Linear Probing with 5-wise Independence

Abstract

Hashing with linear probing dates back to the 1950s and is among the most studied algorithms for storing (key, value) pairs. In recent years it has become one of the most important hash table organizations since it uses the cache of modern computers very well. Unfortunately, previous analyses rely either on complicated and space consuming hash functions, or on the unrealistic assumption of free access to a hash function with random and independent function values. Carter and Wegman, in their seminal paper on universal hashing, raised the question of extending their analysis to linear probing. However, we show in this paper that linear probing using a 2-wise independent hash function may have expected logarithmic cost per operation. Recently, Ptraşcu and Thorup have shown that 3- and 4-wise independent hash functions may also give rise to logarithmic expected query time. On the positive side, we show that 5-wise independence is enough to ensure constant expected time per operation. This resolves the question of finding a space and time efficient hash function that provably ensures good performance for hashing with linear probing.