Abstract: Universal hash functions that exhibit (c log n)-wise independence are shown to give a performance in double hashing, uniform hashing and virtually anyreasonable generalization of double hashing that has an expected probe count of 1/(1-alpha)+O(1/n) for the insertion of the (alpha n)-th item into a table of size n, for any fixed alpha < 1. This performance is optimal. These results are derived from a novel formulation that overestimates the expected probe count by underestimating the presence of local items already inserted into the hash table, and from a very sharp analysis of the underlying stochasticstructures formed by colliding items.

Analogous bounds are attained for the expected r-th moment of the probe count, or any fixed r, and linear probing is also shown to achieve a performance with universal hash functions that is equivalent to the fully random case.