Several Remarks on Index Generation Functions
Dan A. Simovici Marius Zimand
Univ. of Massachusetts Towson University
Boston Dept. of Comp. and
Dept. of Comp. Science Information Sciences
Boston, Massachusetts Towson, Maryland
dsim@cs.umb.edu mzimand@towson.edu
Dan Pletea
Univ. of Massachusetts Boston
Dept. of Comp. Science
Boston, Massachusetts
dpletea@cs.umb.edu
Abstract—We propose a probabilistic greedy algorithm for
decomposing partially specified index generation functions.
These functions have numerous applications in a variety of
circuit design problems. We show that finding an optimal
decomposition is an intractable problem, which motivates our
approach.
Keywords-index generation function; NP-complete; proba-
bilistic algorithm;
I. I NTRODUCTION
Partially specified index generation functions (PSIGFs)
play an important role in circuit design due to their multiple
applications. As pointed by T. Sasao [4], [5], [6], PSIGFs
play are used in implementing address tables for routers, in
building terminal access controllers for local area networks
and they have other applications in memory patch circuits,
electronic dictionaries, password lists, etc.
A partial function f between the sets A and B is denoted
as f : A ❀ B. Its definition domain, which is a subset of
A is denoted as Dom
f
.
A PSIGF is an injective partial function whose definition
domain Dom(f ) is a subset of {0, 1}
k
and whose set of
values is {1,...,n}, where n = | Dom(f )|. The equality
n = | Dom(f )| together with the injectivity of f means that
f is a bijection between Dom(f ) and {1,...,n}. The mem-
bers of Dom(f ), written as row vectors in {0, 1}
k
are re-
ferred to as registered vectors. Our purpose is to formulate an
algorithm that allows us to replace the variables x
1
,...,x
k
with a smaller number ℓ of linear combinations of these
variables y
1
,...,y
ℓ
(over GF(2)) defined by linear functions
of the form y
i
= g
i
(x
1
,...,x
n
)= x
ij1
⊕···⊕ x
ijp
i
such
that we can express (for some suitable F )
f (x
1
,...,x
k
)= F (g
1
(x
1
,...,x
k
),...,g
ℓ
(x
1
,...,x
k
))
for all (x
1
,...,x
k
) ∈ Dom(f ). We mention that linear
techniques have been used frequently in circuit design (see,
for example [3]).
As we shall see in Section III, this is an intractable
problem. We propose a greedy algorithm that achieves a
satisfactory solution.
II. BOOLEAN MATRICES AND COLLECTIONS OF SETS
Let GF(2) = ({0, 1}, +, ·, 0, 1) be the two-element Galois
field, where the addition is the exclusive-or operation defined
by
a + b =
0 if a = b,
1 otherwise.
It is common to also write a⊕b instead of a+b. Vectors from
GF(2)
p
will be denoted by bold letters; a vector that has
only one component equal to 1, located in the i
th
place will
be denoted by e
i
. The transpose of a matrix A is denoted
by A
′
. The components of a vector x ∈ GF(2)
p
will be
denoted by x
1
,...,x
p
. The one-column vector in GF(2)
k
whose components are equal to 1 is denoted by 1
k
; the one-
column vector in GF(2)
k
whose components are equal to 0
is denoted by 0
k
.
The set of n × k-matrices over GF(2) is denoted by
GF(2)
n×k
. The standard product of two matrices P ∈
GF(2)
m×n
and Q ∈ GF(2)
n×p
, where the elements of B
are regarded as the real numbers 0 and 1 will be denoted by
AB. The inner product of vectors x ∈ GF(2)
n
, y ∈ GF(2)
n
,
is denoted xy.
Suppose that Dom(f )= {m
1
,..., m
n
}, where f is a
PSIGF. The problem that we are discussing can be formu-
lated in matrix terms as follows. Given an n × k-matrix over
GF(2),
M
f
=
⎛
⎜
⎝
m
1
.
.
.
m
n
⎞
⎟
⎠
we look for a minimum number ℓ and a matrix A =
(a
1
··· a
ℓ
) ∈ GF(2)
k×ℓ
such that for all i = j we have
2012 IEEE 42nd International Symposium on Multiple-Valued Logic
0195-623X/12 $26.00 © 2012 IEEE
DOI 10.1109/ISMVL.2012.17
179