Abstract not found

Hashing with linear probing dates back to the 1950s, and is among the most studied algorithms. 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 truly random hash function. Already 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 pairwise independent family may have expected logarithmic cost per operation. 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 linear probing.

Universal hash functions that exhibit c log n-wise independence are shown to give a performance in double hashing and virtually any reasonable generalization of double hashing that has an expected probe count of 1/(1-alpha) + epsilon for the insertion of the alpha n-th item into a table of size n, for any fixed alpha 0. This performance is within epsilon of optimal. These results are derived from a novel formulation that overestimates the expected probe count by underestimating the presence of partial items already inserted into the hash table, and from a sharp analysis of the underlying stochastic structures formed by colliding items.

Linear probing is one of the most popular implementations of dynamic hash tables storing all keys in a single array. When we get a key, we first hash it to a location. Next we probe consecutive locations until the key or an empty location is found. At STOC’07, Pagh et al. presented data sets where the standard implementation of 2-universal hashing leads to an expected number of Ω(log n) probes. They also showed that with 5-universal hashing, the expected number of probes is constant. Unfortunately, we do not have 5-universal hashing for, say, variable length strings. When we want to do such complex hashing from a complex domain, the generic standard solution is that we first do collision free hashing (w.h.p.) into a simpler intermediate domain, and second do the complicated hash function on this intermediate domain. Our contribution is that for an expected constant number of linear probes, it is suffices that each key has O(1) expected collisions with the first hash function, as long as the second hash function is 5-universal. This means that the intermediate domain can be n times smaller, and such a smaller intermediate domain typically means that the overall hash function can be made simpler and at least twice as fast. The same doubling of hashing speed for O(1) expected probes follows for most domains bigger than 32-bit integers, e.g., 64-bit integers and fixed length strings. In addition, we study how the overhead from linear probing diminishes as the array gets larger, and what happens if strings are stored directly as intervals of the array. These cases were not considered by Pagh et al. 1

)-independent hash functions are required, matching an upper bound of [Indyk, SODA’99]. We also show that the multiply-shift scheme of Dietzfelbinger, most commonly used in practice, fails badly in both applications. Abstract. We show that linear probing requires 5-independent hash functions for expected constant-time performance, matching an upper bound of [Pagh et al. STOC’07]. For (1 + ε)-approximate minwise independence, we show that Ω(lg 1 ε 1

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, Pǎtra¸scu and Thorup have shown that also 3- and 4-wise independent hash functions may 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.