This paper demonstrates how a broad collection of hash function families can be expressed as dynamical systems. We then show that this representation can be useful for analysis. In particular, we provide an analysis which proves that a widely-used family of double hash functions will transform any initial distribution of keys into the uniform distribution over the table space

Abdallah, Chaouki T.

Heileman, Gregory L.

Moret, Bernard M.

Smith, Bradley J.

English

University of New Mexico

University of New Mexico
UNM Digital Repository
Electrical & Computer Engineering Faculty
Publications Engineering Publications
1-1-1999
Dynamical system representation of open address
hash functions
Chaouki T. Abdallah
Gregory L. Heileman
Bernard M. Moret
Bradley J. Smith
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Recommended Citation
Abdallah, Chaouki T.; Gregory L. Heileman; Bernard M. Moret; and Bradley J. Smith. "Dynamical system representation of open
address hash functions." Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms (1999): 919-920.
https://digitalrepository.unm.edu/ece_fsp/188
5919 
Dynamical System Representation of Open Address ,Hash Functions 
Gregory L. Heileman* Chaouki T. Abdallah’ Bernard M. E. Morett Bradley J. Smith* 
Abstract 
This paper demonstrates how a broad collection of hash 
function families can be expressed as dynamical systems. We 
then show that this representation can be useful for analysis. 
In particular, we provide an analysis which proves that a 
widely-used family of double hash functions will transform 
any initial distribution of keys into the uniform distribution 
over the table space. 
1 Introduction. 
Number theory and probabilistic analyses have typically 
been used to justify the use of various probing strategies 
for open address hashing. For example, Guibas and Sze- 
meredi [l], and later Lueker and Molodowitch [3], have 
shown that double hashing offers a good approximation 
to uniform hashing if the hash function is assumed to 
select table entries with equal probability. In this paper 
we use an analysis based on nonlinear dynamical sys- 
tems theory to provide a similar result, without making 
any probabilistic assumptions about the data distribu- 
tion. The key to our approach is the ability to express 
hash functions as dynamical systems. 
We consider four important families of hash func- 
tions: linear probing, quadratic hashing, linear double 
hashing, and exponential double hashing. We assume 
that each of these hashing strategies implements the 
Find(k), Insert(z), and Delete(k) operations, where k 
is a key drawn from a totally ordered universe U, and 
x is a data element that has a key associated with it. 
We assume that these operations are implemented on 
a table of size m, and that a table index is computed 
from the key using a ordinary hash function h, that per- 
forms the mapping h : U -+ Z,, where Z, denotes the 
integers modulo m. An open address hash function H 
makes use of one or more ordinary hash functions to 
successively probe the table space in the event of colli- 
sions. 
2 Hashing Functions as Dynamical Systems. 
The family of linear hash functions can be expressed as 
pa.rtment of Electrical and Computer Engineering, 
+Department of Computer Science, University of New Mexico, 
Albuquerque, NM 87131, USA. 
(2.1) He, = (h(k) + cm) mod m, 
where h is an ordinary hash function, c is a constant, 
and n = 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 order to construct an equivalent iterator 
for (2-l), it must be rewritten as a recurrence relation 
so that it has explicit dependence on previous values of 
the iteration sequence. Since 
&+I = (h(k) + cn + c) mod m, 
and using the fact that for a, b, m E R, 
(a + b) mod m = (a mod m + b mod m) mod m, 
we may rewrite (2.1) as the following linear time- 
invariant first-order iterator 
HL+1 = (Hc, + c) mod m 
HE0 = h(k). 
An equivalent description in one-dimensional state 
space is given by 
xn+l = (x, + c) mod m, 
where 20 = h(k), and Hr, = xn is the output equation 
used to iterate over the table space. Note that the 
dependence on k is specified in the initial condition. 
Quadratic hashing is an extension of linear probing 
that makes the probe sequence nonlinearly dependent 
on the probe number n. For any ordinary hash function 
h, the family of quadratic hash functions is given by 
HQ, = (h(k) + cln + cgz*) mod m, 
where cl and c2 are constants. Using the technique 
applied previously, we obtain the following iterator 
Hem+1 = (HQ, + cl + c2(2n + 1)) mod m, 
HQO = h(k). 
This can be rewritten as the following set of coupled 
linear (modulo m) time-invariant first-order iterators in 
twc4imensiona.l state space 
GLfl = (cc, + 1) mod m, 
Yn+l = (yn + 2~22~ + c2 + cl) mod m, 
5920 
where ze 
equation 
= 0, Ye = h(k), and HQ, = YT( is the output 
used to iterate over the table space. 
Given two ordinary hash functions g and h, the 
family HL~ of linear double hash functions is given by 
P-2) H.CP, = (g(k) + nh(k)) mod m, 
where the initial probe H&D, = g(k). Thus the probe 
sequence depends on k through both g and h, and is 
linear in g(k) and h(k). It can be shown that (2.2) is 
equivalent to the following iterator 
HW&+1 = (HLD, + h(k)) mod m, 
HO, = g(k), 
which can be rewritten as a set of coupled linear (mod- 
ulo m) time-invariant first-order difference equations in 
two-dimensional state space: 
(2.3) G&+1 = xn, 
(2.4) Yn+1 = (xn + yn) mod m, 
where xo = h(k), yo = g(k), and HCD, = yn is the 
output equation used to iterate over the table space. A 
widely used member of Htv, proposed by Knuth [2], 
has g(k) = k mod m and h(k) = k mod (m - 2), where 
both m and m - 2 are prime. 
We propose a new family of double hash functions 
HE; which we call exponential double hashing, that uses 
two ordinary hash functions g and h to compute a probe 
sequence according to 
(2.5) HE, = (g(k) + h(k)n) mod m. 
The exponentiation does not have to be explicitly im- 
plemented, but can be computed via successive multi- 
plications during the probing process. For this reason, 
the number of mathematical operations needed to im- 
plement (2.5) is identical to the number needed to im- 
plement (2.2). To obtain an iterator for the family HE 
described in (2.5)) we write 
H&.+1 = [h(k)h(k)” + g(k)] mod m 
= [((WY + g(k)) mod m . h(k) mod m) 
modm + ((1 - h(L)) g(k)) mod m] mod m, 
and obtain 
H-L+, = (%,h(k) + Cl- h(k))&)) mod m, 
H&O = g(k) + 1. 
This is a first-order time-varying nonlinear system. An 
equivalent system of first-order time-invariant iterators 
can be obtained by using a three-dimensional state 
space: 
G&+1 = xn, 
Yn+l = Yn, 
%+1 = (YA + xn Cl- Y,J) mod m, 
where x0 
H-c,, 
= g(k), YO = h(k), 
= z, is the output equation 
20 = x0 + 1, and 
used to iterate over 
the table space. A particular member of HE, recently 
introduced by Smith et al. [4], has g(k) = k mod m, and 
h(k) = k mod (m - 2), where m = 2p f 1 is selected so 
that both m and p are prime. Smith et. al [4] showed 
experimentally that on average this member of HE tends 
to outperform any member of H&v. We suggest here 
that this may be due to the fact that HE hash functions 
operates in a more complicated state space than H&D 
hash functions. 
3 Analysis. 
We now consider an analysis that makes use of the 
dynamical system representation of Hr=. This analysis 
demonstrates the potential usefulness of representing 
hash functions as dynamical systems. Given (2.3) 
and (2.4), the output equation can be rewritten in 
terms of the initial probes as Yn = ye + nzc (mod m). 
Let us denote the joint density function for the initial 
prob= us% fzo,yo, and without loss of generality (by 
appropriate scaling) we may assume m = 2~. In this 
case the characteristic function of the output equation 
is given by 
J 
2x 
= eit(v+nu)fzO,YO (u, V) du &I. 
0 
By the Riemann-Lebesgue Lemma, it follows that for 
t # 0, (am, -+ 0 weakly as 72 + 00. Thus, 
since fro ,yo is a density function, for n sufficiently 
large ‘py, (t) = 1 when k = 0, and (py, (t) = 0 when 
k # 0, which is in fact the characteristic function 
for the uniform distribution on the circle. Since the 
characteristic function uniquely determines the density 
function for a given random variable, this demonstrates 
that for appropriately chosen m, and after a sufficient 
number of probes, hash functions in the linear double 
hashing family will transform any initial density into 
the uniform distribution over the table space. 0 
References 
PI 
PI 
[31 
[41 
L. Guibas and E. Szemerecii, The analysis of double 
ha.&ing, J. Comput. and Sys. Sci., 16 (1978), pp. 226- 
274. 
D. E: Knuth, The Art of Computer Programming, vol- 
ume 3, Searching and Sorting. Addison-Wesley, Read- 
ing, MA, 1973. 
G. Lueker and M. Molodotitch, More analysis of 
double hcrshing, in ACM STOC, 20 (1988), pp. 354- 
359. 
B. 3. Smith, G. L. Heileman, and C. T. Abdallah, The 
ezponential hash function, ACM 3. Exper. Alg., 2(3) 
(1997), wu.jea.acm.org/1997/SmitbErp0nential/. 


Dynamical system representation of open address hash functions

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Dynamical system representation of open address hash functions

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