Data structures and algorithms 3: multi-dimensional searching and computational geometry
Data structures and algorithms 3: multi-dimensional searching and computational geometry
Unbiased bits from sources of weak randomness and probabilistic communication complexity
SIAM Journal on Computing - Special issue on cryptography
A reliable randomized algorithm for the closest-pair problem
Journal of Algorithms
On the cell probe complexity of membership and perfect hashing
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Cuckoo hashing: further analysis
Information Processing Letters
Journal of Algorithms
Linear probing with constant independence
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Why simple hash functions work: exploiting the entropy in a data stream
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Weaknesses of Cuckoo Hashing with a Simple Universal Hash Class: The Case of Large Universes
SOFSEM '09 Proceedings of the 35th Conference on Current Trends in Theory and Practice of Computer Science
Orientability of random hypergraphs and the power of multiple choices
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
The Power of Simple Tabulation Hashing
Journal of the ACM (JACM)
Sharp load thresholds for cuckoo hashing
Random Structures & Algorithms
Explicit and efficient hash families suffice for cuckoo hashing with a stash
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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Cuckoo hashing, introduced by Pagh and Rodler [10], is a dynamic dictionary data structure for storing a set S of n keys from a universe U, with constant lookup time and amortized expected constant insertion time. For the analysis, space (2+ε)n and Ω(log n)-wise independence of the hash functions is sufficient. In experiments mentioned in [10], several weaker hash classes worked well; however, a certain simple multiplicative hash family worked badly. In this paper, we prove that the failure probability is high when cuckoo hashing is run with the multiplicative class or with the very common class of linear hash functions over a prime field, even if space 4n is provided. The key set S is fully random, but it must be relatively dense in the universe U of all keys (like |S| ≥ |U|11/12). The bad behavior and the fact that this effect depends on the density of S in U can also be observed in experiments. The result transfers to larger universes if the keys are chosen from a suitable smaller domain. Viewed from a different perspective, our result illustrates that care must be taken when applying a recent result of Mitzenmacher and Vadhan ([12], SODA 2008) proving good behavior of universal hash classes in combination with key sets that have some entropy. Their result is applicable to cuckoo hashing. A technical hypothesis in [12], namely the assumption that either the "collision probability" or the "maximum probability" is small, translates into the condition that |S| is relatively small in comparison to |U|. Our result shows that the result from [12] on 2-universal classes ceases to hold if |S|/|U| is not small enough, even for very common 2-universal hash classes and fully random key sets.