Journal ArticleDOI

# Cayley Digraphs of Finite Abelian Groups and Monomial Ideals

01 Jul 2007-SIAM Journal on Discrete Mathematics (Society for Industrial and Applied Mathematics)-Vol. 21, Iss: 3, pp 763-784

TL;DR: The description of the diagrams known as L-shapes provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs and presents a family of Cayley digraphs parametrized by their diameter, all of them associated to irreducible monomial ideals.

AbstractIn the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph's node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with a finite abelian group as vertex set, we define a monomial ideal and consider its representations via its minimal system of generators or its irredundant irreducible decomposition. From this last piece of information, we can compute the graph's diameter and average minimum distance. That monomial ideal is the initial ideal of a certain lattice with respect to a graded monomial ordering. This result permits the use of Grobner bases for computing the ideal and finding an optimal routing. Finally, we present a family of Cayley digraphs parametrized by their diameter $d$, all of them associated to irreducible monomial ideals.

Topics: Monomial ideal (70%), Monomial (68%), Vertex (graph theory) (55%), Abelian group (54%)

### 1. Introduction.

• In section 2 the authors collect several known facts about monomial ideals, presenting examples and fixing notation for later use.
• Section 6 presents an algorithm specifically tailored for degree three circulants.
• Section 7 is dedicated to providing formulae to find the diameter and the average minimum distance.

### 9. Conclusions.

• In this paper the authors have proposed monomial ideals as a natural tool for studying Cayley digraphs with a finite abelian group as vertex set.
• The authors have generalized the L-shape concept in the plane to L-shape in the r-dimensional affine space.
• From a more practical point of view, it would be interesting to investigate the implementation in computer networks of the family of circulant graphs of degree three under parameters such as routing, fault tolerance, etc. Downloaded 02/05/13 to 193.144.185.28.

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SIAM J. DISCRETE MATH.
c
2007 Society for Industrial and Applied Mathematics
Vol. 21, No. 3, pp. 763–784
CAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS AND
MONOMIAL IDEALS
DOMINGO G
´
OMEZ
, JAIME GUTIERREZ
, AND
´
ALVAR IBEAS
Abstract. In the study of double-loop computer networks, the diagrams known as L-shapes arise
as a graphical representation of an optimal routing for every graph’s node. The description of these
diagrams provides an eﬃcient method for computing the diameter and the average minimum distance
of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each
Cayley digraph with a ﬁnite abelian group as vertex set, we deﬁne a monomial ideal and consider
its representations via its minimal system of generators or its irredundant irreducible decomposition.
From this last piece of information, we can compute the graph’s diameter and average minimum
distance. That monomial ideal is the initial ideal of a certain lattice with respect to a graded
monomial ordering. This result permits the use of Gr¨obner bases for computing the ideal and ﬁnding
an optimal routing. Finally, we present a family of Cayley digraphs parametrized by their diameter d,
all of them associated to irreducible monomial ideals.
Key words. monomial ideals, Cayley digraph, Gr¨obner bases, multiloop networks
AMS subject classiﬁcations. 13P10, 05C25, 68M10
DOI. 10.1137/050646056
1. Introduction. Let Γ be a group and S Γ a subset. The Cayley digraph
associated to ,S) is a directed graph whose vertex set is Γ and whose edge set is
{(g, h) Γ
2
| g
1
h S}. Every Cayley digraph is vertex-symmetric and its degree
equals the number of elements in S. These graphs are connected if and only if the
set S generates the group. We are dealing with digraphs associated to ﬁnite abelian
groups, but we are mainly interested in those associated to cyclic groups. Let N be
a positive integer and Z
N
the integers modulo N. For any subset S = {j
1
,...,j
r
} of
this abelian group we denote by C
N
(S)=C
N
(j
1
,...,j
r
) the corresponding Cayley
digraph (see Figure 1.1), which is called the circulant digraph or multiloop computer
network of jumps j
1
,...,j
r
. It is connected if and only if gcd(j
1
,...,j
r
,N)=1.
If S is a subset of Z
N
such that for every element in S its inverse also lies in S,
then C
N
(S) is an undirected graph called a circulant graph or distributed multiloop
computer network.
Multiloop networks were ﬁrst proposed in [32] for organizing multimodule memory
services and have a vast number of applications in telecommunication networking,
VLSI design, and distributed computation. Their properties, such as diameter and
reliability, have been the focus of much research in computer network design; see, for
instance, [5, 7, 12, 13, 19, 21, 25, 33].
The single-loop network or ring network is mathematically trivial. Digraphs with
r =2ordouble-loop networks and their corresponding undirected graphs (distributed
double-loop networks, with degree four) have been extensively studied; see the sur-
veys [3, 20] and the references therein. When C
N
(j
1
,j
2
) is connected, one can deﬁne a
minimum distance diagram (MDD) as an array with vertex 0 in cell (0, 0) and vertex c
Received by the editors November 25, 2005; accepted for publication (in revised form) March
22, 2007; published electronically September 26, 2007. This research was partially supported by the
Research Project MTM2004-07086 of the Spanish Ministry of Science. An extended abstract of part
of this work appeared in [14].
http://www.siam.org/journals/sidma/21-3/64605.html
University of Cantabria, E–39071 Santander, Spain (domingo.gomez@unican.es, jaime.gutierrez
@unican.es, alvar.ibeas@unican.es).
763

764 DOMINGO G
´
OMEZ, JAIME GUTIERREZ, AND
´
ALVAR IBEAS
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Fig. 1.1. C
18
(3, 8).
0 5
14
10
19
28
15
24
9
20
29
23
25
1
4
30
6
18
2
11
32
7
16
13
12
21
27
17
26
8
22
31 3
Fig. 1.2. MDD of C
33
(5, 14).
in cell (x, y)(x is the column index and y the row index), for a particular choice satis-
fying j
1
x + j
2
y c mod N, and x + y minimum. One example is shown in Figure 1.2.
The classical work of Wong and Coppersmith [32] presents an algorithm for con-
structing an MDD of C
N
(j
1
,j
2
)inO(N
2
) steps and shows it has an “L” shape.
Several characterizations and applications of this idea for describing circulants with
desirable properties appear in [1, 8, 9, 13]. However, they do not focus on higher
degree digraphs.
Two notable parameters in a graph are the diameter d and the average minimum
distance
¯
d. The former represents the worst delay in the communication between two
nodes, and the latter represents the average delay. Given an L-shape, it is easy to
compute d and
¯
d.
On the other hand, let d
r
(N):=min{d(C
N
(j
1
,...,j
r
) | j
1
,...,j
r
Z
N
}. An im-
portant problem is to determine this value and ﬁnd a speciﬁc C
N
(j
1
,...,j
r
) attaining
this minimum. The network C
N
(j
1
,...,j
r
) is said to be optimal if its diameter equals
d
r
(N). In some cases, it is diﬃcult to obtain optimal networks; however, one can ﬁnd
general simple functions serving as upper and lower bounds for d
r
(N); see [3]. The
paper [32] shows d
2
(N)
3N 2 and presents a family of circulant digraphs with
diameter 2
N 2.
In this article we present monomial ideals as a natural tool for studying the
MDDs of arbitrary Cayley digraphs, provided that the vertex group is ﬁnite and

CAYLEY DIGRAPHS AND MONOMIAL IDEALS 765
abelian. Given a graded monomial ordering and a Cayley digraph ,S), we build a
monomial ideal in the polynomial ring K[X
1
,...,X
r
], where K is an arbitrary ﬁeld
and r =#S. We obtain some properties of this monomial ideal: in particular, a
certain generalization of the two-dimensional L-shape is shown. On the other side, it
is the initial ideal of a certain lattice. This result permits the use of Gr¨obner bases
for computing the ideal and ﬁnding an optimal routing for each pair of nodes. Given
the representation of the monomial ideal via its irreducible decomposition, we provide
formulae to compute d and
¯
d. We also show a family of circulant digraphs of degree
two which coincides with the family obtained in paper [32]. Finally, we present a
new and attractive family of circulant digraphs of arbitrary degree parametrized by
the diameter d, with average minimum distance d/2, and whose associated monomial
ideals are irreducible.
The paper is divided into nine sections. In section 2 we collect several known
facts about monomial ideals, presenting examples and ﬁxing notation for later use.
Section 3 presents the key idea of associating monomial ideals to digraphs in order to
obtain an MDD, and it also provides an algorithm to construct an MDD for Cayley
digraphs with a ﬁnite abelian group as vertex set. Section 4 is devoted to present-
ing the relation between MDDs and the ideal of a lattice. In section 5 we present
an algorithm to compute a shortest path between two vertices by means of Gr¨obner
bases. Section 6 presents an algorithm speciﬁcally tailored for degree three circulants.
It computes the minimal system of generators in O(s log N) arithmetic operations,
where s is the number of generators and N is the number of nodes. Section 7 is dedi-
cated to providing formulae to ﬁnd the diameter and the average minimum distance.
Then section 8 presents a family of multiloop computer networks with an arbitrary
number of jumps, parametrized by the diameter d, and all of them associated to ir-
reducible monomial ideals. We conclude with a short summary and a discussion of
open questions.
2. Monomial ideals. Monomial ideals form an important link between com-
mutative algebra and combinatorics. Here we review several basic related results and
deﬁnitions concerning monomial ideals; see, for instance, [2, 30].
Let K be an arbitrary ﬁeld and K[X
1
,...,X
r
] the polynomial ring in the variables
X
1
,...,X
r
. Throughout the paper, we very often identify monomials of K[X
1
,...,X
r
]
with vectors of N
r
and use the following notation:
x
a
= X
a
1
1
···X
a
r
r
←→ a =(a
1
,...,a
r
),
x
a
|x
b
⇐⇒ a =(a
1
,...,a
r
) b =(b
1
,...,b
r
)
def
⇐⇒ a
i
b
i
i =1,...,r,
a =(a
1
,...,a
r
) b =(b
1
,...,b
r
)
def
⇐⇒ (b
i
> 0 a
i
<b
i
) ,
e
i
:= (0,...,
i
1
,...,0), m
a
:= (X
a
i
i
| a
i
> 0), 1 := (1,...,1).
The deﬁnition of suits the characterization in (2.2), and when it is employed
(in expressions like a b), we usually have 1 b.
A monomial ideal is an ideal generated by monomials, i.e., I K[X
1
,...,X
r
]is
a monomial ideal if there is a subset A N
r
such that
I =(x
a
| a A)=(A).

766 DOMINGO G
´
OMEZ, JAIME GUTIERREZ, AND
´
ALVAR IBEAS
y
3
x
2
y
3
x
2
y
2
x
4
y
2
x
4
Fig. 2.1. Staircase diagram and Buchberger’s graph.
There are two standard ways of describing a nontrivial monomial ideal:
Via the (unique) minimal system of monomial generators I =(x
a
1
,...,x
a
s
),
we have
(2.1) x
u
I ⇐⇒ i ∈{1,...,s}|a
i
u.
Via the (unique) irredundant decomposition by irreducible monomial ideals
I = m
b
1
···m
b
n
,wehave
(2.2) x
u
∈ I ⇐⇒ i ∈{1,...,n}|u b
i
.
The so-called staircase diagram is a useful graphical representation of monomial
ideals.
Example 2.1. The monomial ideal I
1
:= (x
4
,x
2
y
2
,y
3
)=(x
2
,y
3
) (x
4
,y
2
)is
represented on the left in Figure 2.1.
There is an algorithm for ﬁnding the irredundant irreducible decomposition of a
monomial ideal based on Alexander duality; see [27]. An irreducible component m
a
can be associated to lcm(X
a
1
1
,...,X
a
r
r
)=x
a
. On the other hand, if K[X
1
,...,X
r
]/I
is an artinian ring, then the monomial x
a
associated to the irreducible component m
a
must coincide with the least common multiple of a subset of the minimal generators
of I. In the above Example 2.1 we have
x
2
y
3
= lcm(x
2
y
2
,y
3
),x
4
y
2
= lcm(x
4
,x
2
y
2
).
The diagram on the right in Figure 2.1 is called Buchberger’s graph of the mono-
mial ideal I
1
; see [28]. At any stage in Buchberger’s algorithm for computing Gr¨obner
bases, one considers the S-pairs among the current polynomials and removes those
which are redundant; the minimal S-pairs deﬁne a graph on the generators of any
monomial ideal.
Theorem 2.2. Let I be a nontrivial monomial ideal given by a minimal system
of generators I =(x
a
1
,...,x
a
s
) and by the irredundant irreducible decomposition
I = m
b
1
···m
b
n
. The following are equivalent:
1. K[X
1
,...,X
r
]/I is an artinian ring.
2. i =1,...,r, one of the generators’ exponents is a
j
= α
i
e
i
for some α
i
N.
3. i =1,...,n, j =1,...,r, b
i,j
> 0.
Proof. We need to prove that the number of monomials outside I is ﬁnite if and
only if either of the two last items is satisﬁed. We do that using the characterizations
in (2.1) and (2.2).

CAYLEY DIGRAPHS AND MONOMIAL IDEALS 767
8,0,0 4,2,0 0,5,0
7,0,1
3,2,2
0,3,1
3,0,4
0,0,5
8,2,1
7,2,4
3,3,5
4,3,2
4,5,1
Fig. 2.2. Planar graph associated to I
2
.
If the second item is true, then the number of monomials which do not lie in the
ideal is bounded by the product
α
i
. Conversely, if that item is false, there exists
an index i ∈{1,...,r} such that X
α
i
∈ I α N.
The third item is obviously equivalent to #{u N
r
| u b
i
for some i
{1,...,r}}< .
We conclude this section by illustrating those facts in the following example.
Example 2.3. In [28], a planar graph is associated to every monomial ideal in three
variables satisfying the conditions in Theorem 2.2. The monomial x
b
associated to
an irreducible component m
b
is identiﬁed with a connected component in the graph’s
complement and can be obtained as the least common multiple of the generators in
its boundary. In Figure 2.2 we show this construction for the ideal:
I
2
:= (x
8
,x
4
y
2
,y
5
,y
3
z,z
5
,x
3
z
4
,x
7
z,x
3
y
2
z
2
)
=(x
8
,y
2
,z) (x
7
,y
2
,z
4
) (x
4
,y
3
,z
2
) (x
4
,y
5
,z) (x
3
,y
3
,z
5
).
The description of those relations permits the simpliﬁcation of some computations
on Cayley digraphs, as pointed out in section 7.
3. Minimum distance diagrams. There are diﬀerent ways to relate monomial
ideals with graphs (see, for instance, [30]). In this section we propose a new approach
to studying Cayley digraphs in which we associate a graph with a monomial ideal.
The routing problem for Cayley digraphs reduces to studying paths originating at
a ﬁxed vertex, as these graphs are vertex-symmetric. Given a graph associated to
, {s
1
,...,s
r
}), where Γ is ﬁnite and abelian, we are looking for the shortest path
from node 0
Γ
to node c c Γ, i.e., a minimum distance diagram (MDD). We can
construct the routing mapping R:
(3.1)
R : N
r
−→ Γ
a → a
1
s
1
+ ···+ a
r
s
r
.
Thus, we need to ﬁnd a right inverse map of R:
D −→ N
r
,
such that
R(D(c)) = c c Γ and D(c)
1
= min{x
1
| x R
1
(c)}.
In general, map D is not unique; see Figure 3.1. This happens when the set R
1
(c)
contains two or more elements with minimum
1
-norm for some c Γ.

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##### References
More filters

Book
01 Dec 1986
TL;DR: Introduction and Preliminaries.
Abstract: Introduction and Preliminaries. Problems, Algorithms, and Complexity. LINEAR ALGEBRA. Linear Algebra and Complexity. LATTICES AND LINEAR DIOPHANTINE EQUATIONS. Theory of Lattices and Linear Diophantine Equations. Algorithms for Linear Diophantine Equations. Diophantine Approximation and Basis Reduction. POLYHEDRA, LINEAR INEQUALITIES, AND LINEAR PROGRAMMING. Fundamental Concepts and Results on Polyhedra, Linear Inequalities, and Linear Programming. The Structure of Polyhedra. Polarity, and Blocking and Anti--Blocking Polyhedra. Sizes and the Theoretical Complexity of Linear Inequalities and Linear Programming. The Simplex Method. Primal--Dual, Elimination, and Relaxation Methods. Khachiyana s Method for Linear Programming. The Ellipsoid Method for Polyhedra More Generally. Further Polynomiality Results in Linear Programming. INTEGER LINEAR PROGRAMMING. Introduction to Integer Linear Programming. Estimates in Integer Linear Programming. The Complexity of Integer Linear Programming. Totally Unimodular Matrices: Fundamental Properties and Examples. Recognizing Total Unimodularity. Further Theory Related to Total Unimodularity. Integral Polyhedra and Total Dual Integrality. Cutting Planes. Further Methods in Integer Linear Programming. References. Indexes.

6,747 citations

Book
01 Jan 1988
Abstract: This book develops geometric techniques for proving the polynomial time solvability of problems in convexity theory, geometry, and - in particular - combinatorial optimization. It offers a unifying approach based on two fundamental geometric algorithms: - the ellipsoid method for finding a point in a convex set and - the basis reduction method for point lattices. The ellipsoid method was used by Khachiyan to show the polynomial time solvability of linear programming. The basis reduction method yields a polynomial time procedure for certain diophantine approximation problems. A combination of these techniques makes it possible to show the polynomial time solvability of many questions concerning poyhedra - for instance, of linear programming problems having possibly exponentially many inequalities. Utilizing results from polyhedral combinatorics, it provides short proofs of the poynomial time solvability of many combinatiorial optimization problems. For a number of these problems, the geometric algorithms discussed in this book are the only techniques known to derive polynomial time solvability. This book is a continuation and extension of previous research of the authors for which they received the Fulkerson Prize, awarded by the Mathematical Programming Society and the American Mathematical Society.

3,634 citations

### "Cayley Digraphs of Finite Abelian G..." refers methods in this paper

• ...This object has been used to solve many problems in mathematics and computer science (see, for instance, [4, 16, 24, 26])....

[...]

01 Jan 1982
Abstract: In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

3,248 citations

Journal ArticleDOI
Arjen K. Lenstra
TL;DR: This paper presents a polynomial-time algorithm to solve the following problem: given a non-zeroPolynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q (X).
Abstract: In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

3,207 citations

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• ...This object has been used to solve many problems in mathematics and computer science (see, for instance, [4, 16, 24, 26])....

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Book
14 Dec 1995
Abstract: Grobner basics The state polytope Variation of term orders Toric ideals Enumeration, sampling and integer programming Primitive partition identities Universal Grobner bases Regular triangulations The second hypersimplex $\mathcal A$-graded algebras Canonical subalgebra bases Generators, Betti numbers and localizations Toric varieties in algebraic geometry Some specific Grobner bases Bibliography Index.

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### "Cayley Digraphs of Finite Abelian G..." refers background in this paper

• ...There are different ways to relate monomial ideals with graphs (see, for instance, [30])....

[...]

• ...Here we review several basic related results and definitions concerning monomial ideals; see, for instance, [2, 30]....

[...]

• ...The theory of Gröbner bases is related to several areas in mathematics and computer science; see, for instance, [2, 17, 30]....

[...]