Cayley Digraphs of Finite Abelian Groups and Monomial Ideals

22 Pages·2007·0.53 MB·English

SIAMJ.DISCRETEMATH. (cid:2)c 2007Societ yforIndustrialandAppliedMathematics
Vol.21,No.3,pp.763–784

CAYLEY DIGRAPHS OF FINITE ABEL IAN GROUPS AND
hp MONOMIAL IDEAL S∗
p
sa. DOMINGO GO´MEZ†, JAIME GUTIERREZ†, AND A´LVAR IBEAS†
s/oj
al Abstract. Inthestudyofdouble-loopcomputernetworks,t hediagramsknownasL-shapesarise
n
r asagraphicalrepresentationofanoptimalroutingforeverygra ph’snode. Thedescriptionofthese
u
o diagramsprovidesanefficientmethodforcomputingthediamete randtheaverageminimumdistance
g/j of the corresponding graphs. We extend these diagrams to mul tiloop computer networks. For each
r
o Cayley digraph with a finite abelian group as vertex set, we d efine a monomial ideal and consider
m. itsrepresentationsviaitsminimalsystemofgeneratorsoritsirr edundantirreducibledecomposition.
a
si From this last piece of information, we can compute the grap h’s diameter and average minimum
w. distance. That monomial ideal is the initial ideal of a certa in lattice with respect to a graded
w monomialordering. ThisresultpermitstheuseofGr¨obnerbasesforcomputingtheidealandfinding
w
p:// aalnloopfttihmeamlraosustoicniga.teFdintoalliyrr,ewdeucpirbelseenmtoanfoammiiallyiodfeCalas.yleydigrap hsparametrizedbytheirdiameterd,
e htt Key words. monomialideals,Cayleydigraph,Gr¨obnerba ses,multiloopnetworks
e
s
ht; AMS subject classifications. 13P10,05C25,68M10
g
yri
p DOI. 10.1137/050646056
o
c
r
e o 1. Introduction. Let Γ be a group and S ⊆ Γ a subset. The Cayley digraph
ns associated to (Γ,S) is a directed graph whose vertex set is Γ and whose edge set is
ce {(g,h) ∈ Γ2 | g−1h ∈ S}. Every Cayley digraph is ve rtex-symmetric and its degree
M li equals the number of elements in S. These graphs ar e connected if and only if the

A
set S generates the group. We are dealing with digraphs associated to finite abelian
I
S
o groups, but we are mainly interested in those associat ed to cyclic groups. Let N be
ct t a positive integer and ZN the integers modulo N. For any subset S ={j1,...,jr} of
n subje dthigisraapbhel(isaenegFrioguuprew1e.1d)e,nwohteichbyisCcNal(lSed) t=heCcNir(cju1l,a.n.t. ,djirg)ratphheocrormreuslptiolonodpincgomCpayulteeyr
butio InfetSwoisrkaosfujbusmetpsofj1Z,N...s,ujcrh. tIhtatisfocronenveecrtyedeleifmaenndt ionnlSy iiftsgicndv(ejr1s,e..a.l,sjor,lNies) i=n S1,.
distri tchomenpuCtNer(Sn)etiwsoarnk.undirected graph called a circulant graph or distributed multiloop
e
R Multiloopnetworkswerefirstproposedin[32]fororganizingmultimodulememory
8. services and have a vast number of applications in t elecommunication networking,
2
5. VLSI design, and distributed computation. Their pro perties, such as diameter and
8
4.1 reliability, have been the focus of much research in com puter network design; see, for
4 instance, [5, 7, 12, 13, 19, 21, 25, 33].
1
3. The single-loop network or ring network is mathem atically trivial. Digraphs with
9
o 1 r =2 or double-loop networks and their corresponding undirected graphs (distributed
3 t double-loop networks, with degree four) have been ex tensively studied; see the sur-
1
5/ veys[3,20]andthereferencestherein. WhenCN(j1,j2 )isconnected,onecandefinea
2/0 minimumdistancediagram(MDD)asanarraywithve rtex0incell(0,0)andvertexc
0
ed ∗Received by the editors November 25, 2005; accepted for p ublication (in revised form) March
d
a 22,2007;publishedelectronicallySeptember26,2007. Thisresearchwaspartiallysupportedbythe
nlo ResearchProjectMTM2004-07086oftheSpanishMinistryofSc ience. Anextendedabstractofpart
w ofthisworkappearedin[14].
Do http://www.siam.org/journals/sidma/21-3/64605.html
†UniversityofCantabria,E–39071Santander,Spain(domin [email protected],jaime.gutierrez
@unican.es,[email protected]).

763

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764 DOMINGOGO´MEZ,JAIMEGUTIERREZ,A NDA´LVARIBEAS

0
1 17
hp 2 16
p
a. 3 15
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m.
7 11
a
w.si 8 9 10
w
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p:// Fig. 1.1. C18(3,8).
e htt
e 8
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M li 28

A
I 14 19 24 29 1 6 11 16 21 26 31 3
S
o 0 5 10 15 20 25 30 2 7 12 17 22
ct t
ubje Fig. 1.2. MDD of C33(5,14) .
n s
butio finyincegllj(xx,+y)j(yx≡istchmeocdoluNm,naninddxex+aynmdiyntimheumro.wOinndeee xx)a,mfoprlaepisasrhtiocwulnarinchFoigicueresa1t.i2s-.
ri 1 2
Redist strucTthinegclaanssiMcaDlDwoorkf oCfNW(jo1n,gj2a)nidnCOo(pNpe2r)smstietphs[3a2n]d prsehsoewntssiatnhaalsgoarnith“mL”fosrhcaopne-.
8. Several characterizations and applications of this idea for describing circulants with
2 desirable properties appear in [1, 8, 9, 13]. However , they do not focus on higher
5.
8 degree digraphs.
1
4. Two notable parameters in a graph are the diameter d and the average minimum
4
3.1 distanced¯. Theformerrepresentstheworstdelayinth ecommunicationbetweentwo
9 nodes, and the latter represents the average delay. G iven an L-shape, it is easy to
1
o compute d and d¯.
05/13 t portOannttphreoboltehmerihsatnodd,elteetrmdri(nNe)th:=isvmailnu{eda(nCdNfi(jn1d,.a.s. p,jerc)ifi|cj1C,N..(.j,1j,r..∈.,ZjrN)}a.ttAaniniimng-
02/ thisminimum. ThenetworkCN(j1,...,jr)issaidtob eoptimalifitsdiameterequals
d d (N). Insomecases, itisdifficulttoobtainoptimaln etworks; however, onecanfind
e r
wnload gpeanpeerra[l3s2i]m√sphloewfsundc2t(iNon)s≤se√rv3iNng−as2uapnpderpraensednltoswaerfabm oiulyndosf fcoirrcdurla(nNt)d;isgereap[3h]s. wTihthe
Do 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

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CAYLEYDIGRAPHSANDMONOMI ALIDEALS 765

abelian. Given a graded monomial ordering and a Cayley digraph (Γ,S), we build a

monomial ideal in the polynomial ring K[X1,...,Xr] , where K is an arbitrary field
hp and r = #S. We obtain some properties of this mo nomial ideal: in particular, a
a.p certain generalization of the two-dimensional L-shape is shown. On the other side, it
s
als/oj ifsorthcoeminpiutitainlgidtehael iodfeaalcaenrtdaifinndlaitntgicaen. Tophtiismraelsuroltutpienr gmfiotrsetahcehupsaeirofofGnroo¨dbense.rGbiavseens
rn therepresentationofthemonomialidealviaitsirreduc ibledecomposition,weprovide
u
o formulae to compute d and d¯. We also show a family of circulant digraphs of degree
rg/j two which coincides with the family obtained in pap er [32]. Finally, we present a
o
m. new and attractive family of circulant digraphs of arbitrary degree parametrized by

sia the diameter d, with average minimum distance d/2, a nd whose associated monomial
w. ideals are irreducible.
w
w The paper is divided into nine sections. In secti on 2 we collect several known
p:// facts about monomial ideals, presenting examples an d fixing notation for later use.
e htt Sobecttaiionna3npMreDseDn,tsatnhdeikteaylsiodeparoovfiadsessocainatainlggormitohnmomtoia lcoidnesatrlsucttoadnigMrapDhDs ifnororCdaeyrletyo
e
s
ht; digraphs with a finite abelian group as vertex set. S ection 4 is devoted to present-
g ing the relation between MDDs and the ideal of a la ttice. In section 5 we present
yri an algorithm to compute a shortest path between two vertices by means of Gro¨bner
p
o bases. Section6presentsanalgorithmspecificallytailo redfordegreethreecirculants.
c
e or It computes the minimal system of generators in O(s logN) arithmetic operations,
ns where s is the number of generators and N is the num ber of nodes. Section 7 is dedi-
ce cated to providing formulae to find the diameter and t he average minimum distance.
M li Then section 8 presents a family of multiloop comput er networks with an arbitrary

A number of jumps, parametrized by the diameter d, and all of them associated to ir-
I
S reducible monomial ideals. We conclude with a short summary and a discussion of
o
ct t open questions.
e
ubj 2. Monomial ideals. Monomial ideals form an important link between com-
n s mutative algebra and combinatorics. Here we review s everal basic related results and
o definitions concerning monomial ideals; see, for instan ce, [2, 30].
uti LetKbeanarbitraryfieldandK[X ,...,X ]thep olynomialringinthevariables
b 1 r
distri Xwi1t,h..v.e,cXtorr.sTofhrNorugahnoduutstehethpeapfoelrl,owweinvgernyotoaftteionni:dent ifymonomialsofK[X1,...,Xr]
e
R
28. xa =X1a1···Xrar ←→a=(a1, ...,ar),
5.
8
4.1 xa|xb ⇐⇒ a=(a1,...,ar)≤b=(b1,...,br) ⇐d e⇒f ai ≤bi ∀i=1,...,r,
4
1
3.
o 19 a=(a1,...,ar)(cid:2)b=(b1,...,br) ⇐de⇒f ( bi >0⇒ai <bi),
3 t
1
2/05/ ei :=(0,...,(cid:2)1i,...,0), ma :=(Xiai | ai > 0), 1:=(1,...,1).
d 0 The definition of (cid:2) suits the characterization in ( 2.2), and when it is employed
de (in expressions like a(cid:2)b), we usually have 1≤b.
a
nlo A monomial ideal is an ideal generated by monom ials, i.e., I ⊂ K[X1,...,Xr] is
w a monomial ideal if there is a subset A⊆Nr such tha t
Do
I =(xa | a∈A)=(A).

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766 DOMINGOGO´MEZ,JAIMEGUTIERREZ,A NDA´LVARIBEAS

x2y3
p
ph y3
a.
s
s/oj
al x4y2
n
r x2y2
u
o
g/j
r
o
m. x4

a
si
w. Fig. 2.1. Staircase diagram and Buchbe rger’s graph.
w
w
p:// There are two standard ways of describing a nont rivial monomial ideal:
e htt • Via the (unique) minimal system of monomia l generators I =(xa1,...,xas),
e we have
s
ht;
g (2.1) xu ∈I ⇐⇒ ∃ i∈{1,..., s} | a ≤u.
yri i
op • Via the (unique) irredundant decomposition by irreducible monomial ideals
c
e or I =mb1 ∩···∩mbn, we have
ns (2.2) xu (cid:15)∈I ⇐⇒ ∃ i∈{1,...,n } | u(cid:2)b .
ce i
M li The so-called staircase diagram is a useful graphic al representation of monomial

A
I ideals.
S
o Example 2.1. The monomial ideal I1 := (x4,x2 y2,y3) = (x2,y3)∩(x4,y2) is
ct t represented on the left in Figure 2.1.
e
ution subj cimsaonannbToamehreatiarisnelsioiaidcsnieaaartnliendbagla,tgsotoehrdleictnmohnmt(hXAef1aolm1er,xo.fian.nn.odd,mieXnrigaradlrtu)xhaa=eliatixrysrsa;eo.dsceOuieanntd[e2tad7hn]ett.o oiArttrhhneeedriiurrhrcreaeidbdnuludecc,idibibfellceKeoc[cmXoopmm1o,pps.oio.tnn.io,eeXnnnttro]mmf/Iaaa
b
ri must coincide with the least common multiple of a su bset of the minimal generators
dist of I. In the above Example 2.1 we have
e
R
8. x2y3 =lcm(x2y2,y3), x4y2 =lcm (x4,x2y2).
2
5.
18 The diagram on the right in Figure 2.1 is called B uchberger’s graph of the mono-
44. mialidealI1;see[28]. AtanystageinBuchberger’salg orithmforcomputingGro¨bner
1
3. bases, one considers the S-pairs among the current p olynomials and removes those
19 which are redundant; the minimal S-pairs define a g raph on the generators of any
13 to monTomheiaolrideemal.2.2. Let I be a nontrivial monomial id eal given by a minimal system
05/ of generators I = (xa1,...,xas) and by the irredun dant irreducible decomposition
02/ I =mb1 ∩···∩mbn. The following are equivalent:
d 1. K[X ,...,X ]/I is an artinian ring.
e 1 r
ad 2. ∀i=1,...,r, one of the generators’ exponents is a =α e for some α ∈N.
wnlo 3. ∀i=1,...,n, ∀j =1,...,r, bi,j >0. j i i i
Do Proof. We need to prove that the number of mono mials outside I is finite if and
only if either of the two last items is satisfied. We do t hat using the characterizations
in (2.1) and (2.2).

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CAYLEYDIGRAPHSANDMONOMI ALIDEALS 767

0,0,5
p
h
a.p 3,0,4
s/ojs 3,3,5 0,3,1
urnal 7,0,1 7,2,4 3,2,24,3,2
o
g/j 8,2,1 4,5,1
r
o 8,0,0 4,2,0 0,5,0
m.

a
w.si Fig. 2.2. Planar graph associated to I2.
w
w
p:// If the second item is true, the(cid:2)n the number of mon omials which do not lie in the
see htt iadneainlTdihseexbtoihu∈inrd{d1e,dit.e.bm.y,rti}shesoupbcvrhoiodtuhusacltyt Xeqiααu(cid:15)∈ii.vaICle∀onαntv∈etorNse#.ly{,uif ∈thaNtrit|emu i(cid:2)s fablse,fotrhesoremeexiist∈s
ht; {1,...,r} }<∞. i
g
yri We conclude this section by illustrating those fact s in the following example.
op Example2.3. In[28],aplanargraphisassociatedt oeverymonomialidealinthree
c
r variables satisfying the conditions in Theorem 2.2. T he monomial xb associated to
e o an irreducible componentmb is identified with a conne cted component in the graph’s
ns
ce complement and can be obtained as the least commo n multiple of the generators in
M li its boundary. In Figure 2.2 we show this construction for the ideal:

SIA I2 :=(x8,x4y2,y5,y3z,z5,x3z4,x7z,x3y2z2)
ct to =(x8,y2,z)∩(x7,y2,z4)∩(x4,y3,z2)∩(x 4,y5,z)∩(x3,y3,z5).
e
ubj Thedescriptionofthoserelationspermitsthesimp lificationofsomecomputations
n s on Cayley digraphs, as pointed out in section 7.
o
uti 3. Minimum distance diagrams. Therearedi fferentwaystorelatemonomial
b idealswithgraphs(see, forinstance, [30]). Inthissectionweproposeanewapproach
ri
dist to studying Cayley digraphs in which we associate a graph with a monomial ideal.
e The routing problem for Cayley digraphs reduces to studying paths originating at
R
8. a fixed vertex, as these graphs are vertex-symmetric. Given a graph associated to
5.2 (Γ,{s1,...,sr}), where Γ is finite and abelian, we are looking for the shortest path
18 from node 0Γ to node c ∀c ∈ Γ, i.e., a minimum dist ance diagram (MDD). We can
44. construct the routing mapping R:
1
193. (3.1) R: Nr −→ Γ
o a (cid:17)→ a1s1+···+ arsr.
3 t
5/1 Thus, we need to find a right inverse map of R:
0
02/ D : Γ −→ Nr,
d
e
d such that
a
wnlo R(D(c))=c ∀c∈Γ and (cid:18)D(c)(cid:18)1 =min {(cid:18)x(cid:18)1 | x∈R−1(c)}.
Do
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 (cid:3) -norm for some c∈Γ.
1

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768 DOMINGOGO´MEZ,JAIMEGUTIERREZ,A NDA´LVARIBEAS

22
8
p 27
h
a.p 13
s 3 32
s/oj 18
al 8 4
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ur 27 23
rg/jo 3 3123 298
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a 8 4 0 5 10 15 20 25 30 2 7 12 17
ww.si 1237 293
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p:// 18 14 19 24 29 1 6 11 16 21 26 31
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e
ht; s 298
g
yri 14 19 24 29 1 6 11 16 21 26 31
p 0 5 10 15 20 25 30 2 7 12 17
o
c
e or Fig. 3.1. Different MDDs for C33 (5,14).
ns
M lice Let RI¯nbdeigtrhaephesxtoefnddeegdremeatpwoo,f wRefrcoamn cNhrartaoctZerr,izaentdhiL s stihteuakteironnelinofteR¯rm. s of lattices.
A
SI Proposition 3.1. Let D be an MDD for (Γ,{ s1,s2}), where Γ is finite and
o abelian. Then there is a different MDD for the same g raph if and only if there exists
ect t avector(T,−T)∈LwithT >0andT ≤max{a1,a2} forsomea=(a1,a2)∈D(Γ).
n subj {(−1I6n,t1h),e(e−x1a,m−p2l)e}:C33(5,14) from Figure 1.2, the ass ociated lattice is generated by
o
buti (T,−T)=α(−16,1)+β(−1,−2)∈L ⇐⇒ α= −T , β = 5T ∈Z ⇐⇒ T ∈(11).
ri 11 11
dist In consequence, this graph admits exactly four M DDs: the L-shape one given
e
R in the introduction and the three shown in Figure 3.1. However, only two of them
28. have an “L” shape. These correspond with the only t wo graded monomial orderings
85. in K[X,Y].
4.1 Inaccordancewiththepreviousdiscussion,awell- orderinginNr compatiblewith
3.14 the norm (cid:3)1 determines a unique MDD. Then, fixing a graded monomial ordering ≺,
9 the obtained MDD is
1
13 to (3.2) D : Γc −(cid:17)→→ min(RN−r1(c )).
5/
2/0 For each graded monomial ordering we can associate t he bijective map p:N−→Nr,
0
d such that n<m⇒p(n)≺p(m), that is, satisfying
e
d
nloa p(i)=min(Nr\{p(j) | j < i}).
w
Do This map provides a method of constructing the MDD with respect to a fixed mono-
mial ordering. The procedure visits (through p) the elements in Nr corresponding
with vertices (elements in Γ) until all of them are com pleted.

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CAYLEYDIGRAPHSANDMONOMI ALIDEALS 769

Algorithm 3.1: MDD construction.
Input: Γ={c | 0≤i<N}, abelian group, {s ,. ..,s }, generating set; s.
hp Output: D(c )i, i=0,...,N −1. 1 r
sa.p 1 D[c0,...,cN−1i]:=¯∅, S :=0, a:=0;
s/oj 2 while S <N do
al 3 c:=R(a);
n
ur 4 if D(c)=∅ then
o
g/j 5 D(c):=a;
or 6 S :=S+1;
am. 7 end
w.si 8 a:=s(a);
w 9 end
w
p://
e htt We include in the MDD building method’s input the mapping s, such that
e
s
ht; s: Nr −→ Nr
yrig a (cid:17)→ p(p−1(a)+ 1),
p
o a≺s(a), (a≺b⇒s(a)(cid:21) b).
c
r
e o Of course, computing the whole diagram D[0,..., N−1] of a circulant cannot be
M licens ucAonlmgtiolpruritethaamcthioi3nn.ag1lliyptseereffifnocdrimiennsgt,,awnithseexsrhiezaedusbitseiivtnehgeseegxarpracpohnhe’tsnhtdaitiaalmcian enttelhra.estWinaphtuemntosrsizt(cid:22)e.fodrF,u(cid:3)trdht+dahrte(cid:4)rbmloouoornpeds,
A
I is approximately 1dr. The examples in Figure 3.2 illu strate the algorithm’s output.
S r!
o Definition3.2. LetΓbeafiniteabeliangroupand (Γ,S)anassociatedconnected
ct t digraph. Let ≺ be a graded monomial ordering. The m onomial ideal
e
n subj IS :=(Nr\D(Γ))
o
uti is the ideal associated with the graph (Γ,S) and the m onomial ordering ≺.
b
ri IntheexamplesofFigure3.2wehavetwomonomi alideals(J1 andJ2)associated
dist with C104(1,5,31) and with graded lex x≺y ≺z and x≺z ≺y, respectively:
e
R
8. J1 =(x5,xy6,y7,y3z3,z4,xy2z3)=(x5,y2,z4)∩(x5,y 6,z3)∩(x,y7,z3)∩(x,y3,z4),
2
85. J2 =(x5,y4,y3z3,z7,xy2z3)=(x5,y2,z7)∩(x5,y4,z3 )∩(x,y3,z7).
1
44. Proposition 3.3. With the above notation, we ha ve that Nr\D(Γ) is an ideal of
3.1 the semigroup Nr.
19 Proof. Let a be an element in the ideal generated by Nr\D(Γ). Then ∃b ∈ Nr,
13 to u∃z≺∈zN.rS\iDnc(eΓu)+subch≺thza+tbaa=ndbR+zis.aNloinwe,azrm(cid:15)∈aD(cid:3)p(,ΓR)(.uT(cid:4)+hb en)=∃uR∈(aN)ra,nwditah(cid:15)∈RD(u()Γ=). R(z),
05/ Obviously, D is an injective map and # D(Γ) = #Γ < ∞. So, the monomial
d 02/ irdinegalKI[SXal.w.a.,yXs c]o/nItaiinssargteinniearnat(osreseoTfhtehoerefmor2m.2X).1aW1, e..s.a,yXthraar;tathnaMt Dis,Dtbhueilqtuforotimenat
e 1 r S
ad gradedmonomialorderingisdegeneratedifI isanirre ducibleideal,thatis,whenthe
wnlo minimal system of generators of IS containsSonly as m any generators as the cardinal
Do ofS. Ingeneral,itisnotthecaseasillustratedinthea boveexamples. Thepaper[32]
constructedMDDsinL-shapefromcirculantdigraphs ofdegreetwo(i.e.,r =2). The
following concept is the generalization of L-shapes to arbitrary dimension.

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770 DOMINGOGO´MEZ,JAIMEGUTIERREZ,A NDA´LVARIBEAS

Grad. lex. x≺z ≺y

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o
c
e or Fig. 3.2. MDD of C104(1,5,3 1).
ns
ce Definition 3.4. Let I be a monomial ideal and let A be the minimal system
M li of generators of I. We say that I is an L-shape if th ere exists at most one element
IA xa =X1a1···Xrar ∈A such that ai >0 ∀i=1,...,r.
S We say that an MDD built following Algorithm 3.1 is an L-shape if the associated
o
ct t monomial ideal is an L-shape.
e IntheexamplesofFigure3.2thegeneratorinvolv ingeveryvariableisxy2z3. We
ubj will prove that any MDD built with Algorithm 3.1 is an L-shape. First we need the
n s following technical result.
o
buti Lemma 3.5. Let A be the minimal system of gen erators of IS. If the exponent
distri boif=xa0∈. A has some component ai positive, then b=( b1,...,br):=D(R(a)) satisfies
Re Proof. Since a is an element of A, then a(cid:15)∈D(Γ). We must have a−e ∈D(Γ),
8. becauseotherwiseawouldnotbeaminimalgenerator. Now,b≺aandR(bi)=R(a).
2
85. If we suppose bi >0, then
1
4.
14 R(b−ei)=R(a−ei), b−ei ≺ a−ei,
3.
9
o 1 which contradicts a−ei ∈D(Γ).
3 t Now, we state the main result in this section.
1
5/ Proposition 3.6. The output of Algorithm 3.1 is an L-shape.
0
2/ Proof. Let A be the minimal system of generators of I . If a ∈ A is such that
0 S
ed ai >0 ∀i, then by Lemma 3.5 we have R(a)=R(0)= 0Γ.
wnload geneMraotroerosvweri,tha−eveer1y∈coDm(Γpo),naenntdpRo(sait−ivee,1)th=en−as1−. S eo1,=if aD∈(−As1a)n=d bb∈−Ae1a.reTthwaot
Do completes the proof.
Now,theproblemistofindthelistofgeneratorsde scribingtheidealassociatedto
a circulant digraph in a convenient way. The following section answers this question.

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CAYLEYDIGRAPHSANDMONOMI ALIDEALS 771

4. Lattice ideals and L-shapes. In this section we study the initial ideal of

the lattice defined by the kernel of the extended rout ing map R¯ and the monomial
hp ideal associated to a circulant graph.
a.p AnintegrallatticeLofZr isthesetofintegerlinea rcombinationsofsomeintegral
als/ojs vbeecetnoruss;eidn toothseorlvweomrdasn,yanprinotbelgemrasl lianttmicaetihsemaaZt-iscusbam nodducolemopfuZterr. sTciheinsceob(jseecet,hfaosr
rn instance, [4, 16, 24, 26]).
u
o ForanyintegrallatticeL⊂Zr thereisanassociat edbinomialideal(see[10,31]):
g/j
r
m.o IL :=(xa+ −xa− | a∈L) ⊆ K[X 1,...,Xr],
a
w.si where a+ and a− are the positive and negative parts o f vector a, that is, the unique
ww vectors with no negative component and such that a= a+−a−.
p:// The interesting articles [10, 31] study the combina torics, geometry, and complex-
e htt ity of Gro¨bner bases for the ideals IL. In particular, t hey show that
ht; se (4.1) xa−xb ∈IL ⇐⇒ a−b∈ L.
g
yri GivenaCayleydigraph(Γ,S ={s1,...,sr})assoc iatedtoafiniteabeliangroup,
op we can extend the routing map R defined in (3.1) from Nr to Zr:
c
r
e o R¯ : Zr −→ Γ
cens a (cid:17)→ a1s1+···+ arsr.
M li The kernel L of the map R¯, i.e.,
S
A
I
S L :={(a ,...,a )∈Zr | a s +···+a s =0 },
o S 1 r 1 1 r r Γ
ubject t is thGeilvaetnticaenainsstoegciraatleldatttoice(ΓL,{asn1d,.a.m.,osnr}o)m.ialorderin g≺,foreverynonzerobinomial
n s xa−xb ∈IL, the leading or initial monomial with res pect to ≺ is given by
o
buti (cid:5)xa if xa (cid:23) xb,
distri LM(xa−xb):= xb otherw ise.
e
8. R As usual, given a polynomial ideal J in K[X1,... ,Xr] we denote by LM(J) the
2 monomial ideal generated by the leading monomials of all nonzero elements of J,
85. that is,
1
4.
14 LM(J):=(LM(f) | f ∈J ∗).
3.
9
1 The following is one of the main results in this section.
o
3 t Proposition 4.1. For every graded monomial or dering ≺, we have that
1
5/
2/0 LM(ILS)=IS.
0
ed Proof. The ideal IL is generated by binomials of the form xa−xb. Then it has
ad a Gro¨bner basis G alsoSconsisting of that kind of bino mial. Let xa be a monomial in
wnlo LM(ILS). There exists a binomial xa−xb in the basi s G, and by (4.1), a−b∈LS.
Do Now, since a (cid:23) b and both paths have the same im age by R, then a (cid:15)∈ D(ZN).
Conversely, let xa ∈IS. We take b:=D(R(a))≺a. I t is clear that a−b∈LS, and
so xa−xb is a binomial in IL , whose leading monom ial is xa.
S

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772 DOMINGOGO´MEZ,JAIMEGUTIERREZ,A NDA´LVARIBEAS

Gro¨bner bases were introduced by Buchberger in his thesis [6] and their use has

become widespread in commutative algebra and algeb raic geometry. The theory of
hp Gro¨bner bases is related to several areas in mathemat ics and computer science; see,
a.p for instance, [2, 17, 30]. As a consequence of the prev ious result we have that if G is
s
als/oj athmeienleimmaelntosrorefdGucceodnGstrito¨ubtneerabmaisnisimoaflthsyesitdeemaloIfLgSe,n tehreantotrhseolfeaoduirnMg mDDon.oImniaolrsdeorf
rn to apply Buchberger’s algorithm for computing a finit e Gro¨bner basis of an ideal, we
u
o needtostartwithafinitesetofgenerators. Inthissens e,wemustpointoutthatevery
org/j generating set of binomials for ILS corresponds to a g enerating set of LS (see (4.1)),
m. buttheconverseisnottrue. Lemma2.1of[31]providesasufficientconditionforthis

sia converse result to be true.
ww. Proposition 4.2. Let CN(j1,...,jr) be a conne cted circulant graph with asso-
ee http://w ciaUted:=la{tt(iNceαL1,...W.,eNhαavr)e,I(αL1=j1(−X11N,Xα22Nj2·,·.·.X.,rNαr−jr1)(,,α(1α jx11j,a1.+,.α−.,2αjx2ra−−−1j|1r,a−.1.∈,.α,Uαr)jr,rjwr−)h,e1.r).e}.,,
s
ght; and β,αi ∈Z,(i=1,...,r), satisfying 1=α1j1+···+ αrjr+βN.
yri Proof. The proof follows from [31, Lemma 2.1 ] and a simple linear algebra
op exercise.
r c Using Propositions 4.1 and 4.2 we can compute a minimal system of generators
e o ofIS forcirculantdigraphs. Thepaper[31]alsocontai nsresultsonthecomplexityof
ns computingthereducedGro¨bnerbasisoflatticeidealsa ndonitssize. Inparticular,it
M lice pwriothvidexepsoannenutpipalersibzeouinndthfeorbtitheconmupmlbexeirtyofoeflaembeansitssoa fnLd.shNoewvesrathneelexsasm, wpleemlautstticceitLe

A
I program 4ti2, which is extremely efficient in computin g the reduced Gro¨bner basis of
S
o binomials ideals. That software is available at http:// www.4ti2.de; see [18].
ect t 5. Optimal routing. In this section we show an algorithm for computing a
ubj shortestpathbetweentwoverticesforanyCayleydigr aphwithafiniteabeliangroup
n s as vertex set using a finite Gro¨bner basis of IL .
o S
uti Message routing is a basic function in communica tion networks. The problem is
b to find a route along which messages should be sent. The routing algorithm dictates
ri
dist token passing strategies in communication networks.
e Given a pair of nodes (t,s) in a graph, there ar e several paths which join the
R
8. origin t and the destination s. We are interested in optimal paths, i.e., those with
5.2 minimum length. For general graphs, finding a short est path between two vertices
18 is a well-known and important problem. Efficient po lynomial time algorithms have
44. been developed for various routing problems. Howev er, for the family of circulant
1
3. graphs, there is an important distinction to be made, and that concerns the natural
19 input size to a problem. For an arbitrary graph it is common to consider the input
3 to size to be O(N2), which is the number of bits in it s adjacency matrix. However,
1 any circulant graph can be described by only r integers. In this representation the
5/
0 input size is O(rlogN). Thus, polynomial time algo rithms for general graphs may
02/ exhibitexponentialcomplexityinthespecialcaseofcir culantgraphsforthiscompact
d input representation. In [7] it is shown that the shor test path problem is NP-hard
e
ad for this concise representation. The paper [15] presen ts very efficient algorithms for
nlo computing a shortest path for circulants with two jum ps.
w
Do As we have already pointed out, in our case the routing problem is reduced to
pairsofnodes(0Γ,j)wherethestartingpointisfixed. Usingthewell-knownextended
Euclidean algorithm we compute a path c from vertex 0 to vertex j if it exists.
Γ

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