We settle the question of tight thresholds for offline cuckoo hashing. The problem can be stated as follows: we have n keys to be hashed into m buckets each capable of holding a single key. Each key has k ≥ 3 (distinct) associated buckets chosen uniformly at random and independently of the choices of other keys. A hash table can be constructed successfully if each key can be placed into one of its buckets. We seek thresholds c k such that, as n goes to infinity, if n/m ≤ c for some c < c k then a hash table can be constructed successfully with high probability, and if n/m ≥ c for some c > c k a hash table cannot be constructed successfully with high probability. Here we are considering the offline version of the problem, where all keys and hash values are given, so the problem is equivalent to previous models of multiple-choice hashing. We find the thresholds for all values of k > 2 by showing that they are in fact the same as the previously known thresholds for the random k-XORSAT problem. We then extend these results to the setting where keys can have differing number of choices, and make a conjecture (based on experimental observations) that extends our result to cuckoo hash tables storing multiple keys in a bucket.


Hash Function Random Graph Hash Table Edge Density Tight Threshold 
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.
    Azar, Y., Broder, A., Karlin, A., Upfal, E.: Balanced allocations. SIAM J. Comput. 29(1), 180–200 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Batu, T., Berenbrink, P., Cooper, C.: Balanced allocations: Balls-into-bins revisited and chains-into-bins, CDAM Research Report Series. LSE-CDAM-2007-34Google Scholar
  3. 3.
    Cain, J.A., Sanders, P., Wormald, N.C.: The random graph threshold for k-orientiability and a fast algorithm for optimal multiple-choice allocation. In: Proc. 18th ACM-SIAM SODA, pp. 469–476 (2007)Google Scholar
  4. 4.
    Calkin, N.J.: Dependent sets of constant weight binary vectors. Combinatorics, Probability, and Computing 6(3), 263–271 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cooper, C.: The size of the cores of a random graph with a given degree sequence. Random Structures and Algorithms 25(4), 353–375 (2004)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Creignou, N., Daudé, H.: Smooth and sharp thresholds for random k-XOR-CNF satisfiability. Theoretical Informatics and Applications 37(2), 127–147 (2003)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Creignou, N., Daudé, H.: The SAT-UNSAT transition for random constraint satisfaction problems. Discrete Mathematics 309(8), 2085–2099 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dietzfelbinger, M., Goerdt, A., Mitzenmacher, M., Montanari, A., Pagh, R., Rink, M.: Tight Thresholds for Cuckoo Hashing via XORSAT. CoRR, abs/0912.0287 (2009)Google Scholar
  9. 9.
    Dietzfelbinger, M., Pagh, R.: Succinct data structures for retrieval and approximate membership. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 385–396. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Dubois, O., Mandler, J.: The 3-XORSAT threshold. In: Proc. 43rd FOCS, pp. 769–778 (2002)Google Scholar
  11. 11.
    Fernholz, D., Ramachandran, V.: The k-orientability thresholds for G n, p. In: Proc. 18th ACM-SIAM SODA, pp. 459–468 (2007)Google Scholar
  12. 12.
    Fotakis, D., Pagh, R., Sanders, P., Spirakis, P.: Space efficient hash tables with worst case constant access time. Theory Comput. Syst. 38(2), 229–248 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Fountoulakis, N., Panagiotou, K.: Orientability of random hypergaphs and the power of multiple choices. In: Gavoille, C. (ed.) ICALP 2010, Part I. LNCS, vol. 6198, pp. 348–359. Springer, Heidelberg (2010)Google Scholar
  14. 14.
    Fountoulakis, N., Panagiotou, K.: Sharp load thresholds for cuckoo hashing. CoRR, abs/0910.5147 (2009)Google Scholar
  15. 15.
    Frieze, A.M., Melsted, P.: Maximum matchings in random bipartite graphs and the space utilization of cuckoo hashtables. CoRR, abs/0910.5535 (2009)Google Scholar
  16. 16.
    Gao, P., Wormald, N.C.: Load balancing and orientability thresholds for random hypergraphs. In: 42nd ACM STOC (to appear, 2010)Google Scholar
  17. 17.
    Hopcroft, J.E., Karp, R.M.: An n 5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lehman, E., Panigrahy, R.: 3.5-way cuckoo hashing for the price of 2-and-a-bit. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 671–681. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  19. 19.
    Luby, M., Mitzenmacher, M., Shokrollahi, M.A., Spielman, D.: Efficient erasure correcting codes. IEEE Transactions on Information Theory 47(2), 569–584 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Méasson, C., Montanari, A., Urbanke, R.: Maxwell construction: the hidden bridge between iterative and maximum a posteriori decoding. IEEE Transactions on Information Theory 54(12), 5277–5307 (2008)CrossRefGoogle Scholar
  21. 21.
    Mézard, M., Ricci-Tersenghi, F., Zecchina, R.: Two solutions to diluted p-spin models and XORSAT problems. J. Statist. Phys. 111(3/4), 505–533 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Mézard, M., Montanari, A.: Information, Physics, and Computation. Oxford University Press, Oxford (2009)zbMATHCrossRefGoogle Scholar
  23. 23.
    Mitzenmacher, M.: Some open questions related to cuckoo hashing. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 1–10. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  24. 24.
    Molloy, M.: Cores in random hypergraphs and Boolean formulas. Random Structures and Algorithms 27(1), 124–135 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Pagh, R., Rodler, F.F.: Cuckoo hashing. J. Algorithms 51(2), 122–144 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Sanders, P.: Algorithms for scalable storage servers. In: Van Emde Boas, P., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2004. LNCS, vol. 2932, pp. 82–101. Springer, Heidelberg (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Martin Dietzfelbinger
    • 1
  • Andreas Goerdt
    • 2
  • Michael Mitzenmacher
    • 3
  • Andrea Montanari
    • 4
  • Rasmus Pagh
    • 5
  • Michael Rink
    • 1
  1. 1.Fakultät für Informatik und AutomatisierungTechnische Universität Ilmenau 
  2. 2.Fakultät für InformatikTechnische Universität Chemnitz 
  3. 3.School of Engineering and Applied SciencesHarvard University 
  4. 4.Electrical Engineering and Statistics DepartmentsStanford University 
  5. 5.Efficient Computation groupIT University of Copenhagen 

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