BSTJ 60: 8. October 1981: Efficient Realization Techniques for Network Flow Patterns. (Chung, F.R.K.; Graham, R.L.; Hwang, F.K.)

6.2 MB·English

Efficient Realization Techniques for Network
Flow Patterns

By FR. K. CHUNG, R. L.GRAHAM and F.K. HWANG.
(deanvscrit rail January 20, 1981)

In this paper, woe describe several new technigues for use in the
design ofutehed communications netunrba. These techniques apply
to the development of traffic routes which realize network: traffic
{lows inthe contest of an existing optimization method thed wssinns
‘hese flows. The generat ideas mvolve the earful selection of Basie
teariables and the successive reduclion of te problem toone of convex
ult formation in Euclidean n-space end fouling Hamitionian cir-
cuits for a clas of highly structured graphs, We include several
Gxamples ahoning how these techniques are plied

1. wmopUCTION

Recently, R. H, Cardwell" proposed a awieched communications
rntrork design algorithn forthe fyeure stored program control net
trork, The networks under considertion are nonserarchieal instruc.
thre and take advantage of traffic noncoineidence in routing. The hase
‘objective of Cardwell’ algorithm ie design a minimum eoat traning
hetoork whic, by using am appropriate routing strategy can carry the
hneceasary trafic Toad and, the same time, meet the required grade
‘of cervice,

In this paper, we describe an extremely eficiont method for produc
ing an approprinte routing strategy. One of our origina intentions was
to develop a mathematical framework inty which dynamic routing
problems, sich a chase deseribed later, could be placed, Indeed, it
ers likely that the approach uaed here may bo valuable for exam-
ining other clanes of euch routing problems

1, BACKGROUND

In Fig 1 we show a block diagram of Cardwell algorithm. Suppose
we wish co design » network uring the algorithm. (See Tf. 1 for &
more detailed description.) We tart by intialiving the blocking prob
{biti of each link. The routing module selects a set of the most
fconomical paths far each pair of nodes and then aasigns flow tothe
paths, Routes, which aze ordered Tes of paths, are then formed +0
{hat the probability that all pathe in wny lt are basy ie small enough
fo mace the required grade of service. Then, by means of « Tinear
programming formulation, the routing module determines a network
How which minimizes che total cost, considering Tink coat and trafic
roncoincidence, In the engineering module, the lang laa formula is
ten sod ta fc the numberof trunles eine foreach link Inthe
Update modile dhe RCCS method of Trt i ured to help ranimizo
tho network cost’ The blocking probabilities for all inks are thon
‘uplted, The whole process is now ierated until satisfactory conver-
ice ix achieve.

Figure shows a block diagram of the outing module forthe unified
algoithim, A base feature of ut method is chat actual route are not
formed until convergent hae been obtained in an earlier part of the
‘unified algorithm. Only afer thie occurs docs the routing roalzaton
fubmodule generate the routes and provide the eppropriate routing
Strategy. Refer to the wark of Mueay and Wong which gives efiient
heuristic algorthens for aalving the nr programming problems in

"The upper bound module is» ner addivion which helps

[rant Lin ss, oom |

ig, 2 oun moi

1772 THEGELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1961
the iterative procedure to converge more rapidly by setting stronger
‘upper bounds forthe caried loads of the various paths chosen,

‘One of the questions cnn this nlyorthn wns the problem of
spntherivrg route foes Une sexe alls Mw (he route realization
bloc in Fig 2), The following iw season of an efciva technique
hat nconpishs thi

tu, eveute ROUTING

‘Consider the special ease of two nodes A and B. Assume there are
paths Py,1= b= m between A seal B. The minoont af Laie wo be
‘earriod on Py ia denoted by 2, where we have normalized the traffic
Toad so that one unit of afi in attempted between 4 and B. The
blocking probebilics for Py is denoted by ps We vall call the vector
Em ay, vor Za) Che desved tric voctoe wad B= (pay ~~ Pal the
‘locking probability veccor. (The ¥'s are actually ewputs of the linear
programming module)

Fora permutation «of (L,2,---, mh by the route (a) generated
‘by z, we mean the roure in which the path Pred and, i blocked,
pth Pa i re. I thac path ie blacked, shen path Pi red, ee.

"The fise question is: what aze the trafic owe on the various Fy,
shen toute tet ie used? Let gy = 1 ~ py and assure that 7 is the
identity permutation. i, elk} = # forall k. Since une unit of trai is
Aniiallyaltempte an Py, the fia path of Rx}, then 9, units of tafe
are carted on Py al sini of tric nr block. These uni are
how altempted on "Thus, p.geunia got catiad and pps are blocked,
Continuing this process, we see in general that on Po
Pra vo» Baoa units of trafic are carried and p, ps ~~» pa vps are
‘ack, We condense ehis formation into the flow vertor Fix)
(Gyn) alah, PH = (i, ds Emp adneve Paphos tee Node
fn paticuar thot the amoeintof irae which be Mlucked is jan
Pipa ==» Po independent of =

‘The overall plan i suse each toute Ris) a cevain fraction as) of
the tie, ay range ver all periutations of (1, = 8]. 90 a to
achieve the desired taife flow x on each Py. in otber wards, if
roa, fd) with

ate) 80,2 tal =

so Ut

1Fisi,

‘where ranges over all pemutations of (1,2. ---.n)
"This Is exactly the sume problem as deciding whether ¥ is in the

NETWORK FLOW PALIERNS 4773
‘comvex hull of points Fla) (considered ax points in n-dimensional
Faclidean space E") and, if 30, finding « representation of © a5 a
‘conver combination of the Fig, Note that all the (q's aro extreme
pints of che convex hil. ince

Dia

1 pine sss par any =,
then Uhe convex hull i actually fat most) an (7 — 1)-dimensional
polytope. Thus, any point in tw cones bull oan he vepresented as &
onves combination of some choee fm elreine points Fl.

"Anan examaple, we consider in detail the case n = 3. n Tube T, we
list the aix pombe ws and the comesponding F's

"We will denote the permutation 7 which sends ¢ vo n(@) by the
sequence s(i}9(2) =.= in). Thi ahold not be confused with the
‘inary eyele notation for x permutation v (which wl also be uso,
For example, the permutation of {1, 8,4, 5,6} given by w(l} = 2,
#12) = 5 1) ~ 6) = 4, (5) =2, x16} = 1 can be writen both as

(198) 25)(0) and — 35.0421.

Figure ahowsa typical picture when (hee points are plotted in 2
All six poins li on the plane By + Fe-+ Fy ~1~ p.ppe We should
tote here thal we niwaya asnutne 0-< pe <1 forall since any path
trith blocking probably one can be removed without affecting the
Caffe fl, al any path mich carries any tafe stall has patie
‘locking probability (les than one.

Tn genera, we would like to be able to decide if the desired cratic
weelae zien the convex bull ef (a) and. ifs, how to represent ic
fas u convex cowbination of F(s}. A natural choice to vonsider is a
‘cic sl of routes Por example, uppeae we consider the thre routes:

133,

Let ua deverzine whethor #8 inthe gonvex hal of these three points
(OF course, a noceesery condition ie Sy. =1— prompm Ta any case,

Table I-Flow vectors forthe
case n= 3

femeena

1774 THE HELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1981
‘re $Geomennl cera fe ects or the Ca =

since the convex bull i 2dimensioal, any point in it is a convex
‘combination of sexe set of thro extreme pinta. The eyelic sos seem
earobable choices since they apparently span racer Inge portions of
the convex hull, although cevainy not al of it. For example, in Fig. 3
tee have shaded the convex hl of Pizy, Fess) and Ploy) This is
Tauck larger thn, aby, che trace spanned by P28), FGS2) and
Fas,
"Therefore, we are lookin for coefficients a such that

Yaaiea =5, o
with
= 0,¥ amt
Fy en (0), the a mast satiny
Soke) = k= 124

‘xpanding these equations wing Table I, we obtain

ous + asp peta + gPs= HH

mnpige | ager eaprgepam a

a pipega + opin + ts Be @

NETWORK FLOW PATTERNS 175
“Tho determinant ofthe syste eq 2 i given by
© oP as

s=| pm pune
lovoan puns te

1 pp: p

a9 | oT ms
lbpe m1

= qaatl = pp

Solving forthe a, we have

apps ais
fee ge Psa
gma a

= [loum) = (passa. ~ pipes,
oe [tafaa) ~ (waved fC ~ pepe.
n= [Caaf ~ (netalaa/O1~ pops)

Laing
a5, 228,
Carr

‘wo soo thatthe ovate = Of 4, — 45> Un ~ 8, = 0,05 ~ fs =0.
"or general n, 2 similar raleuation shows that the corresponding
ceystemafm equations hs delerminant A given by

Bm ante ees gall = pipes Bad

and coefficient values

ser = (len fgn =) — (pean/aal/(l— pape +++ Pd
for the eyaie set of routes
12M
284-— ml
Bhai?

‘where addition of indices ie modal m,n,
com lo = BV is

‘Consequently, we suoeeed with thia eylic choice of routes if all tbe
a3 are at lant 0, iy

1776 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1981
O17 By eB ae Eh Eby o
Note that 3a: = 1 follows at once from
Be

Hp pe

amd, in particular, nove that che labeling of the Py is arbitrary. Any
arrangurment of Une os and Ja catisying oq, (2 wll give us acyclic set
fof routes which Works, ie, set of routes which conains in the
‘convex hull.

i order fo find these efficiently wo can do the fllering: From che
fiven x and py form

We are just euurching for oyelic portation (Ji, ji ++ al of
(2,00) such that
25,0,

To find thi, form the direcced graph G which has sits verex ot the
‘ul paths Py and an edge from each Fy to P; for which o = 6. If in
{Gwe find « Hamiltonian ciront (i rreuilpassing through each
verre exactly once, say, PP, +++ BlP,), Ohen by the deiiion of
the edges of Gwe muse ha

ee eee

which ie precisely what we want

Thus, We have shown that # can bo relized rom a eylic set of
oures if and only if hes « Hain cicuit. OF course, the
problem of finding a Hamiltonian eirit in an abileary geaph ix
Jnown to be an NP-complele problem (sev Ref. 3 for an exposition of
this corm and, therefore, sina crtuinly computationally ieractabe
fs che graph become lage. Fortunately, however, the graphs Gare
far Grom arbitrary nd, in fact, we can provide ss algonthon foe nding
Lamiltonianeiteuite in them whieh rina in time Oln log n)

First, we snay aasuine without lose of gonerelty (by a suitable
labeling) that

Note that neceasary condition for tho existonee af n Hamiltonian
csteuit in @

NETWORK FLOW PATTERNS 4777
Por all

ken,
ties a3] 20-2, o

where (X) denen the cardinality of che set X.To s2 this, note that
1G hae a Hameonian ciel, then foreach h, ehere i a leust one
fedge frm a vertex in (Pu, Pra ++ Pa} to one jn (Py Pay o>
Pra. Thus

a

which implies
[tio ayn eee

1 fact, 2g (49 ia alo a slfcient condition for @ to be Hamiltonian
"This ean be seen from the following proof (by induetion on n}.

‘Suppose = 2 and eq, (4) holds, Then elonly 0; = &y and Gis
‘Haritonian, Nez, aseume that eq, (4 iv slfcent for ll sich graphs
with n — 1 vertioes. Suppuse Ghia verlicrs and satin eq. (1). Let
1G" be the ndced subgraph on (P,P, > Pa a) (Where we have
‘asumes wy tonal that ey op ++ Sans engy ca ae that Galo
taiies ma (1). By induction, 7 haa a Hansltonian circuit, eny,
P,P, Py , Since G estes 2,

42 6, for some i, SiS 0-1.

Bat also

p> be forall hy C2 he n=
Ts,
Be PPB

isa Hamsitonian circuit in @ and the induction step is vorapoted. This
oven that en (0) i fact, a nacersary and sulficent ondivion for
Geo have a Hamiltonian circuit

‘We rumarize the preceding discuscion in the following

+++, Pa joining two given points A and #, and the
locking probability vector — (ps, + ps) and desired
talc velar # = (2 «+ 2

‘Object: To find permutation w of 1,2,

+n) sting

fare
vith
we ant,

‘where Fis the flow vector function wi x ithe eyetie route 4, et
BPD voe NE ne, Ps tried fst, cet Pye)
Algor:
(0) Caloalace

Pett tor sh
me

(Recall that we are always assuming thac 0< py <2 fr all)

(Gf) Relabel the oy if neceacary, 90 that e1 = 0.2 --- =

Gy Sete (0-2, 98, FL

(Gs) Tae © y, goo (oi. WF = 9, nner i after 2 Jn the cyclo
representation of =

(O) Tey > by ty <8, 2 <i. Hy 8, y andl 2 are unchanged I
fengaeed—t Land goto (ie)

(ei) The desined Hawaiian eieuit i PsPyoyPangy ++ Par 6.
Define

Bea

risks
To pipes

‘eam he reall by uci route my forthe faction ae of the timo, ==
hen,

“eal cannot be cea

End.

[Note that except for (tin which n logs n operations are required
in the ordering of the w o/s all other stepe require at mast O(a}
operations "Ths the computational complerity of the alrite i
‘logs + OF) in ime and tn} in space

We point out char the desired tzulfie flow voetor # can often be
eased by more than one at of evel routes, Ley Che graph C mag
have more than one Hinmillanin vinci (each of which comesponds
to a eyelic routing rulicon), Te preeling aigrittaa will aways
rode ane mar realivation provided any exists at all

Two examples

Example > There are Five paths betwen owe points A and B. The
Asie Urlfie vector and the blocking probability wuelor fae
follne

tiny set eyeic routes

(0.185, 0.281, 0.200, 0072, 02421
Fim 104,07, n6,0, 081
Ton,
4 = (0.924, 0.770, 0.350, 014, 0348)
B= (0740, 0530, 0.330, 0.072, 0.104),
"The correspnnding raph (a shown in Fig. 4
‘Prom the algorchin, we find the Huikonian cicuit, PiPLPSPP,
sormesponding tothe permulation 7 (12854). Therefore, se have the
‘vahies shown in Table TL
‘hw sling strategy ito vse route w for the fraction ef the ime.
Example 2: There are al ive patha between A and B. However,
‘the desired traffic weet 2” and the locking probability veclar pare
slightly itforent from thow in Fxample 1
= (1.191, 0281, 0220, 0072, 0249)

F’~ (07.07.08, 06,03),

Tos,
(9886, 0.770, 0.50, 0.144, 0286)
= (045, 0699, usi0, 0.072, 0.108),

“The enrrespnnding graph is ahown in Fig. 3.
From our algorithm, we find the Hamdlonian vice, PzPiPSP-P.,
‘curmspnnding to the permutation «= (21064). Therefore, we have
Use val chown "Table HL
Tei easly verified that
v= 5 amen

1700 THE BFL SYSTEM TECHNICAL JOURNAL, OCTOBER 1981