Andrew C. Yao
Abstract
It was conjectured by J. Ullman that uniform hashing is optimal in its expected retrieval cost among all open-address hashing schemes [4]. In this paper, we show that, for any open-address hashing scheme, the expected cost of retrieving a record from a large table that is α-fraction full is at least (1/α) log (1/(1 - α)) + o(1). This proves Ullman's conjecture to be true in the asymptotic sense.
- 1 AJTAI, M., KOMLOS, J., AND SZEMEREDI, E.There is no fast single hashing function. Inf. Proc. Lett. 7 (1978), 270-273.Google Scholar
- 2 KNUTH, D.E.Computer science and its relation to mathematics. Am. Math. Monthly 8 (1974), 323-343.Google Scholar
- 3 KNUTH, D.E.The Art of Computer Programming. Vol. 3, Sorting and Searching. Addison- Welsley, Reading, Mass., 1975 (2nd printing). Google Scholar
- 4 ULLMAN, J. D. A note on the efficiency of hashing functions, at. ACM 19, 3 (July 1972), 569-575. Google Scholar
Index Terms
- Uniform hashing is optimal
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Reviews
Alfs T. Berztiss
Open addressing is a method for constructing hash tables. Under this method, the key of a record maps to a sequence that is a permutation of all the locations of a hash table. A record is inserted in the first free location found by following the sequence. In the uniform hashing variant of open addressing, the permutations are random. Yao proves that the average cost of retrieving a record from any open-addressing hash table is no smaller than this cost under uniform hashing. The question of whether uniform hashing is also the optimal open addressing method for insertion remains open.
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