Competitive Exploration of Rectilinear Polygons

MikaelHammar  Bengt J. Nilsson y Mia Persson y Abstra t

Exploringapolygonwithrobots, whentherobotsdonothaveknowledgeofthesurroundings

anbeviewedasanonlineproblem. Typi alforonlineproblemsisthatde isionsmustbemade

basedonpasteventswithout ompleteinformationaboutthefuture. Inour asetherobotsdonot

have ompleteinformationabouttheenvironment. Competitiveanalysis anbeusedtomeasure

the performan eofmethodssolvingonlineproblems. The ompetitiveratio ofsu hamethodis

theratiobetweenthemethod'sperforman eandtheperforman e ofthebestmethodhavingfull

knowledgeof the future. We are interested inobtaining good bounds onthe ompetitiveratio

ofexploringpolygonsandprove onstant ompetitivestrategiesandlowerboundsforexploringa

simplere tilinear polygonintheL

1 metri .

1 Introdu tion

Exploring an environmentis an important and well studied problem in roboti s. In many realisti

situations the robots donotpossess ompleteknowledge aboutthe environment, e.g.,they may not

haveamapofthesurroundings[1,2,4,6,7,8,9℄.

Thesear hoftherobots anbeviewedasanonlineproblem sin etherobots'de isionsaboutthe

sear harebasedonlyonthepartoftheenvironmentthattheyhaveseensofar. Weusetheframework

of ompetitive analysis to measure the performan eof an online sear h strategy S. The ompetitive

ratioofSisdenedasthemaximumoftheratioofthedistan etraveledbytherobotthatmovesthe

farthestusingS to theoptimaldistan eofthesear h.

We are interested in obtaining good bounds for the ompetitive ratio of exploring a re tilinear

polygon. Thesear h ismodeledbyapathor losed tourfollowedbyoneormorepointsizedrobots

inside thepolygon,givenastartingpointfor thesear h. Theonlyinformation that therobots have

aboutthesurrounding polygonisthepartofthepolygonthattheytogetherhaveseensofar.

Forthe ase of exploration with onerobot, Deng et al. [4℄ show adeterministi strategy having

ompetitiveratiotwoforthisproblemifdistan eismeasureda ordingtotheL

1

-metri . Hammaret

al. [5℄ proveastrategy with ompetitiveratio5=3andKleinberg[7℄provesalowerbound of 5=4for

the ompetitiveratioofanydeterministi strategy. Wewillshowadeterministi strategyobtaininga

ompetitiveratioof3=2forsear hingare tilinearpolygonintheL

1

-metri withonerobot.



DepartmentofComputerS ien e,SalernoUniversity,Baronissi(SA)-84081,Italy.

email: hammardia.unisa.it

y

Te hnologyandSo iety,MalmöUniversityCollege,S-20506Malmö,Sweden.

inre tilinearpolygonsand ompetitiveresultsonexplorationwithtwoandthree robots.

Thepaperisorganizedasfollows. Inthenextse tionwepresentsomedenitionsandpreliminary

results. In Se tion 3 we give an overview of the strategy by Deng et al. [4℄. Se tion 4 ontainsan

improvedstrategyfor singlerobot explorationgivinga ompetitiveratioof 3=2. Inse tions5and 6

we onsiderpathexplorationandexplorationwithmultiple robots.

2 Preliminaries

Wewillhen eforthalwaysmeasuredistan ea ordingtotheL

1

metri ,i.e.,thedistan ebetweentwo

pointspandq isdenedby

jjp;qjj=jp x q x j+jp y q y j; wherep x andq x

arethex- oordinatesofpandqandp

y andq

y

arethey- oordinates. Wedene the

x-distan ebetweenpandqtobejjp;qjj

x =jp

x q

x

jandthey-distan etobejjp;qjj

y =jp y q y j.

If C is a polygonal urve, then the length of C, denoted length(C), is dened the sum of the

distan esbetween onse utivepairsofsegmentendpointsin C.

LetPbeasimplere tilinearpolygon. TwopointsinParesaidto see ea hother, orbevisible to

ea hother,iftheline segment onne tingthepointsliesinP. LetpbeapointsomewhereinsideP.

Awat hman routethroughpisdened tobea losed urveC that passesthroughpsu hthat every

pointinPisseenbysomepointonC. Theshortestwat hmanroutethroughpisdenotedbySWR

p .

It anbeshownthattheshortestwat hmanrouteinasimplepolygonisa losedpolygonal urve[3℄.

Sin eweareonly interestedin theL

1

lengthof apolygonal urvewe anassumethat the urve

isre tilinear,thatis,thesegmentsofthe urveareallaxisparallel. Notethattheshortestre tilinear

wat hmanroute throughapointpisnotne essarilyunique.

Forapointpin Pwedene fourquadrants withrespe tto p. Thoseare theregionsobtainedby

uttingPalongthetwomaximalaxisparallellinesegmentsthat passthroughp. Thefour quadrants

are denoted Q 1 (p), Q 2 (p), Q 3 (p), andQ 4

(p) in anti- lo kwiseorder from the toprightquadrantto

thebottomrightquadrant. WeletQ

i;j

(p)denotetheunionofQ

i

(p)andQ

j (p).

Considera reex vertex of P. The two edges of P onne ting at the reex vertex an ea h be

extended inside P until the extensions rea h aboundary point. The segments thus onstru tedare

alledextensions and toea hextensionadire tionisasso iated. Thedire tionisthesameasthatof

the ollinearpolygonedgeaswefollowtheboundaryofPin lo kwiseorder. Weusethefour ompass

dire tionsnorth,west,south, andeast todenotethedire tionofanextension.

Lemma 2.1 (Chin, Ntafos [3℄) A losed urveis awat hmanroute forP ifand only ifthe urve

hasatleastonepointto therightofeveryextensionofP.

Ourrstobje tiveistopresenta ompetitiveonlinestrategythatenablesarobottofollowa losed

urvefromthestartpointsinPandba ktoswiththe urvebeingawat hmanrouteforP.

AnextensionesplitsPintotwosetsP

l andP

r

withP

l

totheleftofeandP

r

totheright. Wesay

apointpistothe leftof eifpbelongstoP

l

. Totheright isdened analogously.

Asafurtherdenitionwesaythatanextensioneisaleft extensionwithrespe ttoapointp,ifp

liestotheleft ofe, andanextensionedominates anotherextensione 0

f=v ext(f) C f 0 (b) ext(f) ( ) q v C p f 0 f 0 prin ipalproje tion s (a) f frontier s f 0 r f 0 l f r fl (d)

Figure1: Illustratingdenitions.

of eare also to therightof e 0

. By Lemma 2.1weare only interestedin the extensionsthat are left

extensions with respe t to the starting point s sin ethe other ones already havea point (the point

s)to the rightof them. Sowithout lossof laritywhen we mentionextensionswewill alwaysmean

extensionsthatareleftextensionswithrespe ttos.

3 An Overview of GO

Considerare tilinearpolygonPthat isnotaprioriknown totherobot. Let sbetherobot'sinitial

positioninsideP. Forthestartingposition softherobotweasso iateapointf 0

ontheboundaryof

Pthatisvisiblefrom sand all f 0

theprin ipal proje tion point ofs. Forinstan e, we an hoosef 0

tobetherstpointontheboundarythatishitbyanupwardraystartingats. Letf betheendpoint

oftheboundarythat therobotseesaswes antheboundaryofPin lo kwise order;seeFigure1(a).

Thepointf is alledthe urrentfrontier.

Let C beapolygonal urvestartingat s. Formallyafrontier f of C is avertexof thevisibility

polygon,VP (C)ofCadja enttoanedgeeofVP (C)that isnotanedgeofP. Extendeuntilithits

apointqonCandletvbethevertexofPthatisrsten ounteredaswemovealongthelinesegment

[q;f℄fromqtof. Wedenotetheleftextensionwithrespe ttosasso iatedtothevertexv byext(f);

seeFigures1(b) and( ).

Dengetal. [4℄introdu eanonlinestrategy alledgreedy-online,GO forshort,toexploreasimple

re tilinearpolygonPintheL

1

metri . IfthestartingpointsliesontheboundaryofP,theirstrategy,

we allitBGO,goesasfollows: fromthestartingpoints antheboundary lo kwiseandestablishthe

rstfrontierf. Moveto the losestpointonext(f)and establishthenextfrontier. Continuein this

fashionuntilallofPhasbeenseenandmoveba ktothestartingpoint.

Deng etal. showthat arobotusing strategyBGOto explore are tilinear polygonfollowsatour

withshortestlength,i.e.,BGOhas ompetitiveratioone. Theyalsopresentasimilarstrategy, alled

IGO,forthe asewhenthestartingpointsliesintheinteriorofP. ForIGOtheyshowa ompetitive

ratio oftwo, i.e., IGO spe ies atour that is at mosttwi e aslongastheshortest wat hmanroute

throughs.

IGOshootsarayupwardstoestablishaprin ipalproje tionpointf 0

andthens anstheboundary

lo kwisetoobtainthefrontier. Next,itpro eedsexa tlyasBGO,movingtothe losestpointonthe

extensionofthefrontier,updatingthefrontier,andrepeatingthepro essuntilallofthepolygonhas

establishingthefrontiers andstill havethesame ompetitiveratio. Hen e,BGO anbeseenastwo

strategies, one s anning lo kwise and the other anti- lo kwise. We antherefore parameterize the

twostrategiessothat BGO(p;orient)isthestrategybeginningatsomepointpontheboundaryand

s anningwithorientationorient whereorient iseither lo kwise w oranti- lo kwiseaw.

SimilarlyforIGO, we annotonly hoose tos an lo kwise oranti- lo kwiseforthefrontierbut

also hoose to shoot the ray giving the rst prin ipal proje tion point in any of the four ompass

dire tionsnorth,west,south,oreast. ThusIGOinfa tbe omeseightdierentstrategiesthatwe an

parameterizeasIGO(p;dir;orient)andtheparameterdir anbeoneofnorth,south,west, oreast.

We further dene partial versions of GO starting at boundary and interior points. Strategies

PBGO(p;orient;region)andPIGO(p;dir;orient;region)applyGOuntileithertherobothasexplored

allof region ortherobot leavestheregionregion. Thestrategiesreturnasresulttheposition of the

robot when it leaves region or when region has been explored. Note that PBGO(p;orient;P) and

PIGO(p;dir;orient;P) arethesamestrategiesasBGO(p;orient)andIGO(p;dir;orient)respe tively

ex eptthattheydonotmoveba ktopwhenallofPhasbeenseen.

4 The Strategy CGO

Wepresentanewstrategy ompetitive-greedy-online(CGO)thatexplorestwoquadrantssimultaneosly

without using uptoomu h distan e. We assume that s lies in the interiorof P sin e otherwisewe

anuseBGO anda hievean optimalroute. The strategyusestwofrontierpointssimultaneouslyto

improvethe ompetitiveratio. However,toinitiatetheexploration,thestrategybeginsbyperforming

a s an of the polygon boundary to de ide in whi h dire tion to start the exploration. This is to

minimizethelossini teduponusbyour hoi eofinitialdire tion.

The initial s an works as follows: onstru t the visibility polygon VP(s) of the initial point s.

Considertheset of edgesin VP (s) not oin idingwith theboundaryof P. Theend pointsof these

edges dene aset of frontier points ea h having an asso iated left extension. Let e denote the left

extension that is furthest from s(distan e beingmeasured orthogonallyto theextension). Letl be

theinnitelinethroughe. Werotatetheviewpointofssothat Q

3 (s)andQ 4 (s)interse tlwhereas Q 1 (s)and Q 2

(s) do not. Hen e, e is a horizontal extension lying below s. Theinitial dire tion of

exploration is upwards throughQ

1

(s)and Q

2

(s). Thetwo frontierpoints used by thestrategy are

obtainedas follows: the left frontier f

l

is established byshooting aray towardsthe left for theleft

prin ipalproje tionpointf 0

l

andthens antheboundaryin lo kwisedire tionforf

l

;seeFigure1(d).

Therightfrontier f

r

isestablishedbyshootingaraytowardstherightfortherightprin ipalproje tion

pointf 0

r

andthens antheboundaryinanti- lo kwisedire tionforf

r

;seeFigure1(d). Toea hfrontier

pointweasso iatealeftextensionext(f

l

)andarightextensionext(f

r

)withrespe ttos.

The strategy CGO, presented in pseudo ode below makes use of three dierent substrategies:

CGO-0, CGO-1,and CGO-2,that ea htakes are ofspe i asesthat ano ur. Subsequentlywe

willprovethe orre tnessand ompetitiveratioforea hofthesubstrategies.

Ourstrategy ensuresthatwheneveritperformsoneofthe substrategiesthisis thelast timethat

the outermost while-loop is exe uted. Hen e, the loop is repeated only when the strategy does not

enteranyof thespe iedsubstrategies. Theloopwill leadthestrategy to followa straightline and

wewillmaintaintheinvariantduringthewhile-loopthat alloftheregionQ

3;4

(p)\Q

1;2

s f l =u (a) s (b) u s ( )

Figure2: Illustratingthekeypointu.

seen.

Wedistinguishfour lassesofextensions. Aisthe lassofextensionsewhosedeningedgeisabove

e,Bisthe lassofextensionsewhosedeningedgeisbelowe. Similarly,Listhe lassofextensionse

whosedeningedgeisto theleftofe,andRisthe lassofextensionsewhosedeningedgeisto the

rightofe. For on iseness,weuseC

1 C

2

asashorthandforthe Cartesianprodu tC

1 C 2 of thetwo lassesC 1 andC 2 .

Wedenetwokeyverti esuandvtogetherwiththeirextensionsext(u)andext(v)thatareusefulto

establishthe orre tsubstrategytoenter. ThevertexuliesinQ

2 (s)andvinQ 1 (s). Ifext(f l )2A[B,

thenuisthevertexissuingext(f

l

)andext(u)=ext(f

l

). Ifext(f

l

)2Landext(f

l

) rossestheverti al

linethroughs,thenuisthevertexissuing ext(f

l

)andagainext(u)=ext(f

l

). Ifext(f

l

)2Ldoesnot

rosstheverti allinethroughs,thenuistheleftmostvertexofthebottommostedgevisiblefromthe

robot,ontheboundarygoingfromf

l

lo kwise untilweleaveQ

2

(s). Theextensionext(u)istheleft

extensionissued byu,and hen e,ext(u)2A; seeFigures 2(a),(b), and( ). The vertexv is dened

symmetri allyin Q

1

(s)withrespe ttof

r .

Ea h of the substrategies is presented in sequen e and for ea h of them we prove that if CGO

exe utes the substrategy, then the ompetitive ratio of CGO is bounded by 3=2. Let FR

s

be the

losedroutefollowedbystrategyCGOstartingataninteriorpoints. LetFR

s

(p;q;orient)denotethe

subpath of FR

s

followedin dire tion orient from point pto pointq, where orient aneither be w

( lo kwise)oraw (anti- lo kwise). Similarly,wedenethesubpathSWR

s

(p;q;orient)ofSWR

s . We

denotebySP(p;q)ashortestre tilinearpathfrom pto qinsideP.

Webeginbyestablishingtwosimplebut usefullemmas.

Lemma 4.1 Ift isapointonsometourSWR

s ,thenlength(SWR t )length(SWR s ): Proof: Sin eSWR s

passesthrought,therouteis awat hmanroute throught. Butsin eSWR

t is

theshortestwat hmanroutethrought,thelemmafollows. 2

Lemma 4.2 LetS beasetofpointsthatareen losedbysometourSWR

s ,andletS 1 =S\Q 1;2 (s), S 2 =S\Q 2;3 (s),S 3 =S\Q 3;4 (s),andS 4 =S\Q 1;4 (s). Then length(SWR s )2max p2S 1 fjjs;pjj y g+2max p2S 2 fjjs;pjj x g+2max p2S 3 fjjs;pjj y g+2max p2S 4 fjjs;pjj x g: Proof: SWR s

en losesallthepointsin S andsin ewe al ulate lengtha ordingto theL

1

metri ,

the smallesttour en losing the pointsis thesmallest re tangle ontaining them. Thelength of the

apointt thatwe anensureispassedbySWR

s

and thateither liesontheboundaryofPor anbe

viewedasto lieontheboundaryofP. Wethen onsider thetourSWR

t

and ompareits lengththe

lengthof FR

s

. ByLemma 4.1 weknowthat length(SWR

t

)length(SWR

s

),hen ethe dieren ein

lengthbetweenFR

s

andSWR

t

isanupperbound onthelossprodu edbyCGO.

Strategy CGO

1 Establishtheexplorationdire tionbyperformingtheinitials anofthepolygonboundary

2 Establishtheleftandrightprin ipalproje tionpointsf 0

l andf

0

r

forQ2(s)andQ1(s)respe tively

3 while Q1;2(s)isnot ompletelyseen do

3.1 Obtaintheleftandrightfrontiers,flandfr

3.2 if f l liesinQ 2 (s)andf r liesinQ 1 (s) then

3.2.1 Updateverti esuandvasdes ribedinthetext

3.2.2 if (ext(u);ext(v))2LRor (ext(u);ext(v))2AR[LAand ext(u) rossesext(v)

then

3.2.2.1 Gotothe losesthorizontalextension

elseif (ext(u);ext(v))2BR[LB or (ext(u);ext(v))2AR[LAand ext(u) doesnot

rossext(v)

then

3.2.2.2 ApplysubstrategyCGO-1

elseif (ext(u);ext(v))2AA[AB[BA[BB then

3.2.2.3 ApplysubstrategyCGO-2

endif

else

3.2.3 ApplysubstrategyCGO-0

endif

endwhile

4 if Pisnot ompletelyseen then

4.1 ApplysubstrategyCGO-0

endif

End CGO

Westart by presenting CGO-0, that does the following: Letp bethe urrentrobot position. If

Q

1

(p) is ompletely seenfrom p then we run PIGO(p;north;aw;P) and move ba k to the starting

points,otherwiseQ

2

(p)is ompletely seenfrom pandwerunPIGO(p;north; w;P) andmoveba k

tothestartingpoints.

Lemma 4.3 IfthestrategyappliessubstrategyCGO-0,thenlength(FR

s

)=length(SWR

s ):

Proof: Assume that CGO-0realizes that when FR

s

rea hesthepointp, then Q

1

(p) is ompletely

seenfrom p. Theother ase,thatQ

2

(p)is ompletelyseenfrom pissymmetri .

Sin ethepathFR

s

(s;p;orient)thatthestrategyhasfollowedwhenitrea hespointpisastraight

line,thepointpisthe urrentlytopmostpointofthepath. Hen e,we anaddaverti alspikeissued

bytheboundarypointimmediatelyabovep,givinganewpolygonP 0

havingpontheboundaryand

furthermore with the same shortest wat hmanroute through p as P. This means that performing

strategyIGO(p;north;orient)inPyieldsthesameresultasperformingBGO(p;orient)inP 0

aboundarypointin P, and orient beingeither w oraw. The tourfollowedistherefore ashortest

wat hmanroute throughthepointpin bothP 0

andP.

Also the point p lies on an extension with respe t to s, by the way p is dened, and it is the

losestpointtossu hthatallofQ

1

(s)hasbeenseenbythepathFR

s

(s;p;orient)=SP(s;p). Hen e,

there isarouteSWR

s

that ontainspand byLemma 4.1length(SWR

p )length(SWR s ). Thetour followedequals FR s =SP(s;p)[SWR p

(p;s;aw);and wehavethat length(FR

s )=length(SWR p ) length(SWR s );andsin eFR s

annotbestri tlyshorterthanSWR

s

theequalityholdswhi h on ludes

theproof. 2

Next wepresentCGO-1. Let uand v be the key verti esas dened earlier. The strategydoes

thefollowing: if(ext(u);ext(v))2LA[LB, we mirrorthe polygon Pat theverti alline throughs

and swapthe namesof uand v. Hen e,(ext(u);ext(v))2AR[BR. We ontinuemovingupwards

updatingf

r

and vuntileither allofQ

1

(s)hasbeenseenorext(v) nolonger rossestheverti alline

throughs.

Ifallof Q

1

(s)hasbeenseenthenweexploretheremainingpartofPusingPIGO(p;east;aw;P),

wherepisthe urrentrobotposition.

Ifext(v)nolonger rossestheverti allinethroughsthenweeitherneedto ontinuetheexploration

bymovingtotherightorreturntouandexploretheremainingpartofthepolygonfromthere.

Ifjjs;pjj y +jjs;ujj x jjs;vjj x

we hooseto returntou. Ifext(u)2AwerunPBGO(u;aw;P)and

ifext(u)2BweusePBGO(u; w;P);seeFigure3. Otherwise,jjs;pjj

y +jjs;ujj x >jjs;vjj x andinthis

asewemovetothe losestpointv 0

onext(v). Bydenition,theextensionofv iseitherinAorBin

this ase.

Ifext(v)2Bthenv=v 0

andwe hoosetorun PBGO(v;aw;P). Otherwise,ext(v)2A. IfQ

1 (v 0 ) isseenfromv 0

thentheentirequadranthasbeenexploredandwerunPIGO(v 0

;east;aw;P)toexplore

the remainder of thepolygon. If Q

1 (v

0

) is notseen from v 0

then there are still things hidden from

therobotinQ

1

(v). WeexploretherestofthequadrantusingPBGO(v 0

;north;aw;Q

1

(v))rea hinga

pointqwherease ondde isionneedsto bemade.

Ifvisseenfromthestartingpointandjjs;qjj

x

jjs;vjj,wegoba ktov andrunPBGO(v;aw;P),

otherwisewerunPIGO(q;east; w;P)fromtheinteriorpointq;seeFigure 5.

Ifv isnotseenfromthestartingpointsthenwegoba ktov andrunPBGO(v;aw;P).

TonishthesubstrategyCGO-1ourlaststepisto returntothestartingpoints.

Lemma 4.4 IfthestrategyappliessubstrategyCGO-1,thenlength(FR

s ) 3 2 length(SWR s ):

Proof: Wehandle ea h aseseparately. Assume fortherst asethat when FR

s

rea hesthepoint

p,thenQ

1

(p)is ompletelyvisible. Hen e,wehavethesamesituationasin theproofof Lemma4.3

andusingthesameproofte hniqueitfollowsthatlength(FR

s

)=length(SWR

s ).

Assumeforthese ond asethatCGO-1de idestogoba ktou,i.e.,thatjjs;pjj

y +jjs;ujj x jjs;vjj x ;

seeFigures3(a)and(b). Thetourfollowedequals oneof

FR s =  SP(s;p)[SP(p;u)[SWR u [SP(u;s) SP(s;p)[SP(p;u)[SWR u (u;r; w)[SP(r;s)

wherer isthelastinterse tion pointofFR

s

withthehorizontalline throughs. Usingthatjjs;pjj

y + jjs;ujj x jjs;vjj x

itfollowsthatthelengthofFR

s

in both asesisboundedby

length(FR s ) = jjs;pjj+jjp;ujj+length(SWR u )+jju;sjj = length(SWR u )+2jjs;pjj y +2jjs;ujj x

s FR s SWR u v u r (b) s FR s u SWR u v (a)

Figure3:Illustratingthe asesinLemma4.4whenjjs;pjj

y +jjs;ujj x jjs;vjj x . s v FRs r SWRv u p (a) s FRs r SWRv p v v 0 u (b)

Figure4:IllustratingtheproofofLemma4.4whenjjs;pjjy+jjs;ujjx>jjs;vjjx.

 length(SWR s )+jjs;pjj y +jjs;ujj x +jjs;vjj x  3 2 length(SWR s ):

Theinequalitiesfollowfrom theassumptiontogetherwithLemmas 4.1and4.2.

Assume forthethird asethat CGO-1goestotheright,i.e.,that jjs;pjj

y +jjs;ujj x >jjs;vjj x . We

beginby handlingthe dierentsub ases that are independent of whether s sees v; see Figures 4(a)

and(b). Thetourfollowedequalsoneof

FR s =  SP(s;v)[SWR v (v;r;aw)[SP(r;s) SP(s;v 0 )[SWR v 0 (v 0 ;r;aw)[SP(r;s) Sin ejjs;vjj x =jjs;v 0 jj x thelengthofFR s

isin bothsub asesboundedby

length(FR s )  length(SWR s )+2jjs;vjj x < length(SWR s )+jjs;pjj y +jjs;ujj x +jjs;vjj x  3 2 length(SWR s );

y x x

indeedseenfroms;seeFigures5(a)and(b). Thetourfollowedin this aseisoneof

FR s =  SP(s;v)[SWR v (v;q; w)[SP(q;v)[SWR v (v;r;aw)[SP(r;s) () SP(s;v)[SWR v [SP(v;s)

whereq istheresultinglo ationafterexploring Q

1

(v). Hereweusethatv isseenfroms,and hen e,

thattheinitials anguaranteesthatthereisapointtofSWR

s inQ 3;4 (s)su hthatjjs;tjj y jjs;vjj x , thusFR s isboundedby length(FR s ) = length(SWR v )+2minfjjs;vjj;jjs;qjj x g  length(SWR s )+jjs;vjj y +jjs;vjj x +jjs;qjj x < length(SWR s )+jjs;vjj y +jjs;tjj y +jjs;qjj x +jjs;ujj x  3 2 length(SWR s ):

Ontheotherhand,whenvisnotseenfroms,thetourfollowsthepathmarkedwith()above;see

Figure5( ). Thus,thepolygonboundaryobs urestheviewfromstov,andhen e,thereisapointq 0

ontheboundarysu hthat theshortestpath froms to v 0

ontainsq 0

. The pathourstrategyfollows

betweens andv 0

is ashortestpathand we anthereforeassume that italso passedthroughq 0 . We usethatjjs;q 0 jj x jjs;vjj x jjs;qjj x

togetthebound.

length(FR s ) = length(SWR q 0 )+2jjs;q 0 jj x  length(SWR s )+jjs;vjj x +jjs;qjj x < length(SWR s )+jjs;vjj y +jjs;ujj x +jjs;qjj x  3 2 length(SWR s ):

TheinequalitiesabovefollowfromLemmas4.1and4.2andthis on ludestheproof. 2

We ontinuetheanalysisbyrstshowingthesubstrategyCGO-2andthenprovingits ompetitive

ratio. Thestrategydoesthefollowing:ifjjs;ujj

x

jjs;vjj

x

thenwemirrorPattheverti allinethrough

salsoswappingthenamesofuandv. Thismeansthatv is losertothe urrentpointpwithrespe t

to x-distan ethanu. Next,go tov 0

, the losestpointonext(v). Ifext(v)2B, run PBGO(v;aw;P)

sin ev=v 0 . Ifext(v)2AandQ 1 (v)isseenfromv 0

thenwerunPIGO(v 0

;east;aw;P). Ifext(v)2A

but Q

1

(v) is not ompletely seenfrom v 0 then we exploreQ 1 (v) using PBGO(v 0 ;north; w;Q 1 (v 0 )). On eQ 1

(v)isexplored wehaverea hedapointqandwemakease ondde ision. Ifjjs;qjj

x

jjs;vjj,

goba ktov andrunPBGO(v;aw;P),otherwiserunPIGO(q;east; w;P). Finallygoba ktos.

Lemma 4.5 IfthestrategyappliessubstrategyCGO-2,thenlength(FR

s ) 3 2 length(SWR s ):

Proof: Assume withoutlossof generalitythatjjs;ujj

x

>jjs;vjj

x

. Theother aseisproved

symmet-ri ally.

Next,assumethat FR

s

passesthroughv; seeFigures6(a),(b), and( ). Thetourfollowedequals

oneof FR s = 8 < : SP(s;v)[SWR v (v;r;aw)[SP(r;s) SP(s;v)[SWR v (v;q; w)[SP(q;v)[SWR v (v;r;aw)[SP(r;s) SP(s;v)[SWR v [SP(v;s)

whereris thelast interse tionpointofFR

s

withthehorizontallinethroughs. ThelengthofFR

s is

inea h asebounded by

length(FR s )=length(SWR v )+2minfjjs;qjj x ;jjs;vjjg:

s FR s r SWRv p v 0 u (a) p v 0 u q FRs s SWRv (b) v s p v v 0 u q SWRv FRs q 0 r ( )

Figure5: IllustratingtheproofofLemma4.4.

Wehavethat minfjjs;qjj x ;jjs;vjjg(jjs;qjj x +jjs;vjj)=2(jjs;qjj x +jjs;vjj y +jjs;ujj x )=2length(SWR s )=4

provingthebound inthis ase. Thelast inequalityfollowsfromLemma 4.2.

If FR

s

doesnotpassthroughv;see Figure6(d); then thetourfollowedequalsFR

s =SP(s;v 0 )[ SWR v 0(v 0

;r;aw)[SP(r;s) where r is the last interse tion point of FR

s

with the horizontal line

throughs. ThelengthofFR

s is length(FR s )=length(SWR v 0 )+2jjs;v 0 jj x length(SWR s )+jjs;v 0 jj x +jjs;ujj x  3 2 length(SWR s ):

Theinequalitiesfollowfrom Lemmas4.1and4.2andthat jjs;v 0

jj

y

0,whi h on ludes theproof. 2

Wehaveprovedthefollowingtheorem.

Theorem 1 CGOis3=2- ompetitive.

5 The Path Problem

Considernowthesituationin whi h we,insteadofa losedtour,wishto obtainashortestpaththat

explorestheinteriorofourre tilinearpolygon,i.e.,thepathfollowedbytherobotdoesnothavetoend

atthestartingpoint. LetOPT P

s

beashortestexplorationpathbeginningats. Itfollowsimmediately

thatlength(OPT P

s

)length(SWR

s

)=2,sin efollowingthepathOPT P

s

toitsendpointandthenba k

s SWR v FRs r v p (a) s SWR v v q FRs p v 0 r jjs;qjjxjjs;vjj (b) s SWR v u FRs q v v 0 p ( ) jjs;qjj x >jjs;vjj s SWR v v FRs p v 0 u r (d)

s p x (b) p s (a) p s ( ) x

Figure7: IllustratingtheproofofTheorem2.

Fromthiswededu ethat thestrategyCGO presentedpreviouslyis 3- ompetitiveforpath

explo-ration. We ontinueto showthatanystrategyforpathexplorationmustbeatleast2- ompetitive.

Theorem 2 Thereisnodeterministi strategyforpathexplorationofare tilinearpolygonthat has

ompetitiveratio2 forany>0.

Proof: We onstru ta ounterexampleasinFigure7. Thestartingpointfortherobotisatthelower

left ornerofthepolygonanditessentiallyseesonlythetwowallsadja enttoit;see Figure7(a).

Therobotnowhastomovetooneoftheextensionsthatitsees. Thesearebothatdistan e1from

thestartingpoint. Assume withoutlossofgeneralitythat itmovesto thehorizontal extension,then

the robot realizesthat there is afurther horizontal extensionat distan e Æ aboveit. Therobot has

theoptionofeither ontinuingupwardsuntilithasvisitedallthehorizontalextensions(ofwhi h the

orresponding frontierpoint is onlyvisible from the previousextension and theextensions are only

separatedbyadistan eofÆ);seeFigure7(b). Thisoptionwillmaketherobotmoveadistan eof1=Æ

upwardsuntilit rea hesthelast extension andmovesto theverti al extensionwhere itrealizesthat

thereisanot hatpointx (whi hliesatdistan eÆbelowthestartingpoints)for ingittomoveba k

downto thispoint. Thetotaldistan emovedis2=Æ+1+Æ.

Theoptimal path is to move to the verti al extension rst, visit the not h at x and then move

upwardsuntil allhorizontal extensionshavebeenvisited, requiringonlyadistan eof1+2Æ+1=Æto

bemoved. Theratiobe omes

2=Æ+1+Æ 1=Æ+1+2Æ =2 Æ+3Æ 2 1+Æ+2Æ 2 2  ifÆ=4<1=4.

Ontheotherhand,iftherobotatsomepointaftermovingtothersthorizontalextensionde ides

tomovetotheverti alextensionitthenrealizesthatithastomoveba ktothenot hatx. Ifitde ides

to ontinueupwardsuntilallhorizontalextensionshavebeenvisitedwehavetheprevioussituation. If

terminatesatpointt;seeFigure7( ). Assume thattherobothasmovedaverti aldistan eofD1

whenitde idestomovedownandvisitthenot hatx. Thetotaldistan emovedisthen3D+1+3Æ

whereastheoptimalpathhaslengthatmostD+1+3Æand theratiobe omes

3D+1+3Æ D+1+3Æ =3 2+6Æ D+1+3Æ 2 

ifÆ=2<1=2,thus on ludingtheproof. 2

6 Exploration with Multiple Robots

Wenowlookatthesituationwhenseveralrobotstogetherarerequiredtoexploreare tilinearpolygon.

Againwelookatthetourvariant,i.e.,ea hrobotmustterminatetheaxplorationatthestartingpoint.

Wegiveupperandlowerbounds forthesituationwith twoandthree robotsallstartingat thesame

point. themeasurethatweoptimizeonisthelengthofthelongesttourthatanyoftherobotsfollow.

LetOPT k

s

bethetouroftherobotthatmovesthelongestlengthofallthekrobots. Sin easingle

robot anfollow ea h of thetours that thek robots followand thus getawat hmanroute, wehave

thatlength(OPT k s )length(SWR s )=k.

We prove lower bounds on the ompetitive ratio of any exploration strategy using twoor three

robots.

Theorem 3 Therearenodeterministi strategiesforexploringare tilinearpolygonwithtwoorthree

robotshavingsmaller ompetitiveratiothan3/2.

Proof: Werstshowthelowerbound fortworobotsandlaterextenditforthreerobots. Thelower

boundisbasedonessentiallythesame ounterexamplethat Kleinbergusesforthelowerbound fora

singlerobot[7℄.

Theinitial polygon is given in Figure 8(a)and onsists of asquare with not hes in the orners.

Thelengthofthesides ofthesquareis 2. Ea hof thetworobots hastomoveadistan eof 2before

itgets to a ornerof thepolygon andfurthermore at mosttwo omplete orners anbeseenby the

robots. Hen epla inganot hinoneofthe ornersthatisnotyet ompletelyseenrequiresoneofthe

robots to move6units whereastheoptimal motion anbedonewith only4units, thus proving the

result.

Thesameproofa tuallygoesthroughforthreerobotson eyourealizethat independentlyofhow

thethreerobotsstarttheirexplorationwe anfor eoneofthemtomove6unitspla ingatmostthree

not hesasshownin Figure8(b). 2

Wealso show astrategy fortworobots that has ompetitive ratio2. We allthe strategyTGO

(two-robot GO) sin e it is based on the GO-strategyof Deng et al; [4℄. The two robots ea h run

IGO(s;north;orient),onewithorient = w andtheotherwithorient=aw untilthetworobotshave

seenthe ompletepolygonafterwhi h theybothmoveba ktothestartingpoint.

Theorem 4 The strategy TGO is 2- ompetitive for exploration of a re tilinear polygon with two

s not h (a) s (b) not h

Figure8: IllustratingtheproofofTheorem3.

Proof: Let r bethe interse tion pointof SWR

s

andthe verti al axisissuingfrom s upwards. The

interse tion point r lies at distan e D 0from s; see Figure 9(a). We an viewthe strategyTGO

asrstmovingthe tworobotsfrom s tor and thenseparating, onemoving lo kwise and theother

ounter lo kwise,followingSWR

r

in twodire tionsuntiltherobotshaveseenallof thepolygonand

moveba kto s.

Considernow the robot that movesthe farthest. Lett bethe pointof interse tion between the

robot'swalkandthelastextensionthatitvisitsbeforeitrealizesthat thewholepolygonisexplored.

The robot thenmoves thedistan e L =D+length(SWR

r

(r;t;dir))+length(SP(t;s));where dir is

thedire tionthattherobotmoves.

SupposethatwefollowthetourSWR

s

fromsin thedire tionthat visitsthepointrbeforet. We

an assume that t is a point on SWR

s

sin e it is an interse tion point with an extension. Assume

withoutlossofgeneralitythatthisdire tion is lo kwise. Wethushavethat

L = D+length(SWR r (r;t;dir))+length(SP(t;s))  length(SWR s (s;r; w))+length(SWR s (r;t; w))+length(SWR s (t;s; w)) = length(SWR s )  2length(OPT 2 s )

whi hprovestheresult. Thattheanalysisistightfollowsfromtheexamplein Figure9(b). 2

7 Con lusions

We havepresented onstant ompetitivestrategies and lowerbounds to explore are tilinear simple

polygonintheL

1

metri withoneormorerobots. Unfortunatelynoneofourresultsaretightsoobvious

open problems are to redu e thegaps betweenthe lowerbounds and the upper bounds. Espe ially

SWRs

t SP(t;s)

s

(a) (b)

Figure9: IllustratingtheproofofTheorem4.

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