Algorithmic Foundations of Robotics VIII: Selected Contributions of the Eighth International Workshop on the Algorithmic Foundations of Robotics (Springer Tracts in Advanced Robotics)

123 Pages·2010·2.84 MB·English

by  Gregory S. Chirikjian

Probabilistic Network Formation through
Coverage and Freeze-Tag
EricMeisner,WeiYang,andVolkanIsler
Abstract. We address the problem of propagating a piece of information among
robotsscatteredinanenvironment.Initially,asinglerobothastheinformation.This
robotsearchesforotherrobotstopassitalong.Whena robotisdiscovered,itcan
participate in the process by searching for other robots. Since our motivation for
studyingthisproblemistoformanad-hocnetwork,wecallittheNetworkForma-
tionProblem.Inthispaper,westudythecasewheretheenvironmentisarectangle
andtherobots’locationsareunknownbutchosenuniformlyatrandom.Wepresent
anefficientnetworkformationalgorithm,Stripes,andshowthatitsexpectedperfor-
manceiswithinalogarithmicfactoroftheoptimalperformance.Wealsocompare
Stripeswithanintuitivenetworkformationalgorithminsimulations.Thefeasibility
ofStripesisdemonstratedwithaproof-of-conceptimplementation.
1 Introduction
Consider the following scenario: a number of robots are performing independent
tasksautonomouslyinanenvironmentwherethereisnocommunicationinfrastruc-
ture.Suppose,atacertainpoint,missionprioritieschangeandapieceofinformation
mustbepropagatedtoallnodes.Forexample,robotscouldbeinitiallystationedto
perform monitoring or surveillance in a large environment. Upon detection of an
event,theymayhavetoformanetworkorperformacollaborativetask.Whatisa
goodstrategytogetallrobotsinvolvedasquicklyaspossible?
EricMeisnerandWeiYang
DepartmentofComputerScience,RensselaerPolytechnicInstitute1108thStreet
TroyNY12180
e-mail:{meisne,yangw}@cs.rpi.edu
VolkanIsler
DepartmentofComputerScience,UniversityofMinnesota200UnionStreet
SEMinneapolis,MN55455
e-mail:[email protected]
G.S.Chirikjianetal.(Eds.):AlgorithmicFoundationsofRoboticsVIII,STAR57,pp.3–17.
springerlink.com (cid:2)c Springer-VerlagBerlinHeidelberg2010
4 E.Meisner,W.Yang,andV.Isler
Inthispaper,westudythisprocessofpropagatinginformationasquicklyaspos-
sible.Specifically,westudythecasewheretheprocessisinitiatedbyasinglerobot.
This robot could, for example, be sent out from the command and control center.
Alternatively,it could be the robotthat detects an intruder. The primarydifficulty
in solving the problem arises from the fact that the robots do not know each oth-
ers’positions.Thefirstrobotmustthereforestartasearch.Oncediscovered,other
robotscanparticipateinpropagatingtheinformation.Sinceourprimarymotivation
forstudyingthisproblemistoformaconnectednetwork,throughoutthepaperwe
willrefertoitastheNetworkFormationProblem.
Our Contributions:Inthispaper,we studya probabilisticscenariowherethe lo-
cationsofrobotsarechosenuniformlyatrandominarectangularenvironment.For
this scenario, we present a network formationstrategy (Stripes) and provethat its
expectedperformance(i.e.networkformationtime)iswithinalogarithmicfactorof
theoptimalperformance.Toobtainthisresult,wealsoobtainalowerboundonthe
expectedperformanceofanynetworkformationstrategy.WealsocompareStripes
withanatural,intuitive“SplitandCover”strategyandshowthatinlargeenviron-
ments, Split and Cover has inferior performance to Stripes. In addition to formal
performancebounds,wedemonstratetheutilityofStripeswithsimulationsandits
feasibilitywithaproof-of-conceptimplementation.Inthenextsection,westartwith
an overviewofrelated workwherewe establish connectionsbetweenthe network
formationproblemandother fundamentalproblemssuchas rendezvous,coverage
andfreeze-tag.
1.1 Related Work
Considerableworkhasbeendoneindesigningdecentralizedprotocolsforpropagat-
inginformationinnetworks(alsoknownasgossipprotocols)[7].Gossipprotocols
are mainly designedfor stationarynetworks.In contrast, we focuson information
propagation among mobile robots. The network formation problem is closely re-
latedtotheFreeze-Tagproblem[4].Infreeze-tag,anumberofplayersare“frozen”.
A single unfrozen player must visit each frozen player in order to unfreeze it, at
whichpointitcanaidinunfreezingotherplayers.Infreezetag,itisassumedthat
theplayersknoweachothers’positions.Innetworkformation,wefocusonthecase
wherethenodelocationsareunknowntoeachother.
Recently, Poduri and Sukhatme explicitly addressed the problem of forming a
connectednetworkunderthenamecoalescence[11,12].Intheirmodel,allofthe
nodes(other than the base station) are performinga randomwalk on a torus. The
authorsobtainboundsonthenetworkformationtime.Theadvantageoftherandom-
walkstrategyisthatitdoesnotrequirelocalizationwithrespecttoaglobalreference
frame.However,thenetworkformationisratherslowbecausenodesvisitmostlo-
cationsmanytimes.Thismaynotbeacceptableintime-criticalapplications.Inthe
presentwork,weaddresstheproblemofexplicitlydesigningmotionstrategiesfor
networkformationwithguaranteedperformance.Sincewefocusonthecasewhere
ProbabilisticNetworkFormationthroughCoverageandFreeze-Tag 5
the robotlocationsareunknown,the networkformationproblemisrelatedto ren-
dezvousandcoverage.
The rendezvoussearch problem[1] asks how two players with the same speed
can locate each other when placed at arbitrary locations in a known environment.
Typically, each player has a radius of detection within which the players can es-
tablish communication.The goalof the playersis to find each otheras quicklyas
possible.Therendezvousproblemhasbeenstudiedextensivelyfortwoplayersona
line.Optimalorprovablygoodstrategiesforvariousversionsofthisproblemhave
beenestablished[2].
The problemof multi-playerrendezvoussearch on a completegraphis studied
in[14].Thisworkaddressesthequestionofwhetherplayersthatmeetshouldstick
togetherorsplitandmeetagainlater.Theresultofthisworkshowsthat,iftheplay-
ershavenomemory,thentheoptimalstrategyistosticktogether.Also,simulations
show thatas the numberofplayersincreases,the splitand meetstrategybecomes
less effective. In general, this work has limited applications because the environ-
mentisacompletegraph.Inotherwords,playerscanteleporttoarbitrarylocations
ateverytimestep.
Work in [3] describes a time optimal strategy for two-player limited visibility
rendezvousintheplanewithknownandunknowndistances.Theoptimalsolution
inthiscaseisforoneoftheplayerstofollowasemi-circularspiraltrajectory.This
workalsorelatesrendezvousinthecontinuousdomaintocoverageproblems.Deter-
ministic andrandomizedmulti-robotrendezvousstrategieswere proposedin [13].
Morerecently,resultsonrendezvousinsimply-connectedpolygonshavebeenob-
tained[8].In[9],theauthorsprovideupperandlowerboundsonthetimecomplex-
ityoftwogeometriclawsthatresultinrendezvous.
Thecoverageorlawnmowingproblem[5,6]hasbeenextensivelystudiedandis
knowntobecloselyrelatedtoTravelingSalespersonProblem.Theprimarydiffer-
encebetweennetworkformationandstandardmulti-robotcoverageisthatincov-
erage,allrobotsparticipateinthecoverageprocessfromthebeginning.Innetwork
formation,theprocessstartswithonerobot,whomust“recruit”otherstoparticipate
incoverage.
In short,the algorithmpresentedin thispapercanbeconsideredanovelalgo-
rithmfor(i)onlinefreeze-tag,(ii)probabilisticmulti-robotcoverage,and(iii)net-
workformation.
2 Problem Formulation
Since the robot locations are unknown, network formation is an online problem.
Onlineproblemsaretypicallyanalyzedusingcompetitiveanalysiswheretheinput
is chosen by an adversary. The competitive ratio of an online algorithm O is the
worst case ratio of the performance of O to the optimal offline performance. In
otherwords,wecompareO withtheoptimalalgorithmthathasaccesstoallofO’s
inputinadvanceandconsidertheworstcasedeviationfromthisoptimalbehavior.
6 E.Meisner,W.Yang,andV.Isler
Itiseasytoseethatthereisnoonlinealgorithmforthenetworkformationprob-
lem with boundedcompetitiveratio:no matter which paththe first robotchooses,
theadversarycanplaceallotherrobotsatthelastlocationthefirstrobotwillvisit.In
thecaseofasquareenvironmentwithareaa2,theonlinealgorithmwouldtaketime
proportionaltoa2whereastheoptimalofflinecostwouldbeatmostintheorderof
a.Thereforethecompetitiveratiowouldbeaanditwouldgrowunboundedlywith
thesizeoftheenvironment.
Since thereare no competitiveonline algorithmsfor networkformation,in this
paperwefocusontheprobabilisticcasewherethelocationsoftherobotsaresam-
pled from a distribution. In the absence of any information,it is reasonableto as-
sumethatthelocationsarechosenuniformlyatrandomfromtheenvironment.The
goalistominimizetheexpectedtimetodiscoverallrobots.Inthispaper,wefocus
onrectangularenvironmentsanduniformdistributions.Webelievethattherectan-
gularenvironmentcasehaspracticalrelevanceforrobotsoperatinginanopenfield
aswellasforUnmannedAerialVehicles.Wediscussextensionstogeneralconvex
environmentsinSection6.
2.1 RobotModel
In network formation,several identical mobile robots are distributed uniformly at
random within the bounded rectangle A with one of these robots possessing in-
formation that needs to be propagated to all of the other robots. The rectangle A
has a width w and height h where w≥h. We assume that the robots are initially
stationary. Once discovered, they can move anywhere within the rectangle A at a
constantspeedandcancommunicatewitheachotherwithinalimitedrange.Once
arobotiswithinthecommunicationrangeofanotherrobot,theycanexchangeany
informationthateitherrobotmighthave,includingtheinformationthatneedstobe
propagated.Inthispaper,weassumethattherobotscanlocalizethemselveswithin
A.We also ignoreenergylimitationsandassumethattherobotswillalwaysmove
attheirmaximumspeedandcanutilizetheirwirelessradioanytime.
NotationandConventions:Wenormalizetheunitssothattherobotsmoveatunit
speedpertime-unit.Intherestofthepaper,weuseAtodenoteboththerectangle
environmentanditsarea.Thisconventionimpliesthatasinglerobotcancoverthe
entireenvironmentinAtimeunits.Hence,Aisatrivialupperboundonnetworkfor-
mationsinceinactivenodesarestationary.Thenumberofrobotsintheenvironment
isk.WeassumethatthefirstrobotenterstheenvironmentatacornerofA.
3 Stripes: AnEfficient Network FormationStrategy
Inthissection,wepresentourmainresult:anefficientnetworkformationstrategy
whichwerefertoas“Stripes”.Thisstrategyreliesondividingtherectangularenvi-
ronmentintonequalsizedverticalstripesS ,...,S (Figure1).Fornow,wetreatn
1 n
ProbabilisticNetworkFormationthroughCoverageandFreeze-Tag 7
Fig.1.Stripesstrategy:Theenvironmentisdividedintoequalverticalstripeswhicharecov-
eredsequentially.Activerobotssplitthecurrentstripeequally.Onceastripeiscovered,all
activerobots(includingnewlydiscoveredrobots)meetattheboundaryofthestripe.
asavariablewhosevaluewillbefixedlater.LetSdenotetheareaofasinglestripe
whichisequaltothetimetocoverasinglestripewithonerobot.
Inthebeginning,asinglerobotisactiveandproceedstocoverS withasimple
1
back and forth sweeping motion. That is, the robotfollows the well-knownbous-
trophedon1 path. When an inactive robot is encounteredand activated, this newly
active robotdoesnotjoin in the coverageof S rightaway2. Instead,it headsto a
1
designatedlocationonthelinethatseparatesS andS andwaitsforthefirstactive
1 2
robot to finish its coverage of S and arrive at the designated location. When the
1
firstrobotisfinished,itmeetsallnewlyactiverobotsatthesamedesignatedloca-
tion.Letk denotethenumberofactiverobots.TherobotsevenlydivideS among
1 2
themselvessothatitcanbecoveredinparallelintimeS /k .
2 1
Whenthesek robotsencounterotherinactiverobotsinS ,thesameprocedure
1 2
is repeated and all active robotsmeet at a designated meetinglocation on the line
whichseparatesS andS .Thisprocessrepeatsfortheremainingstripesuntilallof
2 3
thestripesarecovered,atwhichpoint,theentireboundedareawillbecoveredand
alloftherobotswillbeactivated.
Formostofthepaper,wewillfocusonrectangularenvironmentswhicharelarge
with respectto the numberof robots. In other words, we focuson the case where
w≥h(cid:3)k.
Intheremainingcase,thenumberofrobotsexceedsthesizeoftheshorterside
of therectangle(i.e.k≥h).We referto thiscase asthesaturatedcase. Whenthe
numberofactiverobotsreachh,theremainingm×hareacanbecoveredinmsteps
bytheactiverobots.Wepresenttheanalysisofthealgorithmforthesaturatedcase
intheappendix.
3.1 NetworkFormationwithStripes
Inthissection,weestablishanupperboundonthenetworkformationtimeofStripes
foranunsaturatedenvironment.Wewillusethefollowinglemma.
Lemma3.1.If k balls are assigned to n bins randomly, the expected number of
emptybinsisatmostne−k/n.
Proof. Wesaythatbiniis“hit”ifitisnon-emptyafterthekththrow.Theprobability
p thattheith binisnothitis(1−1)k.LetX bearandomvariablewhichtakesthe
i n i
1Boustrophedon=“thewayoftheox”[6].
2Inpractice,thisrobotcanparticipateincoveringthestripe.However,thisdoesnotimprove
theanalysis.Hence,weignorethisbenefitforanalysispurposes.
8 E.Meisner,W.Yang,andV.Isler
value one if bin i is not hit and zero otherwise. Then, E[X]= p. The number of
i i
empty bins is given by the random variable ∑n X whose expected value is np.
i=1 i i
Thelemmafollowsfromtheinequality(1+u)≤eu.
ToobtainanupperboundonthenetworkformationtimewithStripes,wetreatthe
robotsasballsandstripesasbins.Notethatthecoveragetimeismaximizedwhen
theemptystripesoccuratthebeginning,andnon-emptystripeshavethefollowing
property: let m be the number of non-empty stripes. m−1 stripes have only one
robotinsideandalltheremainingrobotsareinasinglestripewhichoccursafterall
otherstripes.Wecomputetheexpectedcoveragetimeforthisworstcase.
Ifwepickn= k ,byLemma3.1thenumberofemptystripesisatmost 1 ,
logk logk
whichislessthanorequalto1fork>1. (cid:2) (cid:3)
Thisresultsinacoveragetimeof A forthefirstnon-emptystripe, A 1 forthe
(cid:2) (cid:3) n n 2
second, A 1 andsoonforthe remainingstripes.Sincethereisatmostonenon-
n 3
emptystripe,thetotalcoveragetimewillbeboundedby:
(cid:4) (cid:5)
A+∑n A 1 = A+A∑n 1 ≈ A+Alogn (1)
n n i n n i n n
i=1 i=1
Byplugginginn= k ,wehavethefollowingresult.
logk
Lemma3.2.The expected network formation time for k robots in a rectangular
unsaturatedenvironmentwithareaAisO(Alog2k)whenrobotsexecutetheStripes
k
algorithmwithn= k stripes.
logk
Sincerobotsgrouptogetherbeforeandaftercoveringa stripe,thereis someover-
headassociatedwiththedifferentstrategiesthatcanbeusedtocoveranindividual
stripe. In establishing Lemma 3.2, this overheadis ignored.We justify this in the
nextsection,wherewepresentthestrategytocoverindividualstripes.
3.2 Coveringa SingleStripe
TheStripesalgorithmcallsforthesetofactiverobotstomeetandregroupbetween
coveringstripes.Inthissection,wepresentastrategyformultiplerobotstocovera
singlestripeinawaythatminimizestheoverheadofmeetingtoredistributeassign-
ments.Ourproposedsinglestripestrategyensuresthateachrobottravelsthesame
distanceduringanepoch,andarrivesatacommonmeetinglocation.
In orderto minimizethe coveragetime of a single stripe, its area mustbe split
equally between all of the active robots. This can be done by simply dividingthe
stripalongthelongsidetocreateequalsizedrectangularareas,oneforeachofthe
activerobots.However,thiswillresultinsomerobotstravelinglongerdistancesto
reachtheirassignedcoverageobjectives.Sincethisareawilleventuallybecovered
by the other robots, this produces an overhead of 2h per stripe, adding up to 2nh
for the entire bounded area A. Our single stripe strategy divides the active stripe
into sections such that, 1) the area coveredby each robotis equaland 2) the time
ProbabilisticNetworkFormationthroughCoverageandFreeze-Tag 9
Fig.2.Oursinglestripecoveragestrategydividesaverticalstripeintoequallysizedareasthat
takeintoaccountanyadditionalareacoveredbyarobotasittravelsawayfromthemeeting
locations(largeblackdots).Thisfigureshowstheresultingdivisionsforsixactiverobotsand
thedesignatedBivaluesusedtoreferencetheheightofthehorizontalrectangularareas.
requiredto covereachsectionandthenreachthemeetinglocationisequal.Inthe
end,thepathsoftherobotsresemblearchwayswithdifferentsizedhorizontalareas
thatareinverselyproportionaltothedistancearobotneedstotraveltothemeeting
location(Figure2).The“legs”ofthearchwaysrepresenttheareascoveredbyeach
robotastheytraveluptotheirassignedrectangularareas.
Givena particularverticalstripe tocoverwith r activerobots,letA ,A ,...,A
1 2 r
denotethetotalassignedcoverageareaforeachrobot,withroboticoveringA and
i
roboti=1coveringtheareathatisfurthestawayfromthemeetinglocation.Since
eachstripeisvertical,itsareaishw =hw(cid:6).WeuseB todenotethedistancebetween
n i
theupperboundariesofeachsuccessiveregion(seeFigure2).Thisparametercan
bevariedtooffsetthe additionalcostoftraveling.Eachrobotisassumedto cover
one unit area per time step so the additional travel area covered by r is simply
1
twice the distance from the meeting point to its assigned coverage area or 2(h−
B ).ThisproducesthefollowingequationforcalculatingtheareaA ofeachregion
1 i
(equation5):
A = B w(cid:6)+2(h−B )=B (w(cid:6)−2)+2h (2)
1 1 1 1
A = B (w(cid:6)−2)+2(h−B −B )=B (w(cid:6)−4)+2h−2B (3)
2 2 1 2 2 1
A = B (w(cid:6)−4)+2(h−B −B −B )
3 3 1 2 3
= B (w(cid:6)−6)+2h−2B =2B (4)
3 1 2
i
A =B(w(cid:6)−2(i−1))+2(h−∑B ) (5)
i i j
j=1
Since each robot must cover the same area, we can determine the relationship
betweenBiandBi−1 bysettingthegenericareaformulaatiandi−1equaltoeach
other (Equation6). The remainingA and B valuescan be determinedby succes-
i i
sively applyingEquations7 and 8. In Equation7,A is equatedto hw(cid:6), producing
1 r
equation8whereB isexpressedusingthenumberofactiverobots,r,thewidthof
1
astripew(cid:6)= w,andtheheightoftheoverallenvironment,h.
n
10 E.Meisner,W.Yang,andV.Isler
B 4
i =1+ (6)
Bi−1 w(cid:6)−2i
(cid:6)
hw
A = =B (w(cid:6)−2)+2h (7)
1 1
r
hw(cid:6) −2h h(w(cid:6)−2r)
B = r = (8)
1 w(cid:6)−2 r(w(cid:6)−2)
The limitation of this single stripe strategy is that the number of active robots
must be less than half of a single stripe’s width, r< w(cid:6). This does not present a
2
problemfortheanalyzedenvironment,whereitisassumedthatw=h>>k.There-
fore,fromthesinglestripestrategy,eachstripecanbedividedintor equallysized
areaswithminimalrepeatedcoverageandthus,eliminatingtheoverheadcoverage
timefromtheStripesequation.
3.3 Lower Bound
Inthissection,weestablishalowerboundontheexpectednetworkformationtime
thatcanbeachievedbyanyalgorithm.Itiseasytoseethatthebestcoveragetime
foranyareaoccurswhen allofthe robotsareactive atthe beginningandthe area
is evenly split between all of them. Therefore, the absolute lower bound for total
coveragetime,regardlessofalgorithm,isT(A,k)= A.Forthecasewhentherobots
k
areuniformlydistributed,wecanobtainabetterboundusingthefollowingresult.
Lemma3.3([10],pp.45).Whennballsareassigneduniformlyatrandomtonbins,
withprobabilityatleast1−1,nobinhasmorethanα= logn ballsinit.
n loglogn
Now consider any network formation strategy for k robots in area A. During the
execution of this strategy, we divide the coverage process into epochs where ith
epochendswhenallactiverobotscovera(previouslyuncovered)totalareaofA/k.
LetS denotethesubsetofAcoveredduringepochi.LetE betheeventthatnoS
i 1 i
hasmorethatαballs.ByLemma3.3,E happenswithprobability(1−1/k).
1
When E happens, the maximum number of robots in S is α. Therefore, the
1 1
minimum time it takes to cover S is T = A . There will be αnew robots in S ,
1 1 kα 2
thereforeitscoveragetimeT isatleast A .Similarly,theithepochwilllastatleast
2 k2α
T = A steps.Sincetherearekepochs,thetotaltimeforallepochstofinishwillbe:
i kiα
∑k A ≈ A logk= Aloglogk (9)
kiα αk k
i=1
WhenE doesnothappen(withprobability 1),thecoveragetimeisatleastA/kas
1 k
discussed earlier. This givesus the lower boundon the expectedcoveragetime of
anyalgorithm.
ProbabilisticNetworkFormationthroughCoverageandFreeze-Tag 11
Lemma3.4.Theexpectedtimefork robotstocoverrectangularareaAisatleast
(1−1)Aloglogk+(1)A.
k k k k
Ignoring o(1) terms, we establish a lower bound of Ω (Aloglogk). Using Lem-
k2 k
mata3.2and3.4,weestablishourmainresult:
Theorem3.1.TheperformanceoftheStripesstrategyiswithinafactorO( log2k )
loglogk
oftheoptimalperformanceforarectangularenvironmentwherew≥h(cid:3)k.
3.4 Splitand Cover
In this section, we discuss the performance of the following intuitive and nat-
ural network formation strategy: The first robot moves up and down the envi-
ronment following a boustrophedon path as before. When it meets an undiscov-
ered node, the two nodes split the undiscovered parts of the environment evenly
and recursivelycovertheir assigned partitions(see Figure 3). The justification for
this natural “Split-and-Cover” strategy is that since the nodes are scattered uni-
formly in the environment, each split should divide the load equally between
discovered robots, therefore balancing (and intuitively minimizing) the network
formationtime.
ItturnsoutthatSplit-and-Coverisnotaneffectivestrategyforlargeenvironments
forthefollowingreason:supposethatwhenthesplithappens,oneofthepartitions
hasa verysmallnumberofrobots.Eventhoughthiseventhasasmallprobability
forasinglesplit,asthenumberofrobots(andhencethenumberofsplits)increases,
itbecomesprobable.Whensuchanimbalanceoccurs,thealgorithmcannotrecover
fromit.AsmallnumberofrobotsmustcoveralargeenvironmentmakingSplit-and-
Coverinefficient.We willfurtherjustifythis argumentwith simulationsandshow
thattheStripesalgorithmismuchmoreefficientthanSplit-and-Coverinunsaturated
environments.
Fig.3.SplitandCoverStrategy
4 Simulations
In this section, we compare Stripes and Split-and-Cover in simulations. We also
present simulation results that demonstrate the effect of the number of Stripes on
theperformanceoftheStripesalgorithm.Oursimulationswereperformedbyrep-
resentingeachoftheindividualrobotsasapointwithintherectangularworld.Each
robot can be assigned to sweep along a continuous piecewise linear path. As de-
scribed in the Section 3, an active robot can detect an inactive robot when it is
12 E.Meisner,W.Yang,andV.Isler
Fig. 4. This figure shows the simulation of the Stripes algorithm. Green circles are active
robots,andredcirclesareinactive.Theblacklinesrepresentthepathsofactiverobotswithin
thecurrentstripe.
withincommunicationrange.Eachrobotmovesoneunitdistanceperunittimeand
maintainsaninternalclock,torepresentthetimewithrespecttothestartofthefirst
robot.Robotsthat meetcan synchronizeclocks. We place the robotsuniformlyat
random within the environmentat the start of each trial. The simulation runs the
algorithmexactlyasitisstatedinSection3.4.
4.1 Results
Inthefirstsimulation,wecompareStripesandSplit-and-Coverinenvironmentsthat
aresparseanddensewithrespecttothenumberofrobotsperarea.Figure5ashows
theresultsofthe Split-and-CoverandStripesalgorithmsinan environmentwhere
the concentrationof robotsis low. The simulationsuse a 1000x1000environment
and20robots.Theplottedvaluesaretheaveragetimetocompletionof1000trials,
andthevalueforeachnumberofstripesusesthesamesetofinitialrobotdistribu-
tions. Fromthese simulations,we concludethat(i) the performanceofthe Stripes
algorithmissensitivetothenumberofstripes,and(ii)insparseenvironments(with
a“good”stripe-numberselection)StripesoutperformsSplit-and-Cover.Thisisalso
justifiedbyFigures6aand6b,whereweplotthehistogramofrunningtimesofthe
Stripes algorithm (with 25 Stripes) and Split-and-Cover. Stripes not only outper-
formsSplit-and-Coverontheaveragebutalsoitsworstcaseperformanceisconsid-
erablybetter.
On the other hand, when the concentration of robots is high, Split-and-Cover
outperforms Stripes (Figures 5b,6c,6d). There are two reasons that this is true.
First, for Split-and-Cover, the unbalanced split described in the previous section
occurs infrequently when the concentration of robots is high. Second, for Stripes
the overhead cost of meeting up to evenly distribute the coverage of a stripe be-
comes very large compared to the discovery time. In most instances, saturation
took place after the fist stripe due to the high concentration of the robots. There-
fore, the running time of Stripes was often given by the time to discover the first
stripe followed by a parallel scan with 7 robots. In conclusion, simulation results
suggest that Stripes should be used in sparse environments with the number of