The ANTS problem

0.67 MB·

by  Ofer Feinerman

The ANTS Problem∗
OferFeinerman AmosKorman
WeizmannInstituteofScience CNRSandUniv. ParisDiderot
7 [email protected] [email protected]
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0
2

n
a Abstract
J

0 WeintroducetheAntsNearbyTreasureSearch(ANTS)problem,whichmodelsnaturalcooperative
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foraging behavior such as that performed by ants around their nest. In this problem, k probabilistic

] agents,initiallyplacedatacentrallocation,collectivelysearchforatreasureonthetwo-dimensional
C
grid. Thetreasureisplacedatatargetlocationbyanadversaryandtheagents’goalistofinditasfastas
D
possibleasafunctionofbothkandD,whereDisthe(unknown)distancebetweenthecentrallocation
.
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andthetarget. Weconcentrateonthecaseinwhichagentscannotcommunicatewhilesearching. Itis
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[ straightforwardtoseethatthetimeuntilatleastoneagentfindsthetargetisatleastΩ(D+D2/k),even

1 forverysophisticatedagents,withunrestrictedmemory. Ouralgorithmicanalysisaimsatestablishing
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connectionsbetweenthetimecomplexityandtheinitialknowledgeheldbyagents(e.g.,regardingtheir
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5 totalnumberk),astheycommencethesearch. Weprovidearangeofbothupperandlowerboundsforthe
5 initialknowledgerequiredforobtainingfastrunningtime. Forexample,weprovethatloglogk+Θ(1)
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bitsofinitialinformationarebothnecessaryandsufficienttoobtainasymptoticallyoptimalrunningtime,
0
. i.e.,O(D+D2/k).Wealsoweprovethatforevery0<(cid:15)<1,runningintimeO(log1−(cid:15)k·(D+D2/k))
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0 requiresthatagentshavethecapacityforstoringΩ(log(cid:15)k)differentstatesastheyleavethenesttostartthe
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search. Tothebestofourknowledge,thelowerboundspresentedinthispaperprovidethefirstnon-trivial
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: lowerboundsonthememorycomplexityofprobabilisticagentsinthecontextofsearchproblems.
v
i We view this paper as a “proof of concept” for a new type of interdisciplinary methodology. To
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fullydemonstratethismethodology,thetheoreticaltradeoffpresentedhere(orasimilarone)shouldbe
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a combinedwithmeasurementsofthetimeperformanceofsearchingants.
∗Preliminaryresultsofthispaperappearedintheproceedingsofthe31stAnnualACMSIGACT-SIGOPSSymposiumon
PrinciplesofDistributedComputing(PODC),2012, andthe26thinternationalsymposiumonDistributedComputing(DISC)
2012,aspartof[28,29].O.F.,incumbentoftheShlomoandMichlaTomarinCareerDevelopmentChair,waspartiallysupported
bytheCloreFoundationandtheIsraelScienceFoundation(grant1694/10). A.K.waspartiallysupportedbytheANRprojects
DISPLEXITYandPROSE,andbytheINRIAprojectGANG.ThisprojecthasreceivedfundingfromtheEuropeanResearchCouncil
(ERC)undertheEuropeanUnion’sHorizon2020researchandinnovationprogramme(grantagreementNo648032).
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1 Introduction
Theuniversalityofsearchbehaviorisreflectedinmultitudesofstudiesindifferentfieldsincludingcomputer
science, robotics, and biology. We use tools from distributed computing to study a biologically inspired
scenarioinwhichmultipleagents,initiallylocatedatonecentrallocation,cooperativelysearchforatreasure
intheplane. Thegoalofthesearchistolocatenearbytreasuresasfastaspossibleandataratethatscales
wellwiththenumberofparticipatingagents.
Manyanimalspeciessearchforfoodaroundacentrallocationthatservesasthesearch’sinitialpoint,
finaldestinationorboth[58]. Thiscentrallocationcouldbeafoodstoragearea,anestwhereoffspringare
rearedorsimply ashelteredorfamiliarenvironment. Centralplace foragingholdsastrong preferenceto
locatingnearbyfoodsourcesbeforethosethatarefurtheraway. Possiblereasonsforthatare,forexample:
(1)decreasingpredationrisk[54],(2)increasingtherateoffoodcollectiononcealargequantityoffoodis
found[58],(3)holdingaterritorywithouttheneedtoreclaimit[38,54],and(4)facilitatingthenavigating
backaftercollectingthefoodusingfamiliarlandmarks[18].
Searching in groups can increase foraging efficiency [44, p. 732]. In some extreme cases, food is so
scarcethatgroupsearchingisbelievedtoberequiredforsurvival[20,45]. Proximityofthefoodsourceto
thecentrallocationisagainimportantinthiscase. Forexample,inthecaseofrecruitment,anearbyfood
sourcewouldbebeneficialnotonlytotheindividualthatlocatedthesourcebutalsoincreasethesubsequent
retrievalrateformanyothercollaborators[72]. Foragingingroupscanalsofacilitatethedefenseoflarger
territories[69].
Eusocial insects (e.g., bees and ants) often live in a single nest or hive, which naturally makes their
foraging patterns central. They further engage in highly cooperative foraging, which can be expected as
these insects reduce competition between individuals to a minimum and share any food that is found. In
manycases,thiscollaborativeeffortisdonewithhardlyanycommunicationbetweensearchers. Forexample,
mid-searchcommunicationofdesertantsCataglyphysandhoneybeesApismelliferaishighlylimiteddueto
theirdispersionandlackofchemicaltrailmarkings[62]. Conversely,itisimportanttonotethattheseinsects
dohaveanopportunitytointeractamongstthemselvesbeforetheyleavetheirnestaswellassomecapacity
toassesstheirownnumber[61].
Both desert ants and bees possess many individual navigation skills. These include the capacity to
maintain a compass-directed vector flight [15, 42], measure distance using an internal odometer [70, 71],
travel to distances taken from a random power law distribution [67], and engage in spiral or quasi-spiral
movementpatterns[65,66,75]. Finally,thesearchtrajectoriesofdesertantshavebeenshowntoincludetwo
distinguishablesections: alongstraightpathinagivendirectionemanatingfromthenestandasecondmore
tortuouspathwithinasmallconfinedarea[42,74].
TheANTSProblem. Inthispaper,wetheoreticallyaddressgeneralquestionsofcollectivecentralplace
searching. Moreprecisely,weintroducetheAntsNearbyTreasureSearch(ANTS)problem,ageneralization
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of the cow-path problem [21, 47, 48], in which k identical (probabilistic) agents, initially placed at some
central location, collectively search for a treasure in the two-dimensional grid (modeling the plane). The
treasureisplacedbyanadversaryatsometargetlocationatdistanceD fromthecentrallocation,whereD is
unknowntotheagents. Traversinganedgeofthegridrequiresoneunitoftime. Thegoaloftheagentsisto
findthetreasureasfastaspossible,wherethetimecomplexityisevaluatedasafunctionofbothk andD.
Inthisdistributedsetting,weareconcernedwiththespeed-upmeasure(seealso,[6,7,31,32]),which
aimstocapturetheimpactofusingk searchersincomparisontousingasingleone. Notethattheobjectives
ofquicklyfindingnearbytreasuresandhavingsignificantspeed-upmaybeatconflict. Thatis,inorderto
ensurethatnearbytreasuresarequicklyfound,alargeenoughfractionofthesearchforcemustbedeployed
nearthecentrallocation. Inturn,thiscrowdingcanpotentiallyleadtooverlappingsearchesthatdecrease
individualefficiency.
It is a rather straightforward observation that the minimal time required for finding the treasure is
Ω(D +D2/k). Indeed, one must spend D time merely to walk up to the treasure, and, intuitively, even
if the agents divide the domain perfectly among themselves, it will still require them Ω(D2/k) steps to
exploreallpointsatdistanceD. Ourfocusisonthequestionofhowagentscanapproachthisboundiftheir
communicationislimitedorevencompletelyabsent. Specifically,weassumethatnocommunicationcanbe
employedbetweenagentsduringtheexecutionofthealgorithm,but,ontheotherhand,communicationis
almostunlimitedwhenperformedinapreliminarystage, beforetheagentsstarttheirsearch. Theformer
assumption is inspired by empirical evidence that desert ants (such as Cataglyphys Niger or Cataglyphis
savignyi)rarelycommunicateoutsidetheirnest[42],andthelatterassumptionisveryliberal,aimingtocover
manypossiblescenarios.
Tosimulatetheinitialstepofinformationsharingwithinthenestweusetheabstractframeworkofadvice.
Thatis,wemodelthepreliminaryprocessforgainingknowledgebymeansofanoraclethatassignsadvice
toagents. Tomeasuretheamountofinformationaccessibletoagents,weanalyzetheadvicesize,namely,the
maximumnumberofbitsusedinanadvice1. Sincewearemainlyinterestedinlowerboundsontheadvice
sizerequiredtoachieveagiventimeperformance,weapplyaliberalapproachandassumeahighlypowerful
oracle. Morespecifically,eventhoughitissupposedtomodeladistributedprobabilisticprocess,weassume
thattheoracleisacentralizedprobabilisticalgorithm(almostunlimitedinitscomputationalpower)thatcan
assigndifferentagentswithdifferentadvices. Notethat,inparticular,byconsideringidentifiersaspartof
theadvice,ourmodelallowstorelaxtheassumptionthatallagentsareidenticalandtoallowagentstobe
ofseveraltypes. Indeed,inthecontextofants,ithasbeenestablishedthatontheirfirstforagingboutsants
executedifferentprotocolsthanthoseexecutedbymoreexperiencedants[74].
1Notethatalowerboundofx≥1bitsontheadvicesizeimpliesalowerboundof2x−1onthenumberofstatesusedbythe
agentsalreadywhencommencingthesearch.
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1.1 OurResults
Weusetoolsfromdistributedcomputingtostudyabiologicallyinspiredscenariowhichmodelscooperative
foragingbehaviorsuchasthatperformedbyantsaroundtheirnest. Morespecifically,weintroducetheANTS
problem,anaturalprobleminspiredbydesertantsforagingaroundtheirnest,andstudyitinasimple,yet
relativelyrealistic,model.
Technical Contribution. We establish connections between the advice size and the running time2. In
particular,tothebestofourknowledge,ourlowerboundsontheadvicesizeconsistofthefirstnon-trivial
lowerboundsonthememoryofprobabilisticsearchers.
Ontheoneextreme,wehavethecaseofuniformsearchalgorithms,inwhichtheagentsarenotassumed
tohaveanya-prioriinformation,thatis,theadvicesizeiszero. Wecompletelycharacterizethespeed-up
penalty that must be paid when using uniform algorithms. Specifically, we show that for a function Φ
such that P∞ 1/Φ(2j) converges, there exists a uniform search algorithm that is Φ(k)-competitive. On
j=1
the other hand, we show that if P∞ 1/Φ(2j) diverges, then there is no uniform search algorithm that is
j=1
Φ(k)-competitive. Inparticular,weobtainthefollowingtheorem,implyingthatthepenaltyforusinguniform
algorithmsisslightlymorethanlogarithmicinthenumberofagents.
Theorem 1.1. For every constant ε > 0, there exists a uniform search algorithm that is O(log1+εk)-
competitive,butthereisnouniformsearchalgorithmthatisO(logk)-competitive.
Infact,O(logk)-competitivealgorithmsnotonlyrequiresomeadvice,butactuallyrequirelogloglogk+
Θ(1)bitsofadvice. Thatisexpressedinthefollowingtheorem.
Theorem1.2. ThereisanO(logk)-competitivesearchalgorithmusingadvicesizeoflogloglogk+O(1)
bits. Ontheotherhand,anyO(logk)-competitivealgorithmhasadvicesizeoflogloglogk+Ω(1)bits.
Ontheotherextreme,weshowthatadviceofsizeloglogk+Θ(1)bitsisbothnecessaryandsufficientto
obtainanasymptoticallyoptimalrunningtime.
Theorem1.3. ThereexistsanO(1)-competitivesearchalgorithmusingadviceofsizeloglogk+O(1). This
boundistightasanyO(1)-competitivesearchalgorithmmusthaveadvicesizeofloglogk−Ω(1)bits.
Wefurtherexhibitlowerandupperboundsfortheadvicesizeforarangeofintermediatecompetitivenesses.
Specifically,thefollowinglowerboundontheadvicesizeimpliesthatadviceofsizeΩ(loglogk)isnecessary
evenforachievingmuchlargertime-competitivenessthanconstant.
Theorem1.4. Fix0 < (cid:15) ≤ 1. AnysearchalgorithmthatisO(log1−(cid:15)k)-competitivemusthaveadvicesize
of(cid:15)loglogk−Ω(1)bits.
2Tomeasurethetimecomplexity,weadopttheterminologyofcompetitiveanalysis,andsaythatanalgorithmisc-competitive,
if its time complexity is at most c times the straightforward lower bound, that is, at most c(D +D2/k). In particular, an
O(1)-competitivealgorithmisanalgorithmthatrunsinO(D+D2/k)time.
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(cid:16) (cid:17)
Theorem 1.5. Fix 0 < (cid:15) ≤ 1. Any O logk -competitive search algorithm requires advice size of
2log(cid:15)logk
log(cid:15)logk−O(1).
Complementingthisresult,weshowthatthelowerboundontheadvicesizeisasymptoticallyoptimal.
(cid:16) (cid:17)
Theorem1.6. Fix0 < (cid:15) ≤ 1. ThereexistsanO logk -competitivesearchalgorithmusingadvicesize
2log(cid:15)logk
ofO(log(cid:15)logk).
TheaforementionedlowerandupperboundsaresummarizedinTable1.
Competitiveness Advicesize
Upperbound O(log1+(cid:15)k) zero
Lowerbound ω(logk) zero
Tightbound O(1) loglogk +Θ(1)
Lowerbound O(log1−(cid:15)k) 0 < (cid:15) < 1 (cid:15)loglogk −Ω(1)
Tightbound O(logk/2log(cid:15)logk) 0 < (cid:15) < 1 Θ(log(cid:15)logk)
Tightbound O(logk) logloglogk +Θ(1)
Table1: Boundsontheadvicesizeforgivencompetitiveness
Apartfromtheirpurelytheoreticalappeal,thealgorithmswesuggestintheaforementionedupperbound
resultsareusedmainlytoindicatethetightnessofourlowerbounds. Indeed,theseparticularalgorithmsuse
somewhatnon-trivialcomponents,suchasbeingexecutedinrepeatediterations,whichseemtocorrespond
less to realistic scenarios. As our final technical result, we propose a uniform search algorithm that is
concurrentlyefficientandextremelysimpleand,assuch,itmayimplysomerelevanceforactualbiological
scenarios.
Conceptual Contribution. This paper may serve as a “proof of concept” for a new bridge connecting
computerscienceandbiology. Specifically,weillustratethattheoreticaldistributedcomputingcanpotentially
provideanovelandefficientmethodologyforthestudyofhighlycomplex,cooperativebiologicalensembles.
Tothisend,wechooseasettingthatis,ontheonehand,sufficientlysimpletobeanalyzedtheoretically,and,
ontheotherhand,sufficientlyrealisticsothatthissetting,oravariantofit,canberealizableinanexperiment.
Ultimately,ifanexperimentthatcomplieswithoursettingrevealsthattheants’searchistimeefficient,inthe
sensedetailedabove,thenourtheoreticallowerboundresultswouldimplylowerboundsonthenumberof
statesantshavealreadywhentheyexittheirnesttocommencethesearch. Notethatlowerboundsconsisting
ofevenafewstateswouldalreadydemonstratethepowerofthistradeoff-basedmethodology, sincethey
wouldbeverydifficulttoobtainusingmoretraditionalmethodologies.
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1.2 Outline
InSection2weformallydescribetheANTSproblemandprovidebasicdefinitions. Tomakethereadermore
familiarwiththemodel,westartthetechnicalpartofthepaperinSection3byprovidingarelativelysimple
O(1)-competitivesearchalgorithm. Thisalgorithmusesa-prioriknowledgeregardingthetotalnumberof
agents. Specifically,thealgorithmassumesthateachagentknowsa2-approximationofk,encodedasadvice
of size loglogk. In Section 4 we turn to prove lower bounds on the advice size. We start this section by
provingthatthecompetitivenessofanyuniformsearchalgorithmismorethanlogarithmic. Thisresultisa
specialcaseofthemainlowerboundtheorem,namely,Theorem4.6,thatweprovejustnext. Thereason
we prove this special case separately is because the proof is simpler, and yet, provides some of the main
ingredients of the more general lower bound proof. We then describe specific lower bounds that directly
followfromTheorem4.6ascorollaries. WeconcludethetechnicalpartofthepaperinSection5wherewe
provideseveralconstructionsofsearchalgorithms. Wefirstpresentseveralconstructionswhoseperformances
provide tight upper bounds for some of our lower bounds results. We then turn our attention to present a
relatively efficient uniform algorithm, called, the harmonic algorithm, whose simplicity may suggest its
relevance to actual search by ants. In Section 6 we discuss the conceptual contribution of the paper as a
proofofconceptforanewinterdisciplinarymethodologythatcombinesalgorithmicstudieswithbiology
experiments.
1.3 RelatedWork
Ourworkfallswithintheframeworkofnaturalalgorithms,arecentattempttostudybiologicalphenomena
fromanalgorithmicperspective[1,2,14,16,27,29].
Since our preliminary conference publications in [28, 29], the ANTS problem had quickly attracted
attentionfromthecommunityofdistributedcomputing,withcontinuationworksbyotherresearchers,study-
ing various theoretical aspects of our model [25, 26, 37, 53, 55, 56]. Specifically, [25, 26, 55] studied a
variantofourmodelwhereagentsaremodeledasfinitestatemachinesandcancommunicateoutsidethenest
throughouttheexecution. Inparticular,undertheseassumptions,Emeketal.,showedthatagentscansolve
theANTSprobleminO(D+D2/k)time[25]. Thisimpliesthatinthiscase,noa-prioriknowledgeabout
k isrequiredbytheagents. Note,however,thatthisassumptionregardingcommunicationisprobablynot
applicableforthecaseofdesertants,wherenocommunicationbysuchsearchingantshasbeenrecorded. The
ANTSproblemwasfurtherstudiedbyLenzenetal.,whoinvestigatedtheeffectsofboundingthememoryas
wellastherangeofavailableprobabilitiesoftheagents[56]. Fraigniaudetal. consideredthecasewherek
non-communicatingagentssearchonlineforanadversariallyplacedtreasure[37]. Theyalsoarguedforthe
usefulnessofparallelsearchperformedbynon-communicatingagentsduetotheinherentrobustnessofsuch
computation. Acontinuationworkconsideringthecasewherethetreasureisplaceduniformlyatrandomina
largedomainwasstudiedin[53].
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TheANTSproblemisaspecialcaseofcollaborativesearch–aclassicalfamilyofproblemsthathas
beenextensivelystudiedindifferentfieldsofscience. Wenextreviewpreviousworksonsearchproblems,
organizingthemaccordingtotheresearchdisciplinesinwhichtheywereinvestigated.
Search Problems in Computer Science. In the theory of computer science, the exploration of graphs
usingmobileagents(orrobots)isacentralquestion. (Foramoredetailedsurveyrefertoe.g.,[29,30].) When
itcomestoprobabilisticsearching,therandomwalkisanaturalcandidate,asitisextremelysimple,usesno
memory, andtriviallyself-stabilizes. Unfortunately, however, therandomwalkturnsouttobeinefficient
inatwo-dimensionalinfinitegrid. Specifically,inthiscase,theexpectedhittingtimeisinfinite,evenifthe
treasureisnearby.
Mostgraphexplorationresearchisconcernedwiththecaseofasingledeterministicagentexploringa
finitegraph,seeforexample[3,22,39,59,63]. Ingeneral,oneofthemainchallengesinsearchproblemsis
theestablishmentofmemorybounds. Forexample,thequestionofwhetherasingleagentcanexploreallfinite
undirectedgraphsusinglogarithmicmemorywasopenforalongtime;answeringittotheaffirmative[63]
establishedanequalitybetweentheclassesoflanguagesSLandL.Asanotherexample,itwasprovedin[68]
thatnofinitesetofconstantmemoryagentscanexploreallgraphs. Tothebestofourknowledge,thecurrent
paperisthefirstpaperestablishingnon-trivialmemorylowerboundsinthecontextofrandomizedsearch
problems.
Ingeneral,themorecomplexsettingofusingmultipleidenticalagentshasreceivedmuchlessattention.
Exploration by deterministic multiple agents was studied in, e.g., [31, 32]. To obtain better results when
usingseveralidenticaldeterministicagents,onemustassumethattheagentsareeithercentrallycoordinated
orthattheyhavesomemeansofcommunication(eitherexplicitly,orimplicitly,bybeingabletodetectthe
presenceofnearbyagents). Whenitcomestoprobabilisticagents,analyzingthespeed-upmeasurefork
randomwalkershasrecentlygainedattention. Inaseriesofpapers,aspeed-upofΩ(k)isestablishedfor
variousfinitegraphfamilies,including,e.g.,expandersandrandomgraphs[4,23,19]. Incontrast,forthe
two-dimensionaln-nodegrid,aslongask ispolynomialinn,thespeedupisonlylogarithmicink. The
situationwithinfinitegridsisevenworse. Specifically,thoughthek randomwalkersfindthetreasurewith
probabilityone,theexpected(hitting)timeisinfinite.
EvaluatingtherunningtimeasafunctionofD,thedistancetothetreasure,wasstudiedinthecontextof
thecow-pathproblem[9]. Thisproblemconsidersasinglemobileagentplacedonagraph,aimingtofind
atreasureplacedbyanadversary. Itwasestablishedin[9]thatthecompetitiveratiofordeterministically
findingapointonthereallineisnine,andthatinthetwo-dimensionalgrid,thespiralsearchalgorithmis
optimaluptolowerorderterms. Severalothervariantsofthisproblemwerestudiedin[21,47,48,57]. In
particular,in[57],thecow-pathproblemwasextendedbyconsideringk agents. However,incontrasttoour
setting,theagentstheyconsiderhaveuniqueidentities,andthegoalisachievedby(centrally)specifyinga
differentpathforeachofthek agents.
Thenotionofadviceiscentralincomputerscience. Inparticular,theconceptofadviceanditsimpacton
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variouscomputationshasrecentlyfoundvariousapplicationsindistributedcomputing. Inthiscontext,the
mainmeasureusedistheadvicesize. Itisforinstanceanalyzedinframeworkssuchasnon-deterministic
decision[36,51,52],broadcast[33],localcomputationofMST[35],graphcoloring[34]andgraphsearching
byasinglerobot[17]. Veryrecently,ithasalsobeeninvestigatedinthecontextofonlinealgorithms[13,24].
The question of how important it is for individual processors to know their total number has recently
beenaddressedinthecontextoflocality. Generallyspeaking,ithasbeenobservedthatforseveralclassical
localcomputationtasks,knowingthenumberofprocessorsisnotessential[50]. Ontheotherhand,inthe
contextoflocaldistributeddecision,someevidenceexistthatsuchknowledgeiscrucialfornon-deterministic
verification[36].
SearchProblemsinBiologyandPhysics. Grouplivingandfoodsourcesthathavetobeactivelysought
after make collective foraging a widespread biological phenomenon. Social foraging theory [40] makes
use of economic and game theory to optimize food exploitation as a function of the group size and the
degreeofcooperationbetweenagentsindifferentenvironmentalsettings. Thistheoryhasbeenextensively
comparedtoexperimentaldata(see,e.g.,[8,41])butdoesnottypicallyaccountforthespatialcharacteristics
ofresourceabundance. Centralplaceforagingtheory[58]assumesasituationinwhichfoodiscollected
fromapatchyresourceandisreturnedtoaparticularlocation,suchasanest. Thistheoryisusedtocalculate
optimaldurationsforexploitingfoodpatchesatdifferentdistancesfromthecentrallocationandhasalso
been tested against experimental observations [38, 43]. Collective foraging around a central location is
particularlyinterestinginthecaseofsocialinsectswherelargegroupsforagecooperativelywith,practically,
nocompetitionbetweenindividuals.
Actual collective search trajectories of non-communicating agents have been studied in the physics
literature (e.g., [64, 12]). Reynolds [64] achieves optimal speed up through overlap reduction which is
obtained by sending searchers on near-straight disjoint lines to infinity. This must come at the expense
of finding proximal treasures. Harkness and Maroudas [42] combined field experiments with computer
simulationsofasemi-randomcollectivesearchandsuggestsubstantialspeedupsasgroupsizeincreases.
SearchProblemsinRobotics. Fromanengineeringperspective,thedistributedcooperationofateamof
autonomousagents(oftenreferredtoasrobotsorUAVs: UnmannedAerialVehicles)isaproblemthathas
beenextensivelystudied. Thesemodelsextendsingleagentsearchesinwhichanagentwithlimitedsensing
abilitiesattemptstolocateoneorseveralmobileorimmobiletargets[60]. Thememoryandcomputational
capacitiesoftheagentaretypicallylargeandmanyalgorithmsrelyontheconstructionofcognitivemapsof
thesearchareathatincludescurrentestimatesthatthetargetresidesineachpoint[76]. Theagentthenplansan
optimalpathwithinthismapwiththeintent,forexample,ofoptimizingtherateofuncertaintydecrease[46].
Cooperativesearchestypicallyincludecommunicationbetweentheagentsthatcanbetransmitteduptoa
givendistance,orevenwithoutanyrestriction. Modelshavebeensuggestedwhereagentscancommunicate
byalteringtheenvironmenttowhichotheragentthenreact[73]. Cooperationwithoutcommunicationhas
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alsobeenexploredtosomeextent[5]buttheanalysisputsnoemphasisonthespeed-upofthesearchprocess.
Inaddition,tothebestofourknowledge,noworksexistinthiscontextthatputemphasisonfindingnearby
targets faster than faraway one. It is important to stress that in all those engineering works, the issue of
whethertherobotsknowtheirtotalnumberistypicallynotaddressed,asobtainingsuchinformationdoesnot
seemtobeproblematic. Furthermore,inmanyworks,robotsarenotidenticalandhaveuniqueidentities.
2 Model and Definitions
2.1 GeneralSetting
We introduce the Ants Nearby Treasure Search (ANTS) problem, in which k mobile agents are searching
foratreasureonsometopologicaldomain. Here,wefocusonthetwo-dimensionalplane. Theagentsare
probabilisticmobilemachines(robots). Theyareidentical,thatis,allagentsexecutethesameprotocolP,
andtheirexecutiondifferonlyduetotheoutcomeoftheirrandomcoins. Eachagenthassomelimitedfield
ofview,i.e.,eachagentcanseeitssurroundinguptoadistanceofsomeε > 0. Hence,forsimplicity,instead
of considering the two-dimensional plane, we assume that the agents are actually walking on the integer
two-dimensionalinfinitegridG = Z2 (theycantraverseanedgeofthegridinbothdirections). Thesearchis
centralplace,thatis,allk agentsinitiatethesearchfromsomecentralnodes ∈ G,calledthesource. Before
thesearchisinitiated,anadversarylocatesthetreasureatsomenodeς ∈ G,referredtoasthetargetnode.
Oncethesearchisinitiated,theagentscannotcommunicateamongthemselves.
Thedistancebetweentwonodesu,v ∈ G,denotedd(u,v),issimplytheManhattandistancebetween
them, i.e., the number of edges on the shortest path connecting u and v in the grid G. For a node u, let
d(u) := d(s,u)denotethedistancebetweenuandthesourcenode. WedenotebyDthedistancebetweenthe
sourcenodeandthetarget,i.e.,D = d(ς). Itisimportanttonotethattheagentshavenoa-prioriinformation
aboutthelocationofς oraboutD. Wesaythattheagentsfindthetreasurewhenoneoftheagentsvisitsthe
targetnodeς. ThegoaloftheagentsittofindthetreasureasfastaspossibleasafunctionofbothD andk.
Sincewearemainlyinterestedinlowerbounds,weassumeaveryliberalsetting. Inparticular,wedo
not restrict either the computational power or the navigation capabilities of agents. Moreover, we put no
restrictionsontheinternalstorageusedfornavigation. Ontheotherhand,wenotethatforconstructingupper
bounds, the algorithms we consider use simple procedures that can be implemented using relatively few
resources. Forexample,withrespecttonavigation,weonlyassumetheabilitytoperformthefollowingbasic
procedures: (1)chooseadirectionuniformlyatrandom,(2)walkina“straightline”toaprescribeddistance,
(3)performaspiralsearcharoundagivennode3,and(4)returntothesourcenode.
3Thespiralsearchisaparticulardeterministicsearchalgorithmthatstartsatanodevandenablestheagenttovisitallnodesat
√
distanceΩ( x)fromvbytraversingxedges,foreveryintegerx(see,e.g.,[9]). Forourpurposes,sinceweareinterestedwith
asymptoticresultsonly,wecanreplacethisatomicnavigationprotocolwithanyprocedurethatguaranteessuchaproperty. For
simplicity,inwhatfollows,weassumethatforanyintegerx,thespiralsearchoflengthxstartingatanodevvisitsallnodesat
√
distanceatmost x/2fromv.
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2.2 OraclesandAdvice
Wewouldliketomodelthesituationinwhichbeforethesearchactuallystarts,someinformationmaybe
exchangedbetweentheagentsatthesourcenode. Inthecontextofants,thispreliminarycommunication
maybequitelimited. Thismaybebecauseofdifficultiesinthecommunicationthatareinherenttotheants
ortheenvironment,e.g.,duetofaultsorlimitedmemory,orbecauseofasynchronyissues,orsimplybecause
agentsareidenticalanditmaybedifficultforantstodistinguishoneagentfromtheother. Nevertheless,we
consideraveryliberalsettinginwhichthispreliminarycommunicationisalmostunrestricted.
Morespecifically,weconsideracentralizedalgorithmcalledoraclethatassignsadvicestoagentsina
preliminarystage. Theoracle,denotedbyO,isaprobabilistic4 centralizedalgorithmthatreceivesasinputa
setofk agentsandassignsanadvicetoeachofthek agents. Weassumethattheoraclemayuseadifferent
protocol for each k; Given k, the randomized algorithm used for assigning the advices to the k agents is
denoted by O . Furthermore, the oracle may assign a different advice to each agent5. Observe that this
k
definitionofanoracleenablesittosimulatealmostanyreasonablepreliminarycommunicationbetweenthe
agents6.
Itisimportanttostressthateventhoughallagentsexecutethesamesearchingprotocol,theymaystart
the search with different advices. Hence, since their searching protocol may rely on the content of this
initialadvice,agentswithdifferentadvicesmaybehavedifferently. Anotherimportantremarkconcernsthe
fact that some part of the advices may be used for encoding (not necessarily disjoint) identifiers. That is,
assumptionsregardingthesettingsinwhichnotallagentsareidenticalandthereareseveraltypesofagents
canbecapturedbyoursettingofadvice.
Tosummarize,asearchalgorithmisapairhP,OiconsistingofarandomizedsearchingprotocolP and
randomizedoracleO = {O } . Givenk agents,therandomizedoracleO assignsaseparateadviceto
k k∈N k
each of the given agents. Subsequently, all agents initiate the actual search by letting each of the agents
executeprotocolP andusingthecorrespondingadviceasinputtoP. Oncethesearchisinitiated,theagents
cannotcommunicateamongthemselves.
4Itisnotclearwhetherornotaprobabilisticoracleisstrictlymorepowerfulthanadeterministicone.Indeed,theoracleassigning
theadviceisunawareofD,andmaythuspotentiallyusetherandomizationtoreducethesizeoftheadvicesbybalancingbetween
theefficiencyofthesearchforsmallvaluesofDandlargervalues.
5Wenotethateventhoughweconsideraveryliberalsetting,andallowaverypowerfuloracle,theoraclesweuseforourupper
boundsconstructionsareverysimpleandrelyonmuchweakerassumptions.Indeed,theseoraclesarenotonlydeterministicbutalso
assignthesameadvicetoeachofthekagents.
6Forexample,itcansimulatetofollowingveryliberalsetting.Assumethatinthepreprocessingstage,thekagentsareorganized
inacliquetopology,andthateachagentcansendaseparatemessagetoeachotheragent.Furthermore,eventhoughtheagentsare
identical,inthispreprocessingstage,letusassumethatagentscandistinguishthemessagesreceivedfromdifferentagents,andthat
eachofthekagentsmayuseadifferentprobabilisticprotocolforthispreliminarycommunication.Inaddition,norestrictionismade
eitheronthememoryandcomputationcapabilitiesofagentsoronthepreprocessingtime,thatis,thepreprocessingstagetakesfinite,
yetunlimited,time.
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