M ICH A EL J.D IN N EEN and BA K H A DY R K H O U SSA IN OV, D epartm entofCom puterScience,U niversity ofAuckland,Auckland, New Zealand.

U pdate G am esand U pdate N etw orks

M ICH A EL J.D IN N EEN and BA K H A DY R K H O U SSA IN OV, D epartm entofCom puterScience,U niversity ofAuckland,Auckland, New Zealand.

E-m ail:

^f

m jd,bm k

^g

@ cs.auckland.ac.nz

ABSTRACT:In thispaperw em odelinfiniteprocessesw ith finiteconfigurationsasinfinitegam es overfinite graphs.W e investigate those gam es,called update gam es,in w hich each configura- tion occurs an infinite num beroftim esduring a tw o-person play. W e also presentan efficient polynom ial-tim e algorithm (and partialcharacterization) for deciding ifa graph is an update network.

Keywords:Infinite gam es,update netw orks.

1 Introduction

M any real-w orld system s can be view ed as infinite duration processes w ith finite states.Severalexam plescan be found in com puteroperating system s,airtraffic con- trolsystem s,banking system s,and the on-going m aintenance ofcom m unication net- w orks. A functioning system has to be robust(e.g.,an operating system should not crash regardlessofw hatthe userdoes).A term ination ofany ofthese system scan be thoughtofas a failure. Thusw e need an infinite duration m odelto study properties ofsuch system s.In practice these system shave only a finite num berofstates(e.g.,a banking system hasa finite num berofcustom ers,assets,etc.).

O vertim e,each system enters only a finite num berofstates and produces an in- finite sequence ofstates,called a run-tim e sequence. Since the num berofstates is finite,som e ofthe states,called persistentstates,appearinfinitely often in a run-tim e sequence. The success ofa run-tim e sequence is determ ined by w hetherornotthe collection ofpersistentstates satisfies certain specifications. Thus,w e can view the run-tim e sequencesas plays ofa tw o-playergam e w here one player,called the Sur- vivor,triesto ensure thatpersistentstatessatisfy som e property and the otherplayer, called the Adversary,doesthe opposite.

O urproposed m odelforan infinite duration system is based on a finite (directed) graph.The verticesofthe graph representthe states ofthe system and the edges(or arcs)correspondto thelegalstatechanges,called m oves(ortransitions),ofthesystem . DEFIN ITIO N 1.1

A n infinite duration gam e^G is a finite (directed)graph^{G =(V;E)},a fam ily^W of subsetsof^V,and tw o players(the Survivorand the A dversary).W e requirethateach vertex of^Ghas out-degree ofatleastone. A m em berof^W is called a winning set.

A configuration ofa gam e is a pairofthe form ^(v;Survivor⁾or^(v;A dversary⁾for

J.D iscrete Algorithm s,Vol.2 N o.1,pp.55–68,2002 cK ing’sCollege Publications

v2V.

Thegam erulesallow configurationm ovesfrom ^(w;X)to^(w0;X0)such that^(w;w0)

2 Eand^{X 6=X0}. Each play ofan infinite duration gam e is an infinite sequence of configurations^(v0

;X

0 );(v

1

;X

1

);:::;(v

i

;X

i

);::: such thatthe gam e rules are fol- low ed. W e calla finite prefix sequence of a play a history. W e say thata vertex

v is visited in the play if configuration^(v;X)occurs in som e history of the play.

N ote thateither the Survivor or the A dversary m ay begin the play. The Survivor w insa play ifthe setofpersistentverticesofthe play isa w inning set,otherw ise the A dversary w ins. A strategy fora player^Xi ofa gam e is a function from play his- tories^(v0

;X

0

);:::;(v

i

;X

i

)to configurations^(vi+1

;X

i+1

)such thatthe m ove from

(v

i

;X

i

)to^(vi+1

;X

i+1

)isa gam erule.

A given strategy fora player^X m ay eitherw in orloose a gam e w hen starting at an initialconfiguration^(v0

;X

0

),w here^v0

2 V and^X0is eitherplayer. A player’s winning strategy foran initialconfiguration isone thatw insno m atterw hatthe other playerdoes.

EX A M PLE1.2

In Figure 1 w e presenta gam e^G=(G;W).A san exam pleofa w inning strategy for theSurvivorconsiderthe initialconfiguration^(4;A dversary⁾.Ifthe A dversary m oves to vertex³then the Survivorsim ply m ovesto vertex¹and the gam e repeatsbetw een those tw o vertices (w hich is a w inning set). O n the other hand,if the A dversary m ovesto vertex⁵,the Survivorm ovesto vertex⁶forcing the A dversary to m ove to

4,w hich isthen controlled by the Survivor.The Survivorattem ptsto force the vertex set^f4;5;6ginto apersistentsetby m oving to vertex⁵.Ifthe A dversary triesto m ove to³from ⁵then the Survivoris allow ed to change its m ind and force^f1;3gas the persistentsetand w in.Thus,the A dversary loosesno m atterw hatchoice ism ade at vertex⁵.

W e end this section w ith a few related references. Previous w ork on tw o-player infinite duration gam es on finite bipartite graphs is presented in the paper by M c- N aughton [1]and extended by N erode etal. [2]. O urw ork focuseson a subclassof the gam esconsidered by these authors.N erode etal.providean algorithm fordecid- ing M cN aughton gam es. Theiralgorithm runs in exponentialtim e ofthe graph size forcertain inputs.Forourgam esw e provide tw o sim ple polynom ialtim e algorithm s fordeciding updatenetw orks,partially based on thestructuralpropertiesoftheunder- lying graphs. W e also note thatseveralearlierpapershave dealtw ith finite duration gam eson autom ataand graphs(e.g.,see [3,4]).

2 U pdate G am es

W e now m odela naturalcom m unication netw ork problem . Suppose w e have data stored on each node ofa netw ork and w e w antto continuously update allnodesw ith consistentdata. Forinstance,w e are interested in addressing redundancy issues in distributed databases.O ften one requirem entisto share key inform ation betw een all nodesofthe distributed database.W e can do thisby having a data packetofcurrent

2

3

4

5

6

W =ff1;3g;f2;3;4;5g;f4;5;6gg V =f1;2;3;4;5;6g

E =f(1;3);(2;3);(2;4);(3;1);(3;2);(4;3);(4;5);(5;3);(5;6);(6;4)g G=(V;E)

1

FIG.1.Exam pleofan infinite duration gam e.

inform ation continuously go through allnodesofthe netw ork.Thisisessentially an infiniteduration gam ew heretheSurvivor’sobjectiveisto achieveaw inning setequal to allthe nodesofthe netw ork.Thisgam eisform ally defined asfollow s:

DEFIN ITIO N 2.1

A n updategam e isan infinite duration gam e^G=(G;W)w ith the singleton w inning set^{W =fVg}.A n updatenetworkistheunderlying graph^Gofan updategam ew here the Survivorhasa w inning strategy foreach initialconfiguration.

Som etim esw e w illtalk abouta graph^Gbeing an updategam ew ithoutm entioning the w inning set,since itisunderstood that^{W =fVg}.

EX A M PLE2.2

Thegraph displayed below in Figure2 isan updatenetw ork.N oticethatallcyclesare ofodd length so thatthe Survivorand the A dversary alternately controlthe vertices w ith m ore than one possible m ove.The Survivorcan use itsopportunitiesto visitall verticesofthe graph.

3 Bipartite U pdate N etw orks

W e firststudy a specific class ofupdate gam es on bipartite graphs,called bipartite update gam es. Forthese gam esw e restrictthe dom ain ofgraphsto bipartite graphs w here the vertices^V of each graph can be partitioned into tw o disjointsets^Aand

S such thatevery edge is directed from ^Ato^S orfrom ^S to^A. W e also stipulate thateach vertex hasan out-going edge (i.e.,thisensuresthatevery play isofinfinite duration).By definition,w eassum ethattheSurvivorm ovesfrom ^Sand theA dversary

FIG.2.A sim ple exam pleofan updategam ew hich isan updatenetw ork.

m ovesfrom ^A.In essencethevertices(in thesebipartitegam es)areowned by thetw o playersofthe gam e. Thus,forthese gam es,there are only^jVjgam e configurations w here each vertex^vdeterm inesa unique configuration depending on w hether^visin

Sor^A.In the nextsection w e return to the updategam esdefined in D efinition 2.1.

DEFIN ITIO N 3.1

A bipartite update network isa bipartite graph^{(V =A[S;E)}ofa bipartite update gam ein w hich theSurvivorhasaw inning strategy to visitevery vertex of^V infinitely often from every initialconfiguration.(Thatis,the Survivorcan force the persistent setofverticesto be^V.)

W e can easily characterize those bipartite update netw orksw ith only one Survivor vertex.These are the bipartite graphsw here out-degree^(s)=jAjforthe single Sur- vivorvertex^s.W e now deriveseveralpropertiesforallbipartiteupdatenetw orks.

LEM M A3.2

If^{(V = A[S;E)}is a bipartite update netw ork then forevery vertex^{s 2 S} there existsatleastone^a2Asuch that^(a;s)2Eand out-degree^(a)=1.

PRO O F.The idea is to show thatif there exists a vertex^s thatdoes notsatisfy the statem entofthelem m athen theA dversary can alw aysavoid visiting^s.Let^As

=faj

(a;s)2 Egand assum e out-degree^{(a) > 1}forall^{a 2A}s. The A dversary has the follow ing w inning strategy.Iftheplay history endsin configuration^a2Asthen since out-degree^(a)>1,the A dversary m ovesto^s0(of^S),w here^{s0 6=s}and^(a;s0)2E. Thiscontradictsthe assum ption oflem m a.

For the follow ing results let^(A[S;E)be a bipartite update gam e^B. For any Survivorvertex^sdefine

Forced(^s)^=fajout-degree^(a)=1and^(a;s)2Eg;

w hich denotesthe setofA dversary verticesthatare ‘forced’to m ove to^s. N ote,by the previouslem m a,thissetw illbe non-em pty forgam esplayed on bipartite update netw orks.

LEM M A3.3

If^Bisa bipartite update netw ork such that^jSj>1then forevery^s2Sthere exists an^s06=sand an^a2Forced(^s),such that^(s0;a;s)isa directed path.

PRO O F.Take any^{s 2 S} and consider^{F =} Forced(^s). By the Lem m a 3.2^F is not em pty. Ifthere is an^{s0 6= s} adjacentto a vertex (i.e.,an in-neighbor)in^F w e are done.O therw ise,all^{s0 6=s}are notadjacentto any vertex in^F.Thusthere existsan

a

0notin^Ffrom w hich theA dversary hasaw inning strategy by notm oving to^s.This contradicts^Bbeing a bipartite updatenetw ork.

DEFIN ITIO N 3.4

G ivenabipartitegraph^{(S[ A;E)}aforced cycleisa(sim ple)cycle^(ak

;s

k

;:::;a

2

;s

2

;

a

1

;s

1 )for^ai

2Forced(^si)and^si

2S.(N otethatforced cycleshaveeven length since the graph isbipartite.)

W enow presentourpenultim ateingredientthatw illbeused to characterizebipartite updatenetw orks.

LEM M A3.5

If^Bisa bipartiteupdatenetw ork such that^jSj>1then there existsa forced cycle of length atleast4.

PRO O F.Take^s1

2S.From Lem m a 3.3 there existsa path^(s2

;a

1

;s

1

)in^Bsuch that

s

2 6=s

1 and^a1

2 Forced(^s1). N ow for^s2 w e apply the lem m a again to geta path

(s

3

;a

2

;s

2

)in^B such that^s3 6= s

2and^a2

2 Forced(^s2). If^s3

= s

1w e are done.

O therw ise repeatLem m a 3.3 forvertex^s3.If^s4 2fs

1

;s

2

gw e are done.O therw ise repeatthe lem m a for^s4.Eventually^si

2fs

1

;s

2

;:::;s

i 2

gsince^Bisfinite.

Thus,if^Bdoesnothavea forced cycleoflength atleast4 then either^jSj=1or^B isnota bipartite update netw ork.W e now presenta contraction m ethod thathelpsus decideifa bipartite gam eisa bipartite updatenetw ork.

Let^{B=(S[ A;E)}beabipartiteupdategam ew ith aforcedcycle^C=(ak

;s

k

;:::;

a

2

;s

2

;a

1

;s

1

)oflength atleast4.W e can define a contracted bipartite update gam e

B 0

=(S 0

[A 0

;E 0

)asfollow s.Fornew vertices^aand^slet

S 0

=(Snfs

1

;s

2

;:::;s

k

g)[fsg and ^{A0 =(Anfa}1

;a

2

;:::;a

k

g)[fag:

W ith^E00being the induced edgesofthe subgraph^Bnfs1

;a

1

;:::;s

k

;a

k glet

E 0

=f(s;a 0

)ja 0

2A 0and^(si

;a 0

)2E; forsom e^ikg[

f(a 0

;s)ja 0

2A

0and^(a0;si

)2E; forsom e^ikg[

f(s 0

;a)js 0

2S

0and^(s0;ai

)2E; forsom e^{ikg[f(a;s);(s;a)g[E00:} Thenextlem m ashow sthe relationship betw een gam e^Band thereduced gam e^B0. LEM M A3.6

If^{B =(S[A;E)}isa bipartite update gam e w ith a forced cycle^Coflength atleast 4 then the contracted bipartite update gam e^{B0 =(S0[A0;E0)}is a bipartite update netw ork ifand only if^Bisone.

PRO O F.W e show thatif^B0 is an update netw ork then^B is also an update netw ork.

W e firstdefine the naturalm apping^pfrom verticesof^Bonto verticesof^B0by

p(v)=v if^v62C

p(v)=s if^v2C\S

p(v)=a if^v2C\A:

Then any play history of^Bism apped,via the function^p(v)=v0,onto a play history of^B0.Considera play history^v0

;v

1

;:::;v

nof^Bthatstartsatvertex^v0and^vn 2S. Let^f0 be a w inning strategy in gam e^B0 forthe Survivorw hen the gam e begins at vertex^v0

0

.W e use the m apping^pto constructthe Survivor’sstrategy^f in gam e^Bby considering the follow ing tw o cases.

Case^v0

n

=s. The strategy is to extend the play (in^B)by visiting allthe verticesof the cycle^C atleastonce. If^f0(v0

0

;:::;v 0

n ) = a

0 w here^{a0 6= a}w e find a^si 2 C

such that^(si

;a 0

) 2 E then extend the play again w ith^a0 as the lastm ove. O ther- w ise^f0(v0

0

;:::;v 0

n

) = aand the play is extended by picking an^ak

2 C such that

(v

n

;a

k )2E. Case^v0

n

6= s. If^f0(v0

0

;:::;v 0

n ) = a

0

6= athen^f w illalso m ove to^a0. O therw ise

a 0

2Cand the play isextended by picking an^ak

2Csuch that^(vn

;a

k )2E. Itisnothard to seethat^f isaw inning strategy fortheSurvivorin gam e^Bw henever

f

0isa w inning strategy in^B0.

W e now show thatif^B is an update netw ork then^B0 is also an update netw ork.

Takeany vertex^v0

0

from ^B0.W e show thatthereisaw inning strategy fortheSurvivor starting at^v0

0

.Fix any vertex^v0such that^p(v0 )=v

0

0

.W e w illkeep a correspondence betw een positions^vi of a play on^B w ith positions^v0

i

of a play on^B0. W e now sim ulate the w inning strategy^f on^B starting at^v0. The strategy forthe initialplay history^v0

0 2S

0is^f0(v0

0

)=p(f(v

0

))exceptforthe case^v0

0

=s.In thisexceptional casetheSurvivor’sinitialstrategy isto m ovedirectly to any^{a0 6=a}and replace^fw ith thestrategy starting at^a0.N ow let^v00

;v 0

1

;:::;v 0

nbeany play history of^B0thatoccurs afterthe initialplay asdictated above.W e definea strategy for^f0w hen^v0

n

isin^S0by studying tw o cases.

Case^v0

n

= s. Considerthe previousvertex^v0

k

=a 0

6=aofthe play history thatis in^A0and^v0

k +1

=s. Thusin the gam e on^B,the A dversary from ^a0 elects to m ove to som e^si in^C. Since in gam e^B0 the A dversary m ay have few erchoices,w e can pick,w ithoutloss ofgenerality,thatitm oved to^siw hereⁱisthe sm allestallow able index. The choice ofany indexⁱis validated because the cycle^C is a forced cycle.

Thestrategy^f0now sim ulatesw hat^fw ould do from ^si.Tw o cases:(1)if^f m ovesto avertex^ajon^Cthen^f0m ovesto^a,or(2)if^f m ovesto avertex^a0noton^Cthen also

f

0m ovesto^a0.In thefirstinstancethestrategy^f0forcesaplay thattogglesbetw een^a and^sin^B0untilcase (2)holds.(A nd thism usthappen since^f isan updatestrategy.) Case^v0

n

6=s.H ere the strategy issim ply^f0(v0

0

;:::;v 0

n

)=p(f(v

0

;:::;v

n

)).Thatis, the play follow sthe strategy^fon the sim ulated gam ehistory of^B.

The strategy^f0forthe Survivorisan update strategy since^pisa m apping from ^B onto^B0(i.e.,ifallverticesof^Bareinfinitely repeated via^fthen allverticesof^B0are infinitely repeated via^f0).

s

1

s

2

s

3

s

4

a

1

a

2

a

3

a

4

a

5 B

B 0

s

a

a

5

a

4 a

3

s

4 s

3

FIG.3.Show ing the bipartiteupdategam ereduction ofLem m a3.6.

W ith respectto the contraction m ethod above,Figure 3 show show a forced cycle of^Bisreduced to a sm allerforced cycle (oflength 2)in^B0.

Forthenextresultletⁿdenotetheorder(num berofvertices)and^mdenotethesize (num berofedges)ofa graph.

TH EO R EM 3.7

Thereexistsan algorithm thatdecidesw hetherabipartiteupdategam e^Bisabipartite updatenetw ork in tim e^O(nm).

PRO O F.W e show thatfinding a cycle thatisguaranteed to existby Lem m a 3.5 takes tim e atm ost^O(m)and thatproducing^B0from ^B in Lem m a 3.6 takes tim e atm ost

O(n+m).W e can also detectin tim e atm ost^O(m)w hen a forced cycleoflength at least⁴doesnotexist.Since w e need to recursively do thisatm ostⁿtim esthe overall running tim e isshow n to be^O(nm).

The algorithm term inates w henever a forced cycle of length atleastfour is not found. Itdecides w hetherthe currentbipartite graph is a update netw ork by sim ply checkingthat^S=fsgand out-degree^(s)=jAj.Thatis,thesingleton Survivorvertex isconnected to allA dversary vertices.

Letusanalyze the running tim e forfinding a forced cycle^C.Recallthe algorithm im plied by Lem m a 3.5 beginsatany vertex^s1and findsan in-neighbor^a1(of^s1)of out-degree 1 w ith^(s2

;a

1

)2 E w here^s2 6= s

1. This takes tim e proportionalto the num berofedgesincidentto^s1to find such a vertex^a1.Repeating w ith^s2w e find an

a

2in tim e proportionalto the num berofedgesinto^s2,etc.W e keep a boolean array to indicate w hich^siare in the partially constructed forced path (i.e.,the look-up tim e w illbe constanttim e to detecta forced cycle oflength atleast4). The totalnum ber

ofstepsto find the cycle isatm osta constantfactortim esthe num berofedgesin the graph.

Finally,w e can observe thatbuilding the contracted bipartite gam e^B0from ^Band

Crunsin lineartim e by the definition of^S0,^A0and^E0.N otethatifthe data structure forgraphsistaken to be adjacency liststhen^E0isconstructed by copying the listsof

Eand replacing oneorm ore vertices^si’sor^aj’sw ith one^sor^a,respectively.

The above resultindicates the structure of bipartite update netw orks. These are basically connected forced cycles,w ith possibly otherlegalm oves forsom e of the Survivorand the A dversary vertices. Figure 4 show s a constructed exam ple ofone such bipartite update netw ork.The Survivor’sstrategy isto system atically repeatthe forced cyclesand ‘detour’to coverthe rem aining non-forced A dversary verticeson a periodicbasis.

FIG.4: Illustrating the structure ofbipartite update netw orks w ith Survivorvertices (black)and A dversary vertices(w hite).

W e listallthe non-isom orphic bipartite netw orks oforderatm ost5 in Figure 5.

N ote that18 of the 19 netw orks of order 5 w ere generated from three of the net- w orks of order4 (see those displayed on the top row ). This w as done by system - atically adding a new adversary node w ith allpossible com binationsin-degreesand out-degrees of 1 and 2. N ote thatthe firstfour graphs in the firstcolum n and the firstand fourth graphsin the lastcolum n are m inim alin the sense thatalledges are essentialforthese graphsto be bipartiteupdatenetw orks.

4 R ecognizing U pdate N etw orks

W e now w antto presentan algorithm to decide w hethera given update gam e isalso an update netw ork.O uridea isto take an update gam e^Gand im plicitly transform it into a bipartite gam e^BG.(N ote^BGw illnotbe a bipartite update gam e,asdescribed in Section 3.)W e then show how to decideifthe graph^G(of^G)isan updatenetw ork by checking ifthe Survivorhas a w inning strategy forevery initialconfiguration of

B

G. Recallthatin a bipartite gam e the A dversary and the Survivoronly m ove from

FIG.5:A llsm allbipartiteupdatenetw orksw ith Survivorvertices(black)and A dver- sary vertices(w hite).

oneofthe vertex partitionsofthe graph.

W e define the gam e^BG

= (B;W) from an update gam e^{G = (G;fV(G)g)}as follow s:

V(B) = fv

S

jv2V(G)g[fv

A

jv2V(G)g

E(B) = f(v

S

;u

A

)j(v;u)2E(G)g[f(v

A

;u

S

)j(v;u)2E(G)g

W = fY j8v2V(G);9w2Y(w=v

Sor^w=vA )g

N ote thatthe graph^B is only tw ice the size of^Gbutthe explicitstorage forthe w inning sets^W is exponentialin the size of^G’s w inning sets^fV(G)g. Figure 6 show sa sm allexam pleofthe construction of^Bfrom ^G.

0

1 2

3

1

A

2

A

3

A 0

A 0

S

1

S

2

S

3

S

FIG.6.M apping an updategam e (graph^G)to a bipartitegam e(graph^B).

The vertices of^B w illcorrespond to a vertex/playercom bination ofthe gam e^G. W e have the follow ing equivalence.

LEM M A4.1

The gam e^Gis an update netw ork ifand only ifthe Survivorhas a w inning strategy forevery initialconfiguration of^BG.

PRO O F.Firstassum e^Gisan update netw ork.Forany initialconfiguration^(v;X)of the gam e^Gthe Survivorhasa w inning strategy^f.The Survivorcan use thisstrategy

f forthe initialconfiguration^vX in the gam e^BG. (Recallthatin the bipartite gam e

B

G the Survivorcan only startfrom a vertex^vS).Since^f forcesallverticesof^Gto be visited infinitely often,atleastone ofthe^vAor^vS isvisited infinitely often in^B forall^v2G.

N ow assum e thatthe Survivorhas a w inning strategy^f0for^BG starting atvertex

v

X.Every persistentsetofvertices^Y thatoccurw hen the Survivoruses^f0 isin^W. TheSurvivorcan sim ulate^f0(on^BG)forthegam e^Gw ith initialconfiguration^(v;X) and w in the gam e.

W e now define for any subsetof vertices^V0 of a bipartite gam e^G the closure Forced^(V0).Thisisthe setofvertices(containing^V0)thatthe Survivorhasa strat- egy to force the A dversary to visitatleastone vertex of^V0. W e have the follow ing algorithm to com puteForced^(V0).

algorithm FindForced(^V0V(G))forbipartitegraph^G=(S[A;E) 1 Q ueueN ew Verts=^V0

Set^{F =V0}

2 w hile Vertex^vin N ew Verts.head()do N ew Verts.rem ove(^v)

3 if^v2Athen

Set^F0= inN eighbors(^v) N ew Verts.append(^F0nF)

F =F[F 0

endif 4 if^v2Sthen

Set^F0=;

5 forVertex^uin inN eighbors(^v)do

ifoutN eighbors(^u)^F then ^F0=F0[fug endfor

N ew Verts.append(^F0nF)

F =F[F 0

endif endw hile return^F end

W e provethe correctnessofthisalgorithm below.

LEM M A4.2

A lgorithm FindForced com putesForced^(V0)fora bipartitegraph^G=(S[A;E). PRO O F.W e show thatforevery vertex^v,^vis in Forced^(V0)ifand only if^vis re- turned in^F by the algorithm FindForced.To do thisw e assign a num ber,called the rank,to each vertex of the graph. The rank indicates the num berofforced m oves needed to reach^V0 from a vertex. The^rankfunction is inductively defined as fol- low s:

1.If^v2V0then^rank(v)=0.

2.Case^{v 2 S} and^rank(v)is notdefined. A ssum e allout-neighborsof^v ofrank of atm ostⁱhave been defined. Then^{rank(v) = i+1}if there exists an^{u 2} outN eighbors^(v)w ith^rank(u)=i.

3.Case^{v 2 A} and^rank(v) is notdefined. A ssum e allout-neighbors of^v have defined rank.Then^rank(v)=i+1ifeach^u2outN eighbors^(v)has^rank(u)

i.

If^{v 2G}doesnotgetranked in the above processthen w e set^rank(v)=1.W e now show that^{v 2} Forced^(V0)ifand only if^{rank(v) <1}. Suppose^{rank(v) =}

n < 1. If^{n = 0}then^{v 2 V0}. O therw ise considertw o cases. If^{v 2 S} then^v is in the closure since atleastone neighbor^uof^v has sm allerrank (i.e.,the Survivor can m ove to^uand^{rank(u) < rank(v)}). If^{v 2 A}then^v is in the closure since allneighbors of^v have rank less thanⁿ(i.e.,any m ove ofthe A dversary m oves to

a vertex^uofrank less thanⁿ). Suppose^{rank(v) = 1}. W e w antto show thatthe A dversary has a strategy thatdoes notallow the Survivorto reach^V0. W e note that follow ing tw o observations. If^{v 2 S}then allneighborsof^v have rank equalto¹ by definition ofthe rank function (i.e.,the Survivorcan notreach^V0 from ^v). A lso by definition,if^{v 2 A}then there is atleastone neighbor^uof^vw ith rank equalto

1(i.e.,the A dversary can m oveto^uthatisnotin the closure).N ow the A dversary’s strategy to avoid^V0 is the follow ing.Forany^{v 2A}w ith^rank(v)=1m ove to a neighborvertex^usuch that^{rank(u) =1}. Clearly this strategy causes allplays to stay on a subsetofthe set^{V00=fwjrank(w)=1g}and^V00\V0=;.

O necan see thatthe algorithm FindForced addsa vertex^vto^Fifand only ifithas finite rank.The algorithm im plicitly labelsa vertex^vof^S[Aby the iteration count ofthew hileloop atline2 w hen^visadded to^F(thevertices^V0arelabeled w ith count 0). H ence ifa vertex is labeled then ithas finite rank. Statem ent3 ofthe algorithm correspondsto the case^v2Sand^{v 62V0}ofthe definition ofrank w hile Statem ents 4–5 correspond to the case^{v 2A} and^{v 62 V0}. This m eans thatif^vhas finite rank then itw illbe labeled by the algorithm .

LEM M A4.3

Forbipartite gam es,there existsan algorithm thatrunsin tim e^O(m),w here^misthe size ofthe graph,thatcom putesForced^(V0).

PRO O F.W e show how to m odify the algorithm FindForced to run in^O(m)tim e.The algorithm as listed needsto processeach vertex in the queue N ew Vertsatm ostonce and foreach ofthese vertices access its in-neighbors. So excluding the loop atline 5 the algorithm runs in^O(m)steps. The process tim e,as listed,to check w hether outN eighbors^{(u) F} takes atm ost^O(n)tim e. H ence,FindForced runs in tim e

O(nm).

W e now explain how to reduce the running tim e ofthe loop atline 5 ofalgorithm FindForcedto constanttim e.Instead ofcheckingthesetm em bershipoutN eighbors^(u)

F w e do the follow ing.W e keep an array ofintegers^Degthatindicatesforeach vertex how m any neighborsare notcurrently in^F.The entry forvertex^xisinitially defined as the out-degree of^x. W henevera vertex^y is added to^F w e decrem ent the entry for each in-neighbor^z of^y by one. W e can now replace the condition outN eighbors^(u)Fby testing w hether^Deg[u]=0,w hich can be donein constant tim e.

RecallLem m a 4.1 states thata gam e^G is an update netw ork if and only if the Survivorhasa w inning strategy forevery initialconfiguration of^BG.The nexttheo- rem also characterizesupdatenetw orks(notnecessarily bipartite gam es)by using the closureoperator.

TH EO R EM 4.4

A gam e^{G = (G;fV(G)g )}is an update netw ork if and only if for all^{v 2 V(G)} Forced^(fvS

;v

A

g)=V(B)in the corresponding bipartite gam e^BG

=(B;W). PRO O F.Supposethereisan updatenetw ork,w ith graph^G,such thatForced^(fvS

;v

A g)

6= V(B)forsom e^{v 2 V(G)}. Take any vertex^xof^Bthatdoes notbelong to this

closure.U sing theproofofLem m a4.2 w eseethattheSurvivorcan notforcetheplay to visit^vSor^vAfrom vertex^x2V(B).ThustheA dversary w insgam e^BGbeginning from ^x.By Lem m a 4.1 the graph^G(ofthe gam e^G)can notbe an updatenetw ork.

W e now prove the otherim plication ofthe theorem . By Lem m a 4.1,itsuffices to show thattheSurvivorcan w in thegam e^BGfrom any starting vertex.W e use thefact thatForced^(fvS

;v

A

g)=V(B),forall^v2V(G),to build aw inning strategy forthe Survivorin^BG.O rderthe verticesof^Gas^v1;v2;:::;vn.Let^xbe a starting vertex of^B. The Survivorcan use algorithm FindForced to visiteither^v1

S

or^v1

A

. N extthe Survivorcan force the play to visitto either^v2

S

or^v2

A

,then either^v3

S

or^v3

A

,etc.The Survivorthen repeatsthe forced playsbetw een the pairs(^vi

S

or^vi

A

)and (^{vi+1m odn}

S

or

v

i+1m odⁿ

A

)w hich yieldsa w inning setof^W.

U sing the previous lem m a and theorem w e can efficiently recognize update net- w orks.

TH EO R EM 4.5

Thereexistsan algorithm thatdecidesw hetheran updategam e^Gisan updatenetw ork in tim e^O(nm),w hereⁿand^mare the orderand size ofthe underlying graph.

PRO O F.W ecan constructthebipartitegraph^Bfrom thegam e^BG,w hich corresponds to^{G = (G;fV(G)g)},in lineartim e w ith respectto the size of^G. W e then invoke Lem m a 4.3 foreach pairofvertices^fvA

;v

S

gfor^{v 2V(G)}.By using Theorem 4.4, w e acceptthe inputif Forced^(fvS

;v

A

g) = V(B) for all^{v 2 V(G)}. The total running tim e is^n=jV(G)jm ultiplied by the tim e needed to com putethe closure(of tw o vertices^vAand^vS)in^B.Thisproductis^O(nm).

5 C onclusion

In thispaperw ehavepresented agam e-theoreticm odelofinfiniteduration processes.

A particularem phasis is given to a class ofnetw orks w hose objective is to continu- ously update allthe nodesw ith consistentdata.W e have show n thatitisalgorithm i- cally feasible to recognize update netw orks.Thatis,w e have provided an algorithm w hich solvestheupdategam eproblem in^O(nm)tim e.M oreover,ouralgorithm for the case ofbipartite update gam escan be used to give a characterization ofbipartite updatenetw orks.

There are m any open questions thatstillneed to be investigated in this area. For exam ple,onecan try to characterizethoseupdategam esforw hich theupdatenetw ork problem isdecidablein lineartim e.O necan also study thequestion offinding feasible algorithm s forgam es w hose w inning conditions are m ore com plex than the one for updategam es.Forthe lattercase,w e w antto efficiently extractw inning strategies(if they existforthe Survivor)foreach setofverticesin the w inning setofa gam e.

The gam esconsidered in this paperoccuroverfinite graphs. These gam escan be generalized to gam es overdifferentfinite m odels (such as hypergraphs). W e w ould like to know w hich ofthese generalized gam eproblem sare tractable.

A cknow ledgm ent

W ethank thetw o anonym ousrefereesw ho provided detailed and usefulcom m entson an earlierversion ofthispaper.

R eferences

[1] R.M cN aughton.Infinite gam esplayed on finite graphs.AnnalsofPure and Applied Logic,65,149–

184,1993.

[2] A .N erode,J.Rem m eland A .Yakhnis.M cN aughton gam es and extracting strategies forconcurrent program s.AnnalsofPureand Applied Logic,78,203–242,1996.

[3] R.J.B ¨uchiand L.H Landw eber.Solving sequentialconditions by finite-state strategies.Trans.Am er.

M ath.Soc.,138,295–311,1969.

[4] T.J.Sch¨afer.Com plexity ofsom e tw o-person perfect-inform ation gam es.J.Com puter & System Sci- ences,16,185–225,1978.

Received 1 N ovem ber1999.