Insertion Time of Random Walk Cuckoo Hashing below the Peeling Threshold

Author Stefan Walzer

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Author Details

Stefan Walzer
  • Universität zu Köln, Germany


I would like to thank Martin Dietzfelbinger for providing several useful comments that helped with improving the presentation of this paper as well as an anonymous reviewer who gave useful feedback regarding the technical argument.

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Stefan Walzer. Insertion Time of Random Walk Cuckoo Hashing below the Peeling Threshold. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 87:1-87:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Most hash tables have an insertion time of 𝒪(1), often qualified as "expected" and/or "amortised". While insertions into cuckoo hash tables indeed seem to take 𝒪(1) expected time in practice, only polylogarithmic guarantees are proven in all but the simplest of practically relevant cases. Given the widespread use of cuckoo hashing to implement compact dictionaries and Bloom filter alternatives, closing this gap is an important open problem for theoreticians. In this paper, we show that random walk insertions into cuckoo hash tables take 𝒪(1) expected amortised time when any number k ≥ 3 of hash functions is used and the load factor is below the corresponding peeling threshold (e.g. ≈0.81 for k = 3). To our knowledge, this is the first meaningful guarantee for constant time insertion for cuckoo hashing that works for k ∈ {3,…,9}. In addition to being useful in its own right, we hope that our key-centred analysis method can be a stepping stone on the path to the true end goal: 𝒪(1) time insertions for all load factors below the load threshold (e.g. ≈0.91 for k = 3).

Subject Classification

ACM Subject Classification
  • Theory of computation → Bloom filters and hashing
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Random graphs
  • Cuckoo Hashing
  • Random Walk
  • Random Hypergraph
  • Peeling
  • Cores


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