Complex Sy stems 2 (1988) 501- 507
D ecreasin g Energy Functions and Lengths of
Transients for So m e C e llular Automata
Eric Go les"
Departm ent of Mat/l ematics, Engineering School , University of Chile,
CasiJla. 170/3, Correa 3, Santiago, Chile
A n drew M . Odly zko
AT&T Bell Laboratories, M urray Hill, N J 07974, U.S.A.
Abstract . Th e work of Ghiglia, Masti n, an d Rom ero on a "p haseunwrapping" algorith m gives rise to the following operation : for any
undirect ed graph with a rbitra ry int eger values attached to the verti ces,
simultaneous upd at es are perform ed on these values, with t he value
of a verte x being cha nged by one in th e direction of th e average of
t he values of the adjace nt vert ices. (When t he average equals t he
value of a vertex, th e val ue of th e ver tex is incremented by one , unless
all the neighbors have th e same value, in which case no change is
made.] Ea rlier work of Odlyzko and Rand all showed that iterat ing
t his operat ion always leads to a cycle of length one or two, bu t did
not give a bound on how ma ny iterations might be needed to reach
such a cycle. T his paper int rodu ces a new "energy funct ion" which
does yield a bound for the t rans ient . A novel feature of this energy is
that it contains not only linea r an d bilinea r ter ms, as is common, but
also terms involving th e minimum function .
1.
In t r o d u c ti on
Let G b e an undirected graph with ve rt ices lab elled 1, . . . , n, and su pp ose
t hat for each i, an integer Xi (O) is as sign ed at time 0 to vertex i . A se ries of
sy nch ro nous updates is p er forme d on t hese val ue s with t he rule that if Xi (t )
is the valu e o f vertex i at t ime t, t hen
Xi(t + 1)
=
Xi(t ) - 1 if Silt) < 0,
Xi(t)
if Xj(t ) = Xi(t ) for all j E J i,
{ Xi(t) + 1 other wise,
(1 .1)
"Sup ported partially by AT&T Bell Labor a tori es and Fondo Naciona l de Ciencias de
Chile 86·8 7, Sa nt iago, Chile.
©
1988 Complex Syst ems Publi cati ons, Inc.
Eric Goles and Andrew M. Odlyzko
502
where
L x; (t ) -
S;(t)
d;x;(t ),
(1.2)
;EJ,
J,
d;
= (j: vert ex j is connected to vert ex i},
= IJ;I = deg ree of vertex i,
This t ransformation was defined by Ghi glia, Mas ti n , a nd Rom ero {I] for certa in graphs th at a rose in t heir wor k on "phase un wra pping" (de term inat ion
of the phase of an analyt ic functi on from a set of values of t he functi on at a
discrete grid). Since t he max im al value assigned to any of t he vert ices never
increases , and the minimal value never decreases, it is clear t hat iter atin g t he
t rans formation leads to a configura t ion t hat rep eats per iodi cally. Gh iglia. et
al. ob serve d t hat for t heir graphs, the transformati on always ended up in a.
cycle of lengt h l or 2. Their a lgorit hm was in fact des igned on t he ass umption
t hat t he cycle len gth is at most 2. Later, Brickell and Pu rt ill (unpublished)
defined t he t rans format ion for general gra ph s and conje ct ured t hat th e period was a lways at most 2. This conjecture was proved in [6], by uti lizing an
"energy fun ct ion" der ived from t hat used in [4]. However , t he proof of [6]
did not provid e a n upper bound for t he transient. (the numb er of ste ps unti l
t he t ran sformation enters a cyc le]. since t he energy fun ction used t here was
not st rict ly decreasin g.
In t his note we obtain an upper hound for t he t ransient (theorem 2 in
sect ion 2). The most inter estin g asp ect of thi s bou nd is th e method used.
Th e first resul ts a bout period s of certain kinds of discrete iterations ha ving to
be 1 or 2 wer e obtained in [5] by rather cumbersom e com bina to rial methods.
Oth er meth od s were develop ed lat er [2-4,7-11]. The method s in [41 relied
on an "ene rgy functi on;" i.e. , a funct ion associated to a configura t ion t ha t
could be shown to he bounded , yet was strictly decreasing at each iterati on
not in t he cycle. The proof in [6J t hat t he Ghi glia et al. itera tion has period
1 or 2 relied on th e use of the following ene rgy funct ion , adopted from t hose
used in [4J:
n
Eo(l ) = -
L
a;;x ;(t)x;(t - I),
(1.3)
if i -I j butj E J ;,
if i = i ,
if i -I j and j 'Ie 1;.
(1.4)
i,;= l
where
a;;
={
T his ene rgy fun ct ion has th e pro pert y t ha t Eo(t + I ) $ Eo(l) for all t , bu t
unlike t he energy funct ion of [4J, equal ity sometimes holds for t not in t he
cycle , so t hat no upper bo un d for t he t ra nsient can be obtai ned from it . On
the ot he r han d , t his ene rgy fun ct ion involves only bilinear terms.
Decreasing Energy Funct ions and Length s of Transients for Some CA
503
The ene rgy funct ion we will use here is slightly different, an d is given by
n
E(t)
=
c<Eo(t) - 2 I: min(x;(t), Xi( t - 1))
(1.5)
i= l
n
2 I: I: min(x;(t), Xj(t - 1))
i= l
ses,
n
+
I:( x;(t) + x;(t - 1)),
i= l
where
0'
= 4 m,ax di
+ 8.
(16)
The novelty of our method is t ha t we use the m inimum funct ion in t he
defini t ion of (1.5), and t hat t he t erms with t he m inima in t hem appear to be
essential. Th ey allow us to show t hat E( t + 1) ::; E(t) - 1 for all t not in t he
cycle (th eore m 1), and t his th en leads to the upper ho und of theo rem 2 for
the leng th of the transient. (As we discuss in section 3, t his upper bo und is
probably not best possible.)
It is possibl e to byp ass t he use of t he m inimum func tion in (1.5) . T he
funct ion E (t) of (1.5) was derived by st udying an encoding (due to the first
a ut hor and me ntioned bri efly in [6]) of the it er atio n (1.1) into a transformation of a graph with only 0 and 1 values, for which th e method s of [4],
for example, could be applied. However , t he new graph is very large , and it s
size dep end s on th e init ial assign me nt of Xl( O), . . ., xn(O) . T hus t his enc oding
do es no t show what is ha ppening ver y clearl y, and th e bound for t he length
of the t ra nsient it gives is weake r than t hat of theor em 2 .
After an early version of th is paper was circ ulated , Pol jak [7] obt ained a
bound for the lengt h of the tran sient by using methods of con vexity t heory
as well as some of t he result s of thi s pap er. His bound (in th e notation of
th eorem 2) is of the form 24ep2 + 2, so it is some t imes bet ter and sometime s
worse th an t he bound we obtain.
2.
Main results
In thi s secti on we pr ove our main results, t heorems 1 and 2.
T heorem 1. For th e transformation defined by (1.1) a,nd the energy fun ction (1.5), one has
E(t + 1) ::; E (t ) - 1 [or any t 2: 1
unless x;(t
+ 1) = Xi(t -1) [or all i,
Proof. Because of th e symmet ry
in which case E(t
a ij
=
aji,
+ 1) = E(t) .
a qui ck calculat ion shows t hat
n
t>E(t): = E (t + 1) - E(t)
=
I: t> ;(t ),
; =1
(2.1)
Eric Gales and Andrew M. Odlyzka
504
where
L>i(t ) =
+
+
Q(Xi(t + 1) - Xi(t - I ))Si(t)
2 min(xi(t ), xi(t - 1)) - 2 min(xi(t + 1), Xi(t) )
2 L {min(x j(t ), xi(t - 1)) - min(x j(t ), Xi(t + I))}
(2.2)
j EJi
+ Xi (t + 1) - Xi (t -
1)
with Si(t) defined by (1.2). Wh at we will now show is t hat L>i(t) S 0 for all
i and t, and that ~ i (t ) ~ - 1 for at least one i if t is not in a cycle.
The first observation is that if
(Xi( t + I) - Xi(t - 1))Si(t) '" 0,
(2.3)
(Xi(t + 1) - Xi(t - I ))Si(t ) 2: 1,
(2.4)
then
and so
L>i(t ) ~ -1 ,
because Q (given by (1.6)) is large . To see t hat (2.3) impli es (2.4), not e t hat
is Si(t) > 0, t hen Xi (t + 1) = Xi (t) + 1, so Xi(t + 1) 2: Xi(t - 1), and an
analogo us argument applies when Si(t ) < O.
T he next observation is t hat if Xi (t + 1) = Xi(t - 1), t hen L>i(t) = O.
Hence it only remains to consider the case when Xi(t + 1) =J. xi(l - 1), but
Si(t ) = O. We consider two subcases:
Case a . Xj(t )
= Xi(t ) for
all j E J;. In t his case Xi(t
+ 1) =
Xi(t), and
therefore
L>i(t)
=
+
2min(xi(t), Xi( t - 1)) - 2Xi(t)
2 L {min(xj( t) , Xi(t - 1)) - min(xj(t ), Xi( t)))
j e J,
+ xi(t )-xi(t - l) .
If Xi(t) 2: Xi (t - 1), th en , since Xi(t
have Xi (t ) = Xi(t - 1) + 1, so
+ 1) =
Xi(t ) '" Xi(t - 1), we must
min(xj(t ), Xi(t -1 )) - min(xj(t ), Xi(t))
~
0
for all j E J i 1 and t.herefore
L>i(t ) ~ 1 - 2 = -1.
If Xi(t) < Xi(t - 1), t hen Xi( t + 1) = Xi( t) = Xi(t - 1) - 1, so
L>i(t ) =- 1.
Decreasing Energy Functions and Lengths of Transien ts for Som e CA
505
Case b . x;( t) '" Xi(t) for some j E Ji . In t his sit uat ion
Xi (t
+ 1) = Xi(t) + 1
so, since Xi (t + 1) '" Xi(t - 1), we must have
Xi(t + 1)
~
Xi(t - 1) + 1
and therefore for each j E J j ,
min(x;(t ), Xi (t - I)) - min(x;(t ), Xi(t + 1)) ::; 0,
and for at least one j E J i , the difference above must be - 1 or - 2.
Hence
2
L {min(x;(t) , Xi(t -
1)) - min (x;(t ), Xi(t + I ))) ::; - 2.
i E) ,
On t he oth er hand ,
m in(xi( t), Xi(t - 1)) - 2 min (xi(t + 1), Xi(t))
+ Xi(t + 1) - Xi(t - 1) = 0 or 1
2
so th at l'.i(t ) ::; - 1 in t his case also. •
We now turn to an investigation of the length of the transient. If we
ass ume that th e initial values sat isfy - 1' ::; Xi(O) ::; P for all i , th en (1.3)
a nd (1.5) show t ha t IE (t ) 1 = O("p' e), and so t he length of the t ran sient is
O(a p2 e). One can improve this bound somewhat.
T heo rem 2. If - p .:::; Xi(O) :::; p [or all i, then the iteration enters a cycle
after
::; 30" pe
(2.5)
upd ates, where e denotes the num ber o f edges in the graph.
Proof. Vve can clearly assume that the graph is connected, so that e 2:: n- l.
We first not e t hat for all i
ISi(t + 1) - Si( t) 1::; 2di.
(2.6)
If Si(t ) > 2di , t hen Xi(t + 1) = Xi(t ) + 1. Similarly, if Si(t ) < - 2di, t he n
Xi (t + 1) = Xi(t ) - 1. Furth ermore, if ISi (t)1 ~ 2di, t hen Si(t + 1) ::; ISi(t)I .
T he las t obser vation im plies that if ISi( t) 1 < 4di , t hen ISi(t + 1)1 < 4di as
well. •
Vve now show that after at most 2p + 1 iterations, all the Xi (t ) will be
at most 4di in absolu te magn itude. If ISi(u )1 ~ 4di, ISi(u + 1)1 ~ 4di, .. .,
ISi(u +2p+ l )1 ~ 4di , th en hy th e ab ove observations Si(U) , . .. , Si(1<+ 2p+ l)
are all of the same sign, and when they are positi ve, we have
Eric Gales and Andrew M. Odlyzko
506
x ,(u + 2p + 1)
=
x ;(u
+ 2p) + 1 = .. . = x;(u ) + 2p"
which is impossib le, as all th e IXi(t)1 ~ p. The same argument appli es,
m utatis mutandi s, when Si(U) is negative. Vlfe can therefore concl ude t hat
15,(1)1 S 4d ; for all i and all 1 2: 2p + 1.
Vve now proceed to t he pr oof of the t heorem . We have
IE(I)I
n
s
" IEo(t )1+ 4np + 2p
L d;
i= l
"I Eo(t)1
+ 4p(n + e) .
Now
n
Eo(t) = -
L x; (1 -
1)5;( 1),
j=l
so fo r t ~ 2p
+ 1,
n
IEo(t)1 S 4p L dj .
j= l
Hen ce for t 2': 2p
+ 1,
IE (I )I SS "pe+ 4p(n+e ).
T he bound (2.5) of t he theorem follows by an application of t heorem 1. •
3.
Concluding re mar-ks
T he const ant 30 in th e bou nd (2.5) of theorem 2 can easily be im proved .
However, it would be mu ch mo re interesting to improve on the m ai n te rm
ope. It is eas y to show exam ples wher e t he t ra nsient is at least proportional
to 11., t he number of ver t ices , or to p, bu t t h is is probab ly not optim al. Wh en
t he grap h is a sim ple pat h, t he bound of theorem 2 is of the for m O(pn ),
which may well be best possible. Howe ver, when the connect ivity of t he
graph incr eases, one would not nor mally expect t he ma xim al length of t he
t ransient to increase too m uch, if at all. It is even possible t hat a boun d of
the form O (p n ) m ight be valid for all graphs.
Some exam ples of assignments of values to simple pa th s th at yield long
t rans ients are (8, 9, 9, 3, 0,5,5 , 9,8) , which has a t rans ient of length 20 (t his
is known to be maximal for simple path s of len gth 9 with the Xi (O) restri ct ed
to be bet ween and 9), an d (8,8, 6,4, 6, 0, 5, 0,8,8, 5), which has a t rans ient
of lengt h 28. (In each case , th e vector gives the values of t he Xi(O) for
i = l , . .. , n , an d vert ex i is connected only to vert ices j with Ii - jl = 1.)
°
Decreasing En ergy Functions and Length s of Transient s for Some CA
507
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