Asynchronous iterations in ultrametric spaces

0.14 MB·

by  Alexander J. T. Gurney

Asynchronous iterations in ultrametric spaces
AlexanderJ.T.Gurney
January27,2017
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0 Abstract
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Someiterativecalculationscanbecarriedoutbyparallelcommunicatingprocessors,and
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yieldthesameresultswhetherornottheprocessorsaresynchronized. Weshowthatthisis
a
J thecaseifandonlyiftheiterationisacontractionthatisstrictonorbits,withrespecttoan

5 ultrametricdefinedonthestatespace. Themaximumnumberofindependentprocessorsis
2 givenbythedimensionofthespace.
Weapplythistheoremtointerdomainrouting,andareabletoprovidetwoadvancesover
]
I the previous state of the art. Firstly, multipath routing problems have unique solutions, if
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certain conditions are satisfied that are analogous to known correctness conditions for the
. single-pathcase. Secondly,thesesolutionscanbecomputedasynchronouslyinavarietyof
s
c ways,whichgobeyondmethodsthatarecurrentlyused.
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1 Introduction
The theoryof asynchronousiterationsis concernedwith the problem of when an iterative algo-
rithmcanbeimplementedonasetofcommunicatingprocessors, withoutexplicitsynchroniza-
tion,andyetstill computethesameresult. Itisknownthatforthistobepossible, certainchar-
acteristics of the iteration must hold with respect to its state space: several different sufficient
conditions areknown. Thesearespecialcases ofamoregeneralresult, whichgives anecessary
and sufficient condition for asynchronous safety. It requires that the state space have a ‘nested
box’structure,andthatsynchronousiterationsalwaysleadtoamore-inwardbox. Thiscondition
is rather ‘low-level’, and may be difficult to verify in many cases; equally, the various sufficient
conditions areeasiertoworkwith, butdonotaccountforallpossibilities. Inaddition, muchof
thepriorworkon theseiterationsassumesthatdatavalues arerealnumbers, whereasthereare
manyiterativealgorithmsthatworkoverotherkindsofdata.
Inthispaper,wereinterpretthe‘nestedbox’structureintermsofaspecialkindofmetricon
thestatespace. Insuchanultrametric,theballsaroundagivenpointalwaysformnestedboxes.
TheBanachfixedpointtheoremforultrametricspaces,whichstatesthataself-mapofthespace
that is contractive and strict on orbits must have a unique fixed point, is precisely the theorem
neededtoproveasynchronoussafety.Thatis,theapplicationofthistheoremprovesnotonlythat
thereisauniquefixedpoint,butalsothatitcanbefoundbyasynchronousiteration. Conversely,
wheneveraniterationisasynchronouslysafe,thenanultrametriccanbedefinedwithrespectto
which the iteration is a contraction of the required kind. Furthermore, the degree of potential
asynchrony(thenumberofprocessorsacrosswhichtheiterationcanbepartitioned)isgivenby
thedimensionoftheultrametricspace. Thisresultappliestodiscretedataaswellastonumeric
problems.
In the final partof the paper, we apply thisnew theoremto a problem in interdomain mul-
tipath routing. The presence of a unique fixed point, let alone the possibility of asynchronous
implementation,wasnotpreviouslyknown.Existingsufficientconditionsdidnotcoverthiscase,
butitisdealtwithbythenewtheorem.
2 Background
Thissectionexplainsthetwoseparateareasoftheory—asynchronousiterations(Section2.1)and
ultrametricspaces(Section2.2)—whichareinvolvedinthemainresultsofthispaper.
2.1 Asynchronousiterations
Therearemanyalgorithmswhichoperatebyiterativelyapplyingthesamefunctiontosomestate
vector.IfthestatespaceisM=M ×M ×···×M ,thenafunctionσfromM toM canbedecom-
1 2 k
posedastheproductofk functionsσ :M→M ,where
i i
σ(m ,m ,...,m )=(σ (m ,m ,...,m ),σ (m ,m ,...,m ),...,σ (m ,m ,...,m )).
1 2 k 1 1 2 k 2 1 2 k k 1 2 k
On a sequential machine, each σ function must be evaluated in turn in order to produce the
i
newstate vector. But if multiple processors are available, theneach function evaluation can be
assigned to a different processor; once they have all finished, they can mutually communicate
theirresultssothatthenextiterationstepcanbegin.Thetotalexecutiontimeforasingleiteration
isthusboundedbythetimetakenfortheslowestprocessor.
2
It haslong been known thatfor some algorithms, synchronous execution is not required for
convergence.Thatis,thesameanswerasproducedbytheaboveprocesscanalsobegeneratedin
afarlessrestrictiveexecutionmodel. Intheasynchronousexecutionofthesamealgorithm,the
samefunctionisexecutedoneachprocessor,buttheexecutionisnolongerinlock-stepandthe
inputdatamaycomefromapriorroundoftheiteration.
Asynchronousiterationhasbeenappliedtomanyproblems: findingshortestpaths[22],dy-
namicprogramming[3],findingfixedpointsoflinearandnon-linearoperators[1],PageRank[13],
andseveralothers[8].Useofthemethodisoftenmotivatedbythelargequantityofdatainvolved,
orbythedifficultyinensuringsynchronizedexecution.Sometimes,particularlyfornumericprob-
lems,convergencetimecanbeimprovedbydroppingthesynchronyrequirement.Whilethethe-
orywasoriginally developedfor numericiterations [1,2, 5], resultshave also been obtainedfor
iterationsondiscretedata[23].
Inthefollowing,wetaketimetobediscreteandlinear;thesetT containsalltimevalues.
Definition1. ForasetP ofprocessors,anasynchronousexecutionscheduleconsistsoftwofunc-
tionsαandβ,where
• α:T →2P yieldsthesetofprocessorswhichactivateateachtimestep,and
• β :T ×P×P → T yields the delay between two given processors at each time step; so if
β(t,i,j)=t′,thenthedatafrom j usedati attimet wasgeneratedattimet′.
Definition2. Aschedule(α,β)onP isadmissibleif
1. Foralli inP,andt inT,thereexistst′>t suchthati isinα(t′).
2. Forallt inT,andi and j inP,β(t,i,j)>t.
3. Foralli and j inP,andt′inT,thereexistsat inT suchthatift>t ,thenβ(t,i,j)6=t′.
f f
These admissibility conditions may be expressed more informally, as: every node activates
infinitelyoften;informationdoesnotpropagatebackwardsintime;andapastdatavaluecanonly
beusedfinitelyoften. Some weakerversionsoftheseconditionshavealsobeenconsidered; for
example,thefinalaxiommaybereplacedbyanupperboundontheageofanydataitemusedin
acalculation[23].
IfσisanfunctionfromM toM,thenwecandefineanasynchronousiterationcorresponding
to σfor any given schedule, and for a particularstartingpoint in M. For each i, there will be a
seriesofvaluesinM generatedbytheiteration;callthesex (t)fort∈T.Letx(t)betheirproduct,
i i
sox(t)isavectorinM.
Let m be a point in M, and let (α,β) be a schedule on a set of k processors. For each i, let
x (0)=m .Fort>0wedefinex (t)by
i i i
x (t−1) i 6∈α(t)
i
x (t)=
i
(σi x1(β(t,i,1)),x2(β(t,i,2)),...,xk(β(t,i,k)) i ∈α(t).
¡ ¢
So if processor i does not update at time t, its value does not change; when it does update, it
carriesoutitsusualσ operation,butmayusevaluesfromfurtherinthepastthantheimmediately
i
precedingstep.
Notethatifα(t)={1,2,...,k}forallt,andβ(t,i,j)=t−1forallt,i and j,thenthisisjustthe
synchronousiterationfromabove.
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Definition 3. Let M = M , for some sets M . Let σ be a function from M to M that has a
i∈I i i
uniquefixedpointm∗. Thenσisanasynchronouslycontractingoperator(ACO)if,foranyadmis-
Q
sibleschedule(α,β)on|I|,andanystartingpointminM,thereissometimeT suchthatfor
m,α,β
anyt>T ,thestatex(t)isequaltom∗.
m,α,β
TheseACOsmayalsobecharacterizedintermsofa‘nestedbox’structureonthestatespace.
Informally,ifthesynchronousiterationissuchthatitalwaystakespointsintoamore-inwardbox,
thenitisasynchronouslysafe. Theasynchronousiterationswilleventuallyleadtothesamefixed
point,butmaydeviatefromthesynchronouscourseofexecution.
Definition4. AsubsetN ofM= M isaboxif,foreachi,thereisasubsetN ofM suchthat
i∈I i i i
N = N .
i∈I i Q
TheoQrem1. AnoperatorσonM = M isanACOifandonlyifthereexistboxes{C ,C ,...,C }
i∈I i 0 1 k
inM withthefollowingproperties:
Q
1. C = m∗ forsomem∗inM.
0
2. C =©M. ª
k
3. If0≤r <s≤k,thenC ⊂C .
r s
4. IfmisinCr+1,thenσ(m)isinCr;andifmisinC0thenσ(m)isinC0.
Proof. See[23].
Thispowerfultheoremmeansthatwecandetermineasynchronouscorrectness,withouthav-
ingtoreasonaboutasynchronousprocesses. Wemerelyhavetoverifythatthesynchronousiter-
ationconverges,andthatcertainconditionsholdforthestatespace. Unfortunately,thesecondi-
tionsmayberathertrickytoapplyinpractice—particularlyifonewantstodemonstratethatan
operator is not an ACO. A furtherproblem is thatthe low-level way in which the conditions are
statedmakesitdifficulttounderstandtheclassofACOsingeneral.
SeveralsufficientconditionsareknownwhichimplythatthecriteriaofTheorem1arefulfilled.
NumericexamplesincludeweightedmaximumnormsoverBanachspaces[6,8],P-contractions[1],
paracontractions[7,17],andisotonemappings[16].Fordiscretedata,therearealsoresultsabout
isotone mappings [23]. All of these impose further requirements on the state space and itera-
tion;thesituationisespeciallypainfulforiterationsondiscretedata,sincenoneoftheusualreal-
numberapparatusisavailable: wedonothavecontinuity,norms,orevensubtraction. Theresult
ofSection3isanecessaryandsufficientconditionforiterationsondiscretedata,soitcoversiter-
ationsthatarenotnecessarilyisotonebutevensomanagetoconvergetoauniquefixedpoint.
2.2 Ultrametricspaces
Anultrametricisaparticularkindof‘distance’measurementthatdiffersinseveralimportantre-
spectsfrommorefamiliarexamples,butwhichalsohasusefulapplications.
Aconventionalmetricspaceallowsonetomeasurethedistancebetweentwopointsasareal
number, with certain intuitive properties being fulfilled: a zero distance means the points are
identical, distances are symmetric, and the triangle inequality is valid. In an ultrametricspace,
the triangle inequality is strengthened. If l, m, and n are three points in the space, then they
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must form an isosceles triangle: two of the distances d(l,m), d(m,n) and d(n,l) are the same.
Furthermore,theremainingdistancecanbenolongerthantheothers,sothetriangleis‘longand
narrow’asopposedto‘shortandwide’. Forthisreason,thespacesaresometimescalledisosceles
spaces.
Definition5. Anultrametricspace(M,d,Γ)consistsofaset M,atotallyorderedsetΓwithleast
element0,andafunctiond:M×M→Γsuchthat
1. d(m,n)=0ifandonlyifm=n
2. d(m,n)=d(n,m)
3. d(l,n)≤max(d(l,m),d(m,n))
foralll,mandninM.
AcanonicalexampleiswhenM isthesetofallstringsoversomealphabet. Fordistinctx and
y inM,let
m(x,y)=min i ∈N x 6=y
i i
somyieldsthefirstindexatwhichthetwostring©sdiffe¯r.Thenª
¯
0 x=y
d(x,y)=
(2−m(x,y) x6=y
isanultrametricdistancefunction.
Indeed,thisexampleisveryclosetobeinguniversal:anyultrametricspaceM isisometrictoa
spacewheretheelementsarefunctionsfromQ≥0toM,andthedistanceisgivenby
d(f,g)=sup q∈Q≥0 f(q)6=g(q) .
© ¯ ª
See[14]formoredetails. Some otherexamplescom¯e fromBoolean algebra, wherethedistance
betweentwoelementscanbedefinedintermsoftheirsymmetricdifference[20].
IfhisafunctionfromM toΓ\{0},thenadistancefunctioncanbedefinedby
0 m=n
d (m,n)=
h
(max(h(m),h(n)) m6=n.
Thisisclearlyanultrametric.
Definition6. Inanultrametricspace(M,d,Γ),theballaboutapointm inM,ofradiusr inΓ,is
theset
B(m;r)={n∈M|d(m,n)≤r}.
Theballsofanultrametricspacehavesomesurprisingproperties.Anypointinaballwillserve
asitscenter(seethelemmabelow). IfwehavetwoballsB(n;r)andB(m;s),theneitheroneisa
subsetoftheother,ortheyaredisjoint.Consequently,thesetofallballs,orderedbyinclusion,has
atreestructure[15].
Lemma2. IfnisinB(m;r)thenB(n;r)=B(m;r).
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Proof. IfnisinB(m;r)thend(m,n)≤r.ThenforanyzinM,
d(m,z)≤max(d(m,n),d(n,z))
so if d(n,z)≤r then d(m,z)≤r as well; and the same argument applies if m and n are inter-
changed.Hencethetwoballshavethesamecontent.
Wewillneedsomenotionofcompletenessofanultrametricspace,inordertoguaranteethe
existenceoffixedpoints. Otherwise,itcouldbethatforcertainiterations,thesequenceofvalues
convergestoapointthatisnotinthespace. Thefollowingdefinitionissufficienttoensurethat
theBanachfixed-pointtheoremactuallyyieldsafixedpoint.
Definition 7. An ultrametric space is sphericallycomplete if every chain of balls has nonempty
intersection.
Simple examplesof sphericallycomplete spacesinclude anyspace thatis finite, andanyfor
whichtheimageofd isafinitesubsetofΓ.Inbothofthesecases,anychainofballsisguaranteed
tobefinite,anditsintersectionisthenequaltothesmallestballinthechain.
The Banach theorem for ultrametric spaces can be made to work for several different kinds
of contracting operator. The general idea is that, with respect to the ultrametric distance, each
applicationoftheoperatorbringspointsclosertogether.Eventually,theentirespaceiscontracted
intoasinglepoint,whichisthedesireduniquefixedpoint.
Definition8. LetσbeafunctionfromM toM.If(M,d,Γ)isanultrametricspace,thenσis(with
respecttod):
• acontractionifd(σ(m),σ(n))≤d(m,n)forallmandninM
• astrictcontractionifd(σ(m),σ(n))<d(m,n)foralldistinctmandninM
• astrictcontractiononorbitsifd(σ(m),σ2(m))<d(m,σ(m)),orm=σ(m),forallminM.
Notethatifσisastrictcontraction,thenitisnecessarilyacontractionthatisstrictonorbits.
Theorem 3. If σ is a function from M to M, and (M,d,Γ) is a spherically complete ultrametric
space with respect to which σ is a contraction that is strictly contracting on orbits, then σ has a
uniquefixedpoint.
Proof. See[18]and[19].
Wenowdemonstratethatultrametricspacescanbecombinedviaaproductoperation. Fur-
thermore, in the resulting space, every ball is a box. This provides the desired connection with
thetheoryofasynchronousiterations: theballsaboutthefixedpointwillbepreciselytheboxes
demandedbytheasynchronousiterationtheorem.
Definition9. Givenultrametricspaces(M ,d ,Γ)for1≤i ≤k,definetheultrametricproductspace
i i
(M,d,Γ)by
M= M
i
1≤i≤k
Y
d(m,n)= maxd (m ,n )
i i i
1≤i≤k
wheremandnarevectorsinM.
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Wewillrefertok asthedimensionof M. Thisusageisappropriateforthecase when M isa
realorcomplexvectorspace. WhenM isdiscrete,itstopologicaldimensioniszero;buthere,we
willcarryonusingtheterm‘dimension’forthenumberofcomponentsoftheproduct.
Lemma4. Inanultrametricproductspace,everyballisabox. Thatis,if(M,d,Γ)istheproductof
(M ,d ,Γ)for1≤i ≤k,thenforanyminM andr inΓ,
i i
B(m;r)= B
i
1≤i≤k
Y
whereeachsetB isasubsetofM .
i i
Proof. Foreachi,letB beB(m ;r)inM .ForanyelementxofM,wehave:
i i i
x∈B(m;r)
⇐⇒r ≥d(x,m)
⇐⇒r ≥ maxd (x ,m )
i i i
1≤i≤k
⇐⇒∀i :1≤i ≤k =⇒ r ≥d (x ,m )
i i i
⇐⇒∀i :1≤i ≤k =⇒ x ∈B(m ;r)
i i
⇐⇒∀i :1≤i ≤k =⇒ x ∈B
i i
⇐⇒x∈ B .
i
1≤i≤k
Y
3 Themainresult
Thissectionisdedicatedtoprovingthetheorembelow,whichprovidesanecessaryandsufficient
conditionforanoperatortobeanACO,intermsofanultrametricstructureonthestatespace.
Theorem5. LetM beaset,andσ:M →M afunctionfromM toM. Thenσisanasynchronously
contractingoperatoronM ifandonlyifthereexistsanultrametricd onM,withfiniteimage,and
withrespecttowhichσisacontractionthatisstrictonorbits.
We will prove the two directions of this theoremseparately. In order to establish this result,
wewillneedtousethepropertythatanoperatoronM isasynchronouslycontractingifandonly
ifaseriesofnestedboxesinM exist,withcertainproperties. Theexistenceofanultrametricwill
providetheseboxes,andconversely,givenaseriesofboxes,wecandefineasuitableultrametric.
Lemma6. If(M,d,Γ)isaultrametricspace,withrespecttowhichσisacontractionthatisstricton
orbits,andΓisfinite,thenaseriesofboxesexiststhathastherequiredproperties.
Proof. Fromthetheoryofultrametricspaces,thecontractionconditionsonσprovidethatithas
auniquefixedpointinM;callthism∗. Foreverypossibleradiusr inΓ,thereisaballofradiusr
aboutm∗:
B(m∗;r)= m∈M d(m∗,m)≤r .
Theseballswillbetherequiredboxes. © ¯ ª
¯
Firstly,B(m∗;0)= m∗ ,sincenootherpointsareatdistancezerofromm∗itself.
Next,duetofinitenessofΓ,theremustbesomeminimalradiusk suchthatB(m∗;k)=M.Let
© ª
R be theset {r |0≤r ≤k}. Clearly, if r <r forsome r andr in R, thenB(m∗;r )⊆B(m∗;r ).
1 2 1 2 1 2
Becausesomeoftheseballsmaycoincide,despitehavingdifferentradii,wedefineasubsetSofR
suchthat
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1. ifr isinR thenthereissomesinSsuchthats≤r andB(m∗;r)=B(m∗;s);and
2. foranys ands inS,B(m∗;s )6=B(m∗;s ).
1 2 1 2
ThisSnowyieldstherequiredsequenceofboxes;itiswell-definedsinceR isfinite.
Itremainstoshowthatσfulfilstherequiredproperty.ForanyminM,wehavetherelationship
d(m,σ(m))=d(m,m∗)
and,ifmisnotequaltoσ(m),
d(m,σ(m))>d(σ(m),σ2(m)).
Hence
d(m,m∗)>d(σ(m),m∗)
unlessm andσ(m)areequal(inwhichcasetheyarebothequaltom∗ itself). Consequently,ap-
plication of σalways takesapoint toa more-inwardball, unlessthatpointisthefixedpointal-
ready.
Thenextstepistoprovetheconverse:thatifanoperatorisasynchronouslycontracting,then
thereisanultrametricwithrespecttowhichtheoperatorisacontractionthatisstrictonorbits.
Lemma7. LetM beasetendowedwithanasynchronouslycontractingoperatorσ.Thenthereisan
ultrametricd onM withrespecttowhichσisacontractionthatisstrictlycontractingonorbits.
Proof. Sinceσisanasynchronouslycontractingoperator,thereexistsaseriesofnestedboxeswith
certainproperties.Wewilldefineanultrametricdistancefunctionthatusesthisboxstructure.
LetC ,C ,...,C betheboxsequence,whereC isasingletonset,C =M,andC ⊂C when-
0 1 k 0 k i j
ever0≤i <j ≤k.
ForanypointminM,wecanfindtheindexoftheinnermosthypercubethatcontainsm:
C(m)=min{i |m∈C }.
i
ThusC(m∗)=0ifandonlyifm∗istheuniquefixedpoint.Therequireddistancefunctionis
0 m=n
d (m,n)=
C
(max(C(m),C(n)) m6=n
whichisanultrametric.Notethatthereareonlyfinitelymanypossibleradii.
Wecannowshowthattheballsaboutm∗ withrespecttod areprecisely thegivenboxesC .
i
Foranyradiusr,
m∈B(m∗;r) ⇐⇒ d(m,m∗)≤r ⇐⇒ C(m)≤r ⇐⇒ min{i |m∈C }≤r ⇐⇒ m∈C .
i r
Finally, we prove that σ must be a contraction that is strictly contracting on orbits. Note that
C(σ(m)) <C(m) for all m other than m∗. Hence d(σ(m),m∗) < d(m,m∗) for such m. Since
d(m,m∗)=d(m,σ(m))forallm,wehave
d(m,σ(m))>d(σ(m),σ2(m))
8
wheneverm6=σ(m),soσisstrictlycontractingonorbits. Similarly,ifm andn aretwopointsin
M,thenC(σ(m))≤C(m)andC(σ(n))≤C(n).ThereforethelargerofC(σ(m))andC(σ(n))cannot
exceedthelargerofC(m)andC(n);weobtain
d(σ(m),σ(n))≤d(m,n)
asdesired.
Thiscompletestheproof. Wehaveestablishedthatanoperatorisasynchronouslycontract-
ingifandonlyifitfulfilstheultrametricBanachfixedpointtheorem. Furthermore,thedegreeof
asynchronyisgivenbythedimensionoftheultrametricspace. Thisprovidesaconvenientproof
techniqueforasynchronousiterations,subsumingseveralotherpreviously-knownspecialcondi-
tions.
4 Recoveryofprevioustheorems
The general theorem of Section 3 has many specific consequences for particularclasses of iter-
ation. Several of these have previously been studied in the literature. In this section, we relate
previousresultstothenewtheory,therebydemonstratingitsgenerality.
4.1 Paracontractions
Inthecaseofiterationsoverrealvectors,thereisawell-knowntheoryofparacontractingopera-
tors. Inthefollowing,letM beRn ,wheren≥1. Forv inM,letkvkbethevectorwhoseithentry
≥0
istheabsolutevalueofv .Fortwovectorsv andw inM,saythatv≤w ifforalli,v ≤w .
i i i
Definition10. AfunctionσfromM toMiscalledaparacontractionifthereexistsannbynmatrix
P withentriesinR≥0,havingspectralradiuslessthan1,andforwhich
σ(x)−σ(y) ≤P x−y .
° ° ° °
Theorem8. Ifσisaparacontraction°onM,then(°M,d)°isanu°ltrametricspacewithrespecttowhich
σisastrictcontraction,where
d(x,y)= max α x−y .
1≤i≤n i i
° °
Proof. Ifσisparacontracting,thenitisastrictcontr°action°withrespecttoaweightedmaximum
norm[1,2].Sothepreviousresultsufficestoprovethisone.
4.2 Weightedmaximumnorms
Suppose that the set M is a product of sets M through M , each of which is equipped with a
1 n
real-valuednormk·k .Foranyrealnumbersα ,anormcanbedefinedonM by
i i
kxk=maxα kx k .
i i i
i
AnoperatorσonM isLipschitzifthereexistssomep,with0≤p<1,suchthat
σ(x)−σ(y) ≤p x−y
° ° ° °
forallxandy inM. ° ° ° °
9
Theorem9. IfσisLipschitzwithrespecttoaweightedmax-norm,thenitisalsoastrictcontraction
withrespecttoanultrametricdistanceonM,ofdimensionn.
Proof. Foreachi,letd betheultrametriconM givenby
i i
d (x,y)=α x−y .
i i i
° °
° °
4.3 Monotoniccontractions
Let≤denotethedirectproductorderonvectorsin(R≥0)n,sothat
x≤y ⇐⇒ ∀i :x ≤ y .
i i i
AfunctionσonM ismonotonicif
x≤y =⇒ σ(x)≤σ(y)
forallx andy inM. Ifσisamonotonicfunction,withauniquefixedpointx∗,and(othercondi-
tions)thenitisanasynchronouslycontractingoperator.
Thisconclusionalsofollowsfromthetheoremofthispaper. Toconstructtherequiredultra-
metric, we can define ultrametrics on each component, and then take their max-product. The
individualultrametricscanbedefinedintermsofthenaturalorderingonR≥0.
Leth(x)=2−x. Notethatforx greaterthanorequaltozero,h(x)isintherange(0, 1]. There-
fore,thefunction
0 x=y
d(x,y)=
(max(h(x),h(y)) x6=y
isanultrametricdistancefunctiononR≥0;andsotheproductofnoftheseisalsoanultrametric,
on(R≥0)n.Theconditionsonσimplythatitiscontractivewithrespecttothisultrametric.
5 Applicationtointerdomainmultipathrouting
Wewillnowseeanextendedexampleoftheuseoftheultrametrictheoremtoproveasynchronous
safetyofaniteration. Theiterationinquestionissimpletodescribe,buthassomeunusualprop-
ertieswhichmakeitunsuitableforhandlingbypreviously-knownasynchronytheorems.
Theproblemcomesfromtheselection ofpathsattheinterdomainlevelin networkrouting.
ThevariousnetworkswhichcombinetomaketheInternetcarryoutpathselectioninawaywhich
provides a great deal of local autonomy: the paths for a given source-destination pair could in
principleberankedarbitrarily.Becauseofthelocalnatureofpreferencesanddecisions,theover-
allroutingoutcomeisnotaglobaloptimum,asforshortest-pathalgorithms,butisaNashequi-
libriumbetweenthenetworksinvolved[10].Thisproblemisalsoinherentlydistributedandasyn-
chronous:theprivatenatureofpolicymeansthatthecomputationcannotbeperformedcentrally,
andsynchronizationonaglobalscaleisinfeasible.
In execution of the path selection process, even synchronously, various anomalies appear
which would be impossible for shortest path algorithms. It is not necessarily the case that the
pathsselected by a given node improve over time: a node is perfectlycapable of switching toa
10