Tight thresholds for cuckoo hashing via XORSAT

  • Authors:

  • Martin Dietzfelbinger;Andreas Goerdt;Michael Mitzenmacher;Andrea Montanari;Rasmus Pagh;Michael Rink

  • Affiliations:

  • Fakultät für Informatik und Automatisierung, Technische Universität Ilmenau;Fakultät für Informatik, Technische Universität Chemnitz;School of Engineering and Applied Sciences, Harvard University;Electrical Engineering and Statistics Departments, Stanford University;Efficient Computation group, IT University of Copenhagen;Fakultät für Informatik und Automatisierung, Technische Universität Ilmenau

  • Venue:
  • ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
  • Year:
  • 2010

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Abstract

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 ck such that, as n goes to infinity, if n/m ≤ c for some c ck then a hash table can be constructed successfully with high probability, and if n/m ≥ c for some c ck 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.