A Difference in Efficiency between Synchronous and Asynchronous ...

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A DIFFERENCE IN EFFICIENCY BETWEEN SYNCIHRONOUS AND ASYNCHRONOUS SYSTEMS* by

Eshrat Arjowandi Michael J. Fischer Nancy A. Lynch

Technical Report #81-03-01

*This material is based upon research supported by the Office of Naval Research under Contracts N00014-80-C-0221 and N00014-79-C-0873, and by the U.S. Army Research Office Contract Number DAAG29-79-C-0155, also NSF #MCS77-15628 and MCS79-24370.

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r)-41111.~ Arjotnandi' Michael J./Fische NancYA.(Lflch5; /MCS79-2437C;

Department of Computer Science, University of Washington Seattle, WA 98195 11

S. TYPE OF REPORT

IN ' 'FICIENCY BETWEEN SYNCHRONOUS

PERFORMING ORGANIZATION

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COMI I.KTINCI'W

ACCESSION NO. 3. NV.CIPIINT'S CATALOG NUMVER

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DISTRISUTION STATEMENT (of this Repout)

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DistibutOf Uflt11Otd 11161110) 17, oisTmeIuTios STATEMENT (of the ebetree et mdin Ol.cb 20. #1 difleum't heow

1IS SUPPLEMCNTARY NOTES

numbe") 19. 9 EY WORDS (Continue an teve'.. eide H nec*6080Y wid Id.tfy bF 6004011

distributed system, clock, time complexity, synchronization

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all process'!s A system of parallel processes is said to be sXcrn~g its own has process run using the same clock, and it is as c ronous if each e prberi a articuldstbte independent clock. For any s, n1 in solved be orts'*- This problem can fined involving system behavior in n on n log time s by a synchronous system but requires time at least (s-1) any asynchronous system.

S'N 0101-LP-014-6601 AlE$CURITY

-CLASSFICATION OF THIS PAGE (Iftan bet

er4 fSw

Mtl ILNC

A DIFFEICENCE IN

BLIEt.ELSYN(.IIR0.40US AND ASYNCHRPONOUS SYST0IS4

LEftrat Arlumadi

Mlichael J. Fibcher

Nancy A. Lynch

D)ept. of Lompiutcr Sien.e

1)pIL.

York University 470~0 Keele Street

IUnivcr: tr. .i Wasihington 5.-ettI,: Wabililotton 98195

Information & Computer Scjitfne Georgia Institute of Technology Atlanta. Georgia 30332

iluwnuview, Canada 1

of (;onptter Science

Ontar~io lIP3

ABSTRAcT A wyse" of parallel processes io sald In be syelnti-i.s if all processes, ron uitin tile safti clock. .and It is aspie.hroau if cach p .,csb bav Its o'n tillepen-ient clock.- For any b, n. az p..:ticular dtittributed problem is Jefinied Iniv.,vIng system behavior at n -portsl". T i r.cblM C411 be solved In time a by a syno-ranous :.ybis WEt rrquIres time at leaist (b-1) logt noil .1y osynchiro-

ref lecting the communication bound in the model, whose precise definition is given it the next se.cion. If we strengthen the comunication eyetenA Slightly to permit a single designated process to broadcast to all thie others. or If we provide each process with access to a global clock. then the asynchronous model can solve the problem In time OWa.

THE

MIODELS

nous system.

INTht)IUCTiuN A system of parallel pri.crss.' is said to be If alil procehses. run uzsinr the, same clozk. an the pro.:eaies operate in !ltzk-stirp. and

lyne'hrur-ous

Iis

ifIfhrunous pt

rh

ias its own In-.

tems arc distributed ctialuter nuwr,r mystems fur convenatiinaz .7v;ut..r*.

anJ 1/0

In this paper. we compare tine at icency of a &si-ltswidel of a synchs..nuu stel with a siamiar asyncbrionoos mite. We houndj tile nunher of pr'cesses~ that can &c.*s aniy t..rc l.:uiar finMuni..ation Channel, and that reetricti-u:. lIscrucial to our results. For a. n t Ni. we 4e! !tiw a part 1cular distributed problem Involving n "p.-rts". It can be s~otvc4 In time a on a syfi~rorn,,uN System but ve show it requires time at leaut t 5 -1) 1 lotbU on any asynchronous system. Here ItIs a constant

IS An. attosic action whirb consisce of simul

'tep

p, and u. ~v are possible values of variable x. Mie -efine process(o) - p and variable(a) - x and say j Involves p and accessea x. Stop- a Is applicable to any global state in Which process p has lnternotlstate a end variable itcontains value u. The effect of performing a Is to changt: tl~estate of p to t and simultaneously to change the value of x to v. A system is specified by desc.ribing F, ". an initial global state. end a set OKSTWS of rnasibis steps. A process p blocks In a $lobel state g If there Is no' Step a In OKSTEPIS applica'lt to a proceso(o)

_______________________with

This material is based upon resear.-ii supported by Ltes Office of Naval Ammearch uWe- Cointracts NO00014-80-C-0221 and I i.7-CiV nd bv Lte U.S. Army licsearo . Ittlce Cantra~t %aber DAAI9-79-C-0155. 4L$7924370.

lie use a version of the cladel of a concurrent system definedl in [LF79.LF6lJ. Briefly. It constate of collection* It of procsses and I of shared variables. The slobal stats of the System consists of the internal stare of each process togather with the volue of each shared variable. A

also N51, *HCS77-0 6za and

-

p.

In this paper. we r-jeite

our system to be non-blocking for all piacesac and all global states. Let a X and deine locality(O) a (procas(a) 0 c OKSTE1S and variabista) . x). A system is h-b-ounded If Ilacality(x) I b for every x 4 1.

Accession For

!TIS

!TTS USA

U:l&iI30Wteu

d

RA&Zfinite

TAB

flJ~f~ io By-

LIC TAhA

DistributioLn AvSabiy

oc

A a1.3 h 4-11t 7COA0 nn/z

A computarion of a System Is a finite or Insequence of step* In CKSTMS such that the Step is applicable to the Initial global frst #tate, and each succeein"Sg step io applicable to th# state resulting frasm the applicatin of tbs ravious step. The result of a finitea computation Itlobl he tat 9f~rapplying The sequece.c* f every Is admissible cputation infinite masteps of the into intely process appears

sequence. A round is any sequence of steps such that every process appears at least once in the sequence. A win.ul round is a round such t-at no proper prefix 1% a round. Every arqucn.e of steps CA" be uniquely partitioned Into segnots such that every segment is & round. except possibly for the last if the cequence is finite. and every round Is tatnino. We call this a partition into siniIl rounds. even though the last ZVJn=Lat Is rnotait.essarily a round.

I

A aequene uf steps ito .o u unique partition Into minimal rounds:

If in the

(1) ho two steps irkthe same round Involve the game process; (2) No two teps n hs htime vsrlcble

Z

the

a

(1) and (2) together Imply that the steps In each r.'..d Are Independent and can be performed In any

or&ucr. or uimultaneously, with the saw result.

id

HIAIN REtSUL? Us Show that any asynchronous b-bounded system solving the (s.n)-sesastoa problem requlren at least (8-1) iogbalJ to quiescence, whereas there Is a trivial synchronous system which aolves the problem in time exactly a. This is the first example we know of. of a problem for which an asynchronous ayst m i provably slower than a

synchronoua ome. sod It shows that a straightforward Step-by-step and procass-by-procaea *iLmlation of an n-process synchronous system by an n-process asaynchronous one zucessarily loses a factor of lOgba in speed.

The run time for a finite Sequence of steps t defined to be the number of segments in the partition Into minimal rounds. This definition is equivalent to the one in [LFBL] which Says it Is the longest mount of elapsed real time that the system could take to execute te sequence. subject to the constraint that the tlme delay s between two Steps of the same process io at m unity. For synchronous syutcma. this defiuition Is Also equivalent to the more uisaj one which Simply Cocnts the number of synchronus steps of

realiges that the trivial asynchronos systtes with one process per port (and so cOMicaton doeg the processes) In which each prmces does nothing except access a port on each step In fact performs 0 sessions within time s. The difficul-

the system, where one Synchronous step consists of

ty Is that no process knows when time a has

the simultaneous execution of a step by each proCeaS.)

elapsed (due to the lack of a global system "clock"), nor does it know when the a s&"sion have in fact been achieved, so none of the processes knows when to stos accessing its port.

is a concurrent Finally. a Synchronous syste system whose allowable computations are All of ts comttins w infinitet syncrnt syste is A concurrent system whose Ailowable coputations are all of its infinite aJoia*Ible comTHE PROBLF-4 We now define a particular behavior for A concurrent syates. Let T C X be a distinguished rt I~evt is Set of variables called zit. A any step that accesses a pirt. A ve..#san is ay aoie part isn at I..st eaeuCnCy Of step cotai etight eet for every prt. A oep tationt si mtcs, sevaL s If itc i ae rsion I a S each of whichl asesio. n i fIt iite t t n is lt imattely quiescent If ¢ t c~--*.tlt4n

I

problems, concerns possible ordering* of sequences of events rather than the computation of particular outputs. It Is an abstraction of the synLhronization needed In many natural problems. Consider. for example. & simple message distribution system in which a sending process writes a sequence of a messages one at a time on a bjard visible to all and walts after each message until all n other processes have read the message. IAutever protocol insures that the sender has waited sufficiently long will also solve the (a.n)-sesslon problem.

only a finite number of port events. Th t to quiescence of an ultimately qoies...-nt *cquane is the run time of the &hortest prefix cta-in ail * 11, port

vens.

The (s.U) -ses oii £-.*blem i(st)-o Let a. n t M. the problem of finding a concur7ent sys em with n ports such that every allowable cumpautatin perand Is uiLt INAtly least) a sessi0n0 forms (t quiescent Note that the (e.n)-ession *rubles. like the 8mtual exclusio and dining paili rs

The result is

even more surprisl•

when oae

A procedure which does vork is for a proceds associated with each port to perform a port event, broadcast that fact and than walt until it has heard that all other port processes have performed their port events and that the session has been completed. This is repeated s times. -y making the port processes the leaves of a tree network, the necessary commnicatina for one session can be accomplished In time 0(1.l; n); hence, the total time to quiescence for the solution is 0(s log n). It sem very Into weit after each port eVent. and one efficient try to lnvent clever scheme to iacrease the coeconcvy In the system. Our lower bound shows. however. that this method io Optimal o within factor, so only a lmited " wenmta ofconstant iprov ef~ t to ps bleio

We now present the forma results. Torm 1.

For all a.

there is

•

-bouned Synchronous system which olve the problem, such that the tim to quescence for each allowable competet s uroot. The syste s m processes, On Procesa a to eah oroed. access "than

5.

ts prtn c h of Its first a Step ceaes Petnport O tso ito.t Intu

a then cease perfmn; pert e ts. In every infinite sychrontous Computation, seeb of the first v minimal rounds constitutes a session.

and tile Icn.

:iYalez bcOMSc

qutecCnt af ter a rounds.

tile bYbtem Malyc

tile (w:~a~..

fit 0" o tilesmallest such member of D. Define 5, - dep(a,) If a' exists. otherwise. and

problem

i~eoem eaut). (~~n ssue b a.a b -- 2. For every b-bounde ayehonu system whicts solve* the (s~n)-musaiutt problem. tile time to Cluiv-cunce In at lvast (u-I) Llusba.,j for sun.

prpris(1) and (11) and mnatouicity tow It suiffices to show

that dep(o) E I u a" u Wx. thadt B, u U~1dep~t) a'(l TeC

*

allwabe .mpuatin.Us

*

first consider 15. ad assume 01 exists. (If 0;' does not exist. there io nothing to prove.)

1he proof of Theorem 2 involves a series of (Thle orderfing repreaeita a kind of logical dependenicy.) lie breczk up tileproof of Theorem 2 by preventing tilelemmas before tile ain. proof. These lems~ And their proof% are oclf-conta" and depead only on ths

-

wthere

C' * t D : round~t) - round(o') + I and proc(T) t loc(x)). For every T' i C '. there exist$ T 4 C With proc('T) ftproc(T). Property (Ui) shous that T and V are s-comparable; property (III) shows that -t 9 T'. tNonotonilcity Imiplies that dtp(T) Ci dep(T). Thus.

PC-uiet ie: Ove~n below. Fur bettet Intuitive motivation. ha.wever. tilereadr oy wish to rc.d the WmAinptaet before reading tile lcvnjs.

jLet,

dep(r) u W.l

TWC

of steps oi a computation.

I

U

By induction. W' C

three: lemmas about a particular partial ordering

W

U

dep(T)

u (z).

as needed.

T

"round" )(of 1. numbrs), P a finite set (of "procescb" X a set Wo "variables"). Let D)be A cethaving it be the aet

Vinally. ws consider r.,at.' assume 0" exists. Then the properties of I)afilS show that rund( ) a r+1.o that ocC.

sappurs roud 0- i.Kproc: D- P a,4 var : D X. Assume that far every pair (r~p) e t -P, there to exsactly one a a D having round(o) *r and proc(a) - p. Let locixI * 43 t a And var(0) - xi. Let b z 2 and Aasume xbfor all x . . ule~)

Th4us. 2, c

U

dep(r). as needed.

0d

I(prac(2-)

Lema 3.

Forauch a

,.It is thecase

that 0O

Le be a partial order on D. and writ. t3indicate that 0 Iand thereIs nopVdp0I

with C~ < A properties:

-ou()I

Assume that Ishas thle following Ii eitur 01 T~the ar(3

ptoc

a

a

ar~i

Proof.

ork

- rouzid(o). backwards.

proc(.i).

(11) If either var(a) - varMt or proc(o) procit). then a and I are s-cotaparable. .(Iii) If a S 1, then ronil(o) Finally, let dep() !E~a deP(o 2

- (var(t)

(Mnotocity).

W~eproceed by Induction an starting with kt- a and working

kt- a.. flyLme 2, dep(a) S (var(o)). so Idepbi)I : 1. as needed. MSIS:

1 5 t

a.

by Lmia 2. ws have

rui).

IiU'CIN

a s ti.

idsp(et)' + 1. where, C Is dafinsd as .dep(o)t f. -ItC in Lemma 2. Each i c C has round(T) * kt+ It

Ifta01 i 0

then

dep(o.

Proof.

so by Induaction. Idep(I)I Sr Ob.vious from the definition

Lema2. Let

tf4ep.

Proof.

b

Also.,

.,Ie()

aD. roun,!(o) mr.

var(o) - x. Let C - (T c D: raud(!loc(x)). r + 1 and procMr) Than Jcp(.-) cU

~

1 [-.--1'+I-b-

k+1?1 -

amee

1

Ses

dep(T) -j W

Proof is by indct ion on %. beginning

with Pmaximal e!cmnts. Let -3t D)and ainsuse the lemma tiolda for all T 3,0. Assume r. x and C are detied zro& a as In the statement of the lema. If there exists a' t1D with 'rarfo') - x 4nd a" 3, . then fix a' as tE4 smallest such Member of D. (Property (LI) Insures that 0'. If it exists, Is defir-d uniquely.) Similarly, It there V=latm %3"* D WIth ptOC(O) co) an .*.ita

Proof of Theorem 2.

Asme an

achrousu

system which solves the (*.n)-sessioe problem. Enimarate the processes arbitrarily. CoAstruct an Infinite admissible compuzatios O by runn thle proCe&sse round-robin order (em. step Of process 1. one of process 2...,. me step of Process q. one step of process I....). Each round-robin round ie, minimal Sad Contains exactly on. step of ach process. me h iet

4 perton the first r rounds o cx.rtiy in. because we .Jb~unu .a cl-rrV4.9 bolution, thd i..1l-tatto. is5

to apply Lemma 3. to the suborderIng of S defined by restriction to rounds With nunhers

. vt L.t t b%:*I.e t,ulPw.tvly qu PI.nt r-..j u VaeVuuCfor this~ -Milutationl. last rust At Willett illy Po'rt event Veu.u.

k~loginJ Inclusive. The 'i-l)IIusborl .. supping 'rounds" required for the lemmas Is ob-

~

to b ow t~

4

"procss" andvariable" espectively.

dl. where a cantlathe first tzaig

perorm

sessions

rst

trct no po

su

.so

-tIb

~ .w e~ transitivity

~ ofses0

evepryostal ordearlof te vity I sthcspuato unewi

ad t

ist the 1.Sinc ste

(s.

vatbe!.oetesteps ordves . and whia

(1)rt ord Is e

tha

A

-5to

fio

Tkhos an toa

8tep h For eovssten

orer

o

Si

consistent reaidt.hhi In iS thete logslrfxnht

iia

~rounds.

i .lutaeet e bu reigntcnan

#,does not contain any

In either case.

step which accesses

mav. let a - CtiLlog.11 . snd write cniat

thire ace aps.e

notbentiu sfta inresasatilurltsn reig ~~~~~~~~~~~~~coigunder telogstp

global state As a. (Clearly .5Itself defines sos?.- a total ordering.)

heC

hnb

t it ieos tal theaut exs a pot0 an a stpakmuhta n t Thus adigth a1 whic a. ste y reutsiante Iudrtstt Co

the a.u~

il..B

nueteestsapor

astle arch

ntstcru.t -a patiall

ord tesr fn dhe rprsen Ig Jc-caany". lonsists at of e(3 csb parialrle) or (Frmally.the

~ pith~

tlep of

pru

f-Arderwo thesteps iffn rree atM~ t10 toing *ed Ii " Com-st sessormls. thlne du'ain fnt eartiale thit steuair h ~q t/llr, isa pairs pto rd u red we et of(1 pair 8.) Furstep evhr rcs()pr~es tfios inBadete resuifl

of

c that the towrdt

It Itstas

Our all m turmai s U oull oftand Iifnit adasibe coaLac an straegyion toi~ rsat as

_V!

tai..ed by renumbering the rounds In the same order. Mlappings -proc" ano over are obtained frots the

Ug b .

(s

Le a

x q u? t a the LPCwish-

7

k-l and #kdoes not contain

any step which accesses

I I It4 a. Let yobe anlarbitrary port. we define Inductively a Port Yk For kt*1...

B is consistent Le01122'mU with S . but at contains at most a S tlllobfl +

and two sequences of steps kanm I%. as follows. ~sessions. fl~ee ases arFirs~ ift~re~ tidee thoth which Is not accessed by an step cl! 8

on

the null be that port and let Ykune nd~*B~ Oriek e h Sequnce A- anOherise *ka le 1 b.the first stvp In fkwhich accesses yk-1,lWe now wish

since each session Aust Contain steps od~.(I oface

sequence of steps were completely contained! in for exasple, then It would fall to contain aStep acce"sIng ;4rt Yk- 1 .)

k

so access"e

to

Ik

7k-

Ik

Figure 1. A total Ordering Of OtPs Is

Consistent with 9

go

RES~ULTS VoI M4ORE GENERCAL HODELS

assumes values bJ 1 and bjis bounded above by a

it teenealizd sdelis b reovig ~constant. for. fromi the time wheni the clock bound in the number of processes which can access fis sue au i,.i tms ie a shasred variable. then a single communication constant amwunt Of tinme before all port processes variable shared by n port processes can be used to have reed the clock. performed port event. sent construct en easy 0(s) solution, messages containing Clock values I bj- . mad 1

A

Vsent

In fctif te mdelis the clock once again. Thereafter. It It At In oigial oigial fctIf te mdelis oly oly en-read en-most one tine unit before the clock Is iucteeralizved slightly by allowing onte of the shared seated again, thereby asaisalug value b varlabecs to be readby an arbitrary niumber of processes (but only to be changed by one process), then an 0(a + log n) solution Is possible. In Thus, the total *lapsed time until the more deutail. we use a shared variable. the messate clock assumes value b is 0(s). Thereafter, board, which every process can reod1 but icnly noc within time O(log a). the aupervisor has received fixed process, the supervibor. can change. Each al-h9eddmssgsedc dd~ that 4 port has a corresponding port pro~cas and there sessins have occurred and display the "SlT" arc addIiionl commication processes whose job Tretm nt ae.alpr it Is to pd-.& messages through a true network from message. Tretm at aealpr tho port processes back to the %supervisor. Tlprcee will have efread t g ort evT~ esae. message bord contains an Integer which we callwilhvstpeprfmng oreets & "clock" value. The supervisor alternately In~~crements the clock and reads the messeages being kFRIE sent back. Each port process repeatedly performs a cycle of reading the clock, performing a port 179 A yc n .. Fshr s05 event, and sending A message back to the superciigtefeeirsiIln tts visor. through the network. which coutains the of~~n ~hebearnd Ss. R tSs timo clock value Just read and the port Identifier. ocrtC uain d..Khn If Cand C2 are two successive clock values 1 2 by port process 1. then a part event must ocrat prIsoeieafter the clock assumes c + o"Inng . b nauraly hlInfrmaioncribim& c2 + .S aual obnn hsifrainof about all ports, thie supervisor can construct a sequence 0 -b 0 -cb 1 b2 e *...4b 8such that for each J.* a session Is guaranteed to occur between the times when the clock first assiumes values b adb+1.(Specifically,

let c 1 1 c 2.... denote

denote the successive clock values sent by port process 1l1 I I n. Then define b a( I(k+L) 1:Ii Snndkis the smallest Index such that ci a b 1 ). for each J, I I j I a.) After the supervisor constructs this entire sequence. it knows that at least a sessions have in fact occurred. at which tine It puts & "STOP" message on Its messate board. idien the port processes read the "STOP" message, they stop performing port events. It Is easy to sea that this construction solves the (s.n)-sessica problem. Weargue that it satisfies the required 0(s + log n) tlim bound. first. we consider message transmission tie. Since we are not sassuming any upper bound on size of variables, the tree netwurk can guarantee (by conratenating messages) that any message can be sent as sown as a process Is ready to send Ift. and also that any massage sent by time t is received by the supervisor by tiae t + 0(log n). NSext. we claim that for each J. I j S a. the elapsed time between when the clock first

Vol. 70 of Lecture Notes in Cam ter scisnce s"ties. Spr~ager-Verlag, 1971, 147-171.

the Sehevlor end Z~IW1wettioR Distributed Syst s Theoretical Cauter Science. 13. Nrtli-Hsllsnd Fublishig Cmpany. 1931. 17-43.

-

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Hr. E. H. Gleissner Naval Ship Research and Development Center Computation and Mathematics Department Bethesda, MD 20084 (1 copy)

(Icop2y()cpl San Diego, CA (1 copy)

Program Director Theoretical Computer Science National Science Foundation Wash en, D.C. 20550

WsigoDC

05

92152 Department of the Army U.S. Army Research Office P.O. Box 12211 Research Triangle Park, NC

(1 copy)

27709