Load Thresholds for Cuckoo Hashing with Double Hashing

Authors Michael Mitzenmacher , Konstantinos Panagiotou, Stefan Walzer

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Michael Mitzenmacher
  • Harvard University, School of Engineering and Applied Sciences, Cambridge, USA
Konstantinos Panagiotou
  • University of Munich, Institute for Mathematics, Germany
Stefan Walzer
  • Technische Universität Ilmenau, Germany

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Michael Mitzenmacher, Konstantinos Panagiotou, and Stefan Walzer. Load Thresholds for Cuckoo Hashing with Double Hashing. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 29:1-29:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


In k-ary cuckoo hashing, each of cn objects is associated with k random buckets in a hash table of size n. An l-orientation is an assignment of objects to associated buckets such that each bucket receives at most l objects. Several works have determined load thresholds c^* = c^*(k,l) for k-ary cuckoo hashing; that is, for c < c^* an l-orientation exists with high probability, and for c > c^* no l-orientation exists with high probability. A natural variant of k-ary cuckoo hashing utilizes double hashing, where, when the buckets are numbered 0,1,...,n-1, the k choices of random buckets form an arithmetic progression modulo n. Double hashing simplifies implementation and requires less randomness, and it has been shown that double hashing has the same behavior as fully random hashing in several other data structures that similarly use multiple hashes for each object. Interestingly, previous work has come close to but has not fully shown that the load threshold for k-ary cuckoo hashing is the same when using double hashing as when using fully random hashing. Specifically, previous work has shown that the thresholds for both settings coincide, except that for double hashing it was possible that o(n) objects would have been left unplaced. Here we close this open question by showing the thresholds are indeed the same, by providing a combinatorial argument that reconciles this stubborn difference.

Subject Classification

ACM Subject Classification
  • Theory of computation → Bloom filters and hashing
  • Cuckoo Hashing
  • Double Hashing
  • Load Thresholds
  • Hypergraph Orientability


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