Characterizing Open Addressing Hash Functions

In [1], we showed that different open addressing hash functions perform differently when the data elements are not uniformly distributed. So, it is tempting to attribute their difference to some mechanism governing the behavior of the hash functions. In this paper, a simple method of characterizing open addressing hash functions is presented. We showed that, indeed, the nature of data spreading ability characterizes the behavior of different open addressing hash functions. We measured and analyzed the spreading speed of a cluster of data elements under different open addressing hash functions. Our experimental results and theoretical analysis showed that different hash functions have different abilities of spreading out clustered data elements. The hash function, which spreads out clustered data elements over the whole table space more uniformly and faster, has better performance when it is applied to clustered data. Experimental results are presented to support our claims, which is followed by some theoretic analysis. 1 Hash function families In this paper we present hash functions in the form of nonlinear dynamical systems. So, first we derive dynamical system expressions for different hashing families. 1.1 Linear and quadratic hashing The family of linear hash functions can be expressed as HL(k, i) = (h(k) + c1i) mod m, (1) where h is an ordinary hash function, and i = 0, 1, . . . is the probe number. This technique is known as linear hashing because the argument of the modulus operator is linearly dependent on the probe number. In general, c1 needs to be chosen so that it is relatively prime to m if all slots in the hash table are to be examined by the probe sequence. In order to construct an equivalent transformation for equation (1), it must be rewritten as a recurrence relation so that it has explicit dependence on previous values of the iteration sequence. Since we have HL(k, i + 1) = (h(k) + c1i + c1) mod m, and using the fact that for a, b,m ∈ IR, (a + b) mod m = (a mod m + b mod m) mod m, (2) we may rewrite equation (1) as the following linear time-invariant first-order iterator HL(k, i + 1) = (HL(k, i) + (c1 mod m)) mod m HL(k, 0) = h(k). Note that the dependence on k is specified in the initial condition H(k, 0). Quadratic hashing is a simple extension of linear hashing that makes the probe sequence nonlinearly dependent on the probe number. For any ordinary hash function h, the family of quadratic hash functions is given by HQ(k, i) = (h(k) + c1i + c2i) mod m (3) where c1 and c2 are positive constants. Once again, the specific values chosen for the constants are critical to the performance of this method (see [2] for details). To obtain a recurrence relation solution to equation (3) we note that HQ(k, i + 1) = [(h(k) + c1i + c2i) +c1 + c2(2i + 1)] mod m which leads to the time-varying first-order recurrence relation HQ(k, i + 1) = (HQ(k, i) + c1 + c2(2i + 1)) mod m HQ(k, 0) = h(k). (4) 1.2 Double hashing family HD It is well known that linear and quadratic hashing strategies suffer from clustering, because both have probe sequence increments that are independent of the key. Double hashing remedies this problem by introducing a second hash function that is used in the computation of the increment. Given two hash functions g and h, the family HD of linear double hash functions is given by: HD(k, i) = (g(k) + ih(k)) mod m. (5) In this family, the initial probe HD(k, 0) = g(k), and successive probes are offset from previous probes by multiples of h(k) modulo m. Thus the probe sequence depends on k through both g and h, and is linear in g(k) and h(k). A widely used member of HD, proposed by Knuth [1973] , has g(k) = k mod m and h(k) = k mod (m− 2), where both m and m− 2 are prime. A recurrence relation for the family HD described by equation (5) is obtained by first noting that HD(k, i + 1) = [(g(k) + ih(k)) + h(k)] mod m,