Improved Exploration of Rectilinear Polygons

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Improved Exploration of Re tilinear Polygons

∗

MikaelHammar

†

Bengt J. Nilsson

‡

Sven S huierer

§

Abstra t

Weprovea5/3- ompetitivestrategyforexploringasimplere tilinear polygon inthe

L

1

metri . Thisimprovesthepreviousfa tortwoboundofDeng,KamedaandPapadimitriou.

1 Introdu tion

Sear hinginanenvironmentisanimportantandwellstudiedprobleminroboti s. Inmanyrealisti

situations therobot doesnotpossess omplete knowledge aboutitsenvironment,for instan e, it

maynothaveamapofitssurroundings[1,2,5, 6,7,9,8,10,11,12,13,14℄.

Thesear hoftherobot anbeviewedasanon-line problemsin etherobot'sde isionsabout

the sear h are based only on the part of its environment that it has seen so far. We use the

frameworkof ompetitiveanalysis tomeasuretheperforman eofanon-linesear hstrategy

S

[15℄.

The ompetitive ratio of

S

is dened as themaximum of theratio of the distan e traveledby a

robotusing

S

totheoptimaldistan eofthesear h.

Infa t,these sear h problems ome inbasi ally twoavors. Insome aseswewantto sear h

fora targetin the environment[5, 6, 9, 10, 11, 12℄. Thetarget is usuallymodeledasa pointin

theenvironmentthat hasto berea hed by therobot. Anotherproblem typeisthat ofexploring

anenvironment, i.e., withoutaprioriknowledge ofthe environmentletarobot onstru tamap

ofit[7,8,11℄.

Weareinterestedinobtainingimprovedupperboundsforthe ompetitiveratioofexploringa

re tilinearpolygon. Thesear hismodeledbyapathor losedtourfollowedbyapointsizedrobot

insidethepolygon,givenastartingpointforthesear h. Theonlyinformationthattherobothas

aboutits surrounding polygon is thepartof the polygon that it hasseenso far. Deng et al. [7℄

showadeterministi strategyhaving ompetitiveratiotwoforthisproblemifdistan eismeasured

a ordingto the

L

1

-metri . Kleinberg[11℄provesalowerbound of

5/4

forthe ompetitiveratio of any deterministi strategy. We will showa strategy obtaining a ompetitive ratio of

5/3

for

sear hingare tilinearpolygoninthe

L

1

-metri .

∗

Thisresear hissupportedbytheDFG-Proje tDiskreteProbleme,No.Ot64/8-1.

†

DepartmentofComputerS ien e,LundUniversity,Box118,S-22100Lund,Sweden.

email:Mikael.Hammar s.lth.s e

‡

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

email:Bengt.Nilssonte.mah.s e

§

InstitutfürInformatik,AmFlughafen17,Geb.051,D-79110Freiburg,Germany.

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naryresults. InSe tion3wegiveanoverviewofthestrategybyDengetal.[7℄. Se tion4 ontains

animprovedversiongivinga ompetitiveratioof

5/3

.

2 Preliminaries

Wewillhen eforthalwaysmeasuredistan ea ordingtothe

L

1

metri ,i.e.,thedistan ebetween twopoints

p

and

q

isdenedby

||p, q|| = |p

x

− q

x

| + |p

y

− q

y

|,

where

p

x

and

q

x

arethe

x

- oordinatesof

p

and

q

and

p

y

and

q

y

arethe

y

- oordinates. Wedene the

x

-distan ebetween

p

and

q

tobe

||p, q||

x

= |p

x

−q

x

|

andthe

y

-distan etobe

||p, q||

y

= |p

y

−q

y

|

. If

C

isapolygonal urve,thenthelengthof

C

,denoted

length(C)

,isdenedasthesumofthe

distan esbetween onse utivepairsofsegmentendpointsin

C

.

Let

P

beasimplere tilinearpolygon. Twopointsin

P

aresaidtosee ea hother,orbevisible

to ea h other, if theline segment onne ting thepointslies in

P

. Forapoint

p

in

P

wedenote

by

VP(p)

thesetofpointsof

P

that arevisiblebytherobotifitstandsonposition

p

.

VP(p)

is

alledthevisibilitypolygon of

p

. If

C

isa urvein

P

,thevisibilitypolygon

VP

(C)

of

C

isdened

by

VP(C) =

S

p∈C

VP(p)

.

Let

p

be apointsomewhere inside

P

. A wat hman route through

p

is dened tobea losed

urve

C

that passes through

p

with

VP

(C) = P

. The shortest wat hman route through

p

is

denoted by

SWR

p

. It an be shown that theshortest wat hmanroute in asimple polygon is a losedpolygonal urve[3,4℄.

Sin e we are only interested in the

L

1

length of a polygonal urve we an assume that the urveisre tilinear,that is,the segmentsof the urveareallaxisparallel. Notethat theshortest

re tilinearwat hmanroutethroughapoint

p

isnotne essarilyunique.

Forapoint

p

in

P

wedenefourquadrants withrespe tto

p

. Thoseare theregionsobtained

by utting

P

along thetwo maximal axis parallel line segmentsthat pass through

p

. The four

quadrantsaredenoted

Q1

(p)

,

Q2

(p)

,

Q3

(p)

,and

Q4

(p)

intheorder fromthetoprightquadrant

tothebottomrightquadrant;seeFigure1(a).

Considerareex vertexof

P

. Thetwoedgesof

P

onne tingatthereex vertex anea hbe

extendedinside

P

untiltheextensionsrea haboundarypoint. Thesegmentsthus onstru tedare

alledextensionsandtoea hextensionadire tionisasso iated. Thedire tionisthesameasthat

ofthe ollinearpolygonedgeaswefollowtheboundaryof

P

in lo kwiseorder;see Figure1(b).

We use the four ompass dire tions

north

,

west

,

south

, and

east

to denote the dire tion of an

extension. Thereasonforthissomewhatobs ure denition ishopefully lariedbythefollowing

lemma.

Lemma2.1 (Chin,Ntafos [3℄)A losed urveisawat hmanroutefor

P

ifandonlyifthe urve

hasat leastonepointtotherightofeveryextensionof

P

.

Ourobje tive isthusto presenta ompetitiveon linestrategy that enablesarobot tofollow

a losed urvefrom thestartpoint

s

in

P

andba kto

s

withthe urvebeingawat hmanroute

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Q1(p)

Q2(p)

Q3(p)

Q4(p)

(a) (b) extensions

p

e

′

p

e

dominates

e

′

e

isaleft extensionw.r.t.

p

( )

Figure1: Illustratingdenitions.

Anextension

e

splits

P

intotwosets

P

l

and

P

r

with

P

l

totheleftof

e

and

P

r

totheright. We sayapoint

p

istotheleft of

e

if

p

belongsto

P

l

. Tothe right isdenedanalogously.

As afurther denition we saythat an extension

e

is aleft extension with respe t to a point

p

, if

p

lies to theleft of

e

, and anextension

e

dominates anotherextension

e

′

, if allpointsof

P

totherightof

e

arealsoto therightof

e

′

;see Figure1( ). ByLemma2.1 weareonlyinterested

intheextensionsthat areleftextensionswithrespe tto thestartingpoint

s

sin etheotherones

alreadyhaveapoint(thepoint

s

)totherightofthem. Sowithoutlossof laritywhenwemention

extensionswewillalwaysmeanextensionsthat areleftextensionswithrespe tto

s

.

Asapreludetothepresentationofourstrategywegiveashortoverviewofpreviousstrategies.

Inthiswaywe anintrodu e on eptsin theiroriginalsettingthat wewilluse extensivelyin the

sequel.

3 An Overview of GO

Considera re tilinear polygon

P

that is not a priori known to the robot. Let

s

be the robot's

initialpositioninside

P

.

Thenext on eptweintrodu eis ru ialtotheworkingsofpreviouslypresentedstrategiesand

isgoingtobeveryimportantforourstrategyaswell. Forthestartingposition

s

oftherobot we

asso iateapoint

f

0

ontheboundaryof

P

thatisvisiblefrom

s

and all

f

0

theprin ipal proje tion

point of

s

. For instan e,we an hoose

f

0

tobetherstpointontheboundarythatis hitbyan

upwardraystartingat

s

. Assumenowthattherobotmovesalonga urve

C

from

s

tosomepoint

p

. Let

f

betheendpointthat therobot hasseensofaraswemove lo kwisefrom

f

0

alongthe

boundaryof

P

;seeFigure2. Thepoint

f

is alledthe urrentfrontier.

Let

C

be a polygonal urve starting at

s

. Formallya frontier

f

of

C

is a vertexof

VP(C)

adja entto anedge

e

of

VP(C)

that isnot anedge of

P

. Extend

e

until ithits apoint

q

on

C

andlet

v

bethevertexof

P

that isrsten ounteredaswemovealongthelinesegment

[q, f ]

from

q

to

f

. We denotethe left extension with respe t to

s

asso iatedto the vertex

v

by

ext(f )

; see Figures2(b)and( ).

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s

f

0

prin ipalproje tion

f

frontier

VP(s)

f = v

ext

(f )

ext

(f )

C

( ) (b) (a)

q

v

C

p

f

0 f

Figure2: Illustratingdenitions.

theirstrategy,we all itBGO,goesasfollows.

Strategy BGO

1 Using

s

itselfasprin ipalproje tionpointestablishtherstfrontier

f

2

p := s

3 while

P

isnot ompletelyseen do

3.1 Gotothepointon

ext

(f )

losestto

p

,letthispointbethenewpoint

p

3.2 Using the urrent point

f

onthe boundaryasprin ipal proje tion point establish the newfrontier

f

endwhile

4 Taketheshortestpathba kto

s

End BGO

Dengetal.showthatarobotusingstrategyBGOtoexploreare tilinearpolygonfollowsatour

with shortestlength, i.e., BGO has ompetitiveratio one. Theyalso presenta similar strategy,

alled IGO, forthe asewhen the starting point

s

lies in the interior of

P

. ForIGO theyshow

a ompetitiveratiooftwo,i.e., IGOspe iesatourthat isat mosttwi e aslongastheshortest

wat hmanroute. IGOdoesthefollowing.

Strategy IGO

1 From

s

shoot aray upwards until aboundarypoint

f

0

is hit, use

f

0

as prin ipal proje tion

point andestablishtherstfrontier

f

2

p := s

3 while

P

isnot ompletelyseen do

3.1 Gotothepointon

ext

(f )

losestto

p

,letthispointbethenewpoint

p

3.2 Using the urrent point

f

onthe boundaryasprin ipal proje tion point establish the newfrontier

f

endwhile

4 Taketheshortestpathba kto

s

End IGO

ForanexamplerunofstrategyIGOseeFigure3. Thepoints

f

i

,for

1 ≤ i ≤ 19

, arethefrontiers

intheorder thattheyareen ounteredalongtheboundary. Thedire ted linesare theextensions

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s

f

1 f

4 f

14 f

16 f

19 f

6 f

7 f

3 f

0 f

2 f

5 f

8 f

9 f

12 f

13 f

15 f

11 f

17 f

18 f

10

Figure3: Examplerunofthegreedy-onlinestrategy.

Asis easilyseen,thetwostrategiesdieronlyin therststepwherein one aseweestablish

therstfrontierstartingdire tlyfrom

s

,whereasintheother aseweshootarayinordertond

theprin ipalproje tionpointandfromthereweestablishtherstfrontier.

It is lear that BGO ould just as wells an the boundaryanti- lo kwise insteadof lo kwise

when establishing the frontiers and still have the same ompetitive ratio. Hen e, BGO an be

seen as two strategies, one s anning lo kwise and the other anti- lo kwise. We an therefore

parameterize the twostrategies so that BGO

(p, orient )

is the strategy beginning at some point

p

on the boundary and s anning with orientation

orient

where

orient

is either lo kwise

cw

or anti- lo kwise

aw

.

SimilarlyforIGO,we annotonly hoosetos an lo kwiseoranti- lo kwiseforthefrontierbut

also hoose toshoot theraygivingtherstprin ipalproje tionpointin anyofthefour ompass

dire tionsnorth,west,south,oreast. ThusIGOinfa tbe omeseightdierentstrategiesthat we

anparameterizeasIGO

(p, dir , orient )

andthe parameter

dir

anbe oneof

north

,

south

,

west

,

or

east

.

Wefurther dene partial versions of GOstarting atboundaryand interiorpoints. Strategies

PBGO

(p, orient , region)

and PIGO

(p, dir , orient , region)

apply GO untileither therobot has

ex-ploredallof

region

ortherobotleavestheregion

region

. Thestrategiesreturnasresulttheposition

oftherobotwhenitleaves

region

orwhen

region

hasbeenexplored. NotethatPBGO

(p, orient , P)

andPIGO

(p, dir , orient , P)

are thesamestrategiesasBGO

(p, orient )

andIGO

(p, dir , orient )

re-spe tivelyex eptthattheydonotmoveba kto

p

whenallof

P

hasbeenseen.

4 The Strategy DGO

Wepresentanewstrategydoublegreedyonline(DGO)thatstartsobyexploringtherstquadrant

of

s

,

Q1

(s)

,withoutusinguptoomu h distan e. Weassumethat

s

liesin theinteriorof

P

sin e

otherwisewe anuse BGO and a hievean optimalroute. Thestrategy uses two frontier points

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s

f

0 u

f

0 r

f

r

Figure4: Upperandrightfrontierstogetherwiththeirextensions.

sameastheoneusedinGO,i.e., weshootaraystraightuptogettheupperprin ipalproje tion

point

f

0 u

and s an in lo kwise dire tion for

f

u

. Theright frontier

f

r

isestablished byshooting araytowardstherightfortherightprin ipalproje tionpoint

f

0 r

andthens an theboundaryin anti- lo kwise dire tion for

f

r

; seeFigure 4. Toea h frontierpointweasso iate aleft extension

ext(f

u

)

and

ext(f

r

)

withrespe tto

s

inthesamewayasbefore.

ThestrategyDGO,presentedinpseudo odeonPage7,makesuseofvedierentsubstrategies:

DGO-0,DGO-1,DGO-2,DGO-3,andDGO-4,thatea htake areofaspe i asethat ano ur.

Subsequentlywewillprovethe orre tnessand ompetitiveratioforea hofthesubstrategies.

We distinguish four lassesof extensions.

A

is the lass of extensions

e

whose dening edge

isabove

e

,

B

is the lass ofextensions

e

whosedening edge isbelow

e

. Similarly,

L

isthe lass

of extensions

e

whose dening edge is to the left of

e

, and

R

is the lass of extensions

e

whose

dening edge is to the right of

e

. For on iseness, we use

C

1 C

2

asashorthand for the artesian produ t

C

1 × C

2

ofthetwo lasses

C

1

and

C

2

.

We will ensure that whenever the strategy performs one of the substrategies this is the last

timethattheoutermostwhile-loopisexe uted. Hen e,theloopisrepeatedonlywhenthestrategy

doesnotenter anyofthespe iedsubstrategies. Theloopwill leadthestrategyto followan

x

-

y

-monotonepathinside

Q1

(s)

sin ewewill ensurethat

ext(f

u

)

and

ext(f

r

)

alwayslieeitherabove or to the right of the point

p

, the urrent position of the robot, in these ases. The strategy

additionallymaintainsthe invariant during the while-loop that the boundary parts of

Q1(s)

to

theleftofthe

x

- oordinateof

p

andbelowthe

y

- oordinateof

p

havebeenseen.

Wewill presentea h ofthe hosensubstrategies in sequen eand forea h of themprovethat

ifDGOexe utesthesubstrategythenthe ompetitiveratioofDGOisbounded by

5/3

. Let

FR

s

bethe losedroutefollowedbystrategyDGOstartingataninteriorpoint

s

. Let

FR

s

(p, q, orient )

denotethesubpath of

FR

s

followedin dire tion

orient

frompoint

p

to point

q

,where

orient

an eitherbe

cw

( lo kwise)or

aw

(anti- lo kwise). Similarly,wedenethesubpath

SWR

s

(p, q, orient )

of

SWR

s

. Wedenote by

SP(p, q)

ashortestre tilinearpathfrom

p

to

q

inside

P

.

Webeginbyestablishingtwosimplebut usefullemmas.

Lemma4.1 If

t

isapointonsometour

SWR

s

,then

length(SWR

t

) ≤ length(SWR

s

).

Proof: Sin e

SWR

s

passesthrough

t

,therouteisawat hmanroutethrough

t

. Butsin e

SWR

t

istheshortestwat hmanroutethrough

t

, thelemmafollows.

2

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If

f

isafrontierpoint,thenlet

reg(f )

denote theregiontotherightof

ext

(f )

.

StrategyDGO

Establishtheupperand rightprin ipalproje tionpoints

f

u

[s]

and

f

r

[s]

while

Q1(s)

isnot ompletelyseendo

Obtaintheupperandrightfrontiers,

f

u

and

f

r

Case 1:

reg(f

u

) ⊆ reg(f

r

)

Case 1.1:

ext

(f

r

) ∈ A ∪ R

Goto the losestpointon

ext(f

r

)

Case 1.2:

ext

(f

r

) ∈ B ∪ L

ApplysubstrategyDGO-1

Case 2:

reg(f

r

) ⊆ reg(f

u

)

Case 2.1:

ext

(f

u

) ∈ A ∪ R

Goto the losestpointon

ext(f

u

)

Case 2.2:

ext

(f

u

) ∈ B ∪ L

Case 3:

ext

(f

u

)

interse ts

ext

(f

r

)

in onepoint

Case 3.1:

(ext(f

u

), ext(f

r

)) ∈ AR

Goto theinterse tionpointbetween

ext

(f

u

)

and

ext(f

r

)

Case 3.2:

(ext(f

u

), ext(f

r

)) ∈ RB ∪ LA

Case 3.3:

(ext(f

u

), ext(f

r

)) ∈ LB

Case 4:

reg(f

u

)

and

reg(f

r

)

aredisjoint

Case 4.1:

(ext(f

u

), ext(f

r

)) ∈ AA ∪ AR ∪ RR ∪ RA

Case 4.2:

(ext(f

u

), ext(f

r

)) ∈ LR ∪ LA ∪ AB ∪ RB

Case 4.3:

(ext(f

u

), ext(f

r

)) ∈ LB

end while

if

P

isnot ompletelyvisible

thenApplysubstrategyDGO-0

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Lemma4.2 If

q

and

q

′

arepointsonsometour

SWR

s

lyingin

Q1

(s)

,then

length(SWR

s

) ≥ 2 max(||s, q||

x

, ||s, q

′

||

x

) + 2 max(||s, q||

y

, ||s, q

′

||

y

).

Proof: Sin e

SWR

s

passesthrough

q

and

q

′

andwe al ulatelengtha ordingto the

L

1

metri , thesmallestpossibletourthatpassesthroughthetwopointsisthere tanglewithatleasttwoof

thepoints

s

,

q

,and

q

′

ontheperimeter. Sin e

q

and

q

′

liein

Q1(s)

,thepoint

s

hastobethelower

left ornerofthere tangleandthelengthoftheperimeterofthere tangleisthenasstated.

2

The stru ture of the following proofs are very similar to ea h other. In ea h ase we will

establishapoint

t

that we anensureispassedby

SWR

s

andthateither liesontheboundaryof

P

or anbeviewed asto lieon theboundaryof

P

. We then onsider the tour

SWR

t

(spe ied bystrategyBGO) and omparehowmu h longer thetour

FR

s

is. ByLemma 4.1weknowthat

length(SWR

t

) ≤ length(SWR

s

)

,hen etheratio

length(FR

s

)

length(SWR

t

)

isan upperbound on the ompetitiveratioof strategyDGO.Onequirk of theproofsis that we

requirethat there existsa tour

SWR

t

that ontainsthe startingpoint

s

in itsinterior oron the boundaryof

SWR

t

. ThisrequirementwillberemovedinLemma 4.8.

4.1 Strategy DGO-0

WestartbypresentingDGO-0.

Strategy DGO-0

1 Let

p

bethe urrentrobotposition 2 if

||s, p||

y

≤ ||s, p||

x

then

2.1

r := PIGO(p, east , cw , P)

else

/∗

if

||s, p||

y

> ||s, p||

x

then

∗/

2.2

r := PIGO(p, east , aw , P)

endif

3 From

r

goba ktothestartingpoint

s

andhalt

End DGO-0

Lemma4.3 IfthestrategyappliessubstrategyDGO-0,then

length(FR

s

) ≤

3

2 length(SWR

s

).

Proof: Sin ethepath

FR

s

(s, p, orient )

thatthestrategyhasfollowedwhenitrea hespoint

p

is

x

-

y

-monotone, the point

p

isthe urrently topmost andrightmost point of thepath. Hen e, we anaddahorizontalspikeissuing fromthe boundarypointimmediatelyto therightof

p

,giving

anewpolygon

P

′

having

p

on theboundaryand furthermorewith the sameshortestwat hman

routethrough

p

as

P

;seeFigure5. ThismeansthatperformingstrategyIGO

(p, east , orient )

in

P

yieldsthesameresultasperformingBGO

(p, orient)

in

P

′

, p

beingaboundarypointin

P

′

, orient

beingeither

cw

or

aw

. Thetourfollowedisthereforeashortestwat hmanroutethroughthepoint

p

in both

P

′

(9)

(a) (b)

||s, p||

y

≤ ||s, p||

x

s

p

r

||s, p||

y

> ||s, p||

x

p

s

SWR

p

r

SWR

p

Figure5:Illustratingthe asesintheproofofLemma4.3.

Also the point

p

lies on a left extension with respe t to

s

, by the way

p

is dened, and it

is the losest point to

s

su h that all of

Q1(s)

has been seen by the path

FR

s

(s, p, orient ) =

SP(s, p)

. Hen e,thereisaroute

SWR

s

thatpassesthrough

p

andbyLemma4.1

length(SWR

p

) ≤

length(SWR

s

)

.

Assume nowfortherst asethat DGO-0performsStep2.1,i.e., thatwhen

FR

s

rea hesthe point

p

, then

||s, p||

y

≤ ||s, p||

x

; see Figure 5(a). Theresultfor these ond asein whi hDGO-0 performsStep2.2followsbysymmetry. Thetourfollowedequals

FR

s

=

SP(s, p) ∪ SWR

p

(p, r, cw ) ∪ SP (r, s).

(1)

Sin ebyourassumption,that

s

doesnotlieoutside

SWR

p

,we anassumewithoutlossofgenerality thatthepoint

r

isthelast interse tionpointof

SWR

p

withtheverti alboundaryaxisof

Q1(s)

; seeFigure5(a).

We provethis asfollows. Letthe point

r

be asdened in thestrategy DGO-0 and let

r

′

be

the last interse tion point of

SWR

p

with the verti al boundary axis of

Q1(s)

. Bythe way the strategyPIGOisdenedthesubpath

SP(s, p) ∪ SWR

p

(p, r, cw )

seesallof

P

. Hen e,thesubpath

SWR

p

(r, p, cw )

equals

SP(r, p)

with

r

eitherlyingin

Q2

(s)

or

Q3

(s)

. If

r ∈ Q3(s)

then

s ∈ SWR

p

,

and

r

′

_{= s}

. If

r ∈ Q2(s)

thenwe an hoose

r

′

sothat

||r, s||

y

= ||r

′

_{, s||}

y

,and

SP(r, s) = SP (r, r

′

_)∪

SP(r

′

_{, s)}

. In both ases

length(SWR

p

(p, r, cw ) ∪ SP(r, s)) = length(SWR

p

(p, r

′

_{, cw ) ∪ SP(r}

′

_{, s))}

.

Thisprovesthatwe an usethepoint

r

′

insteadofthepoint

r

in (1).

Wehavethat

length(FR

s

) = ||s, p|| + length(SWR

p

) − ||r, p|| + ||r, s||

≤ length(SWR

s

) + ||s, p|| + ||r, s|| − ||r, p||

= length(SWR

s

) + ||s, p||

x

+ ||s, p||

y

+ ||r, s||

x

+ ||r, s||

y

− ||r, p||

x

− ||r, p||

y

= length(SWR

s

) + ||s, p||

y

+ ||r, s||

y

− ||r, p||

y

≤ length(SWR

s

) + 2||s, p||

y

≤ length(SWR

s

) + ||s, p|| ≤

3 length(SWR

s

),

(10)

sin e

length(SWR

s

) ≥ 2||s, p||

byLemma4.2and

||s, p||

y

≤ ||s, p||

x

.

2

4.2 Strategy DGO-1

NextwepresentDGO-1.

Strategy DGO-1

1 if

ext

(f

u

) ∈ L

then Mirror

P

atthediagonalof

Q1(s)

endif 2 Goto

v

,theverteximmediatelytotherightofthe urrentpoint 3

q := PBGO(v, aw , Q1(v))

4 if

||s, q||

y

≤ ||s, q||

x

then 4.1 Goba kto

v

4.2

r := PBGO(v, cw , P)

else

/∗

if

||s, q||

y

> ||s, q||

x

then

∗/

4.3

r := PIGO(q, north, aw , P)

endif

5 Goba ktothestartingpoint

s

andhalt

End DGO-1

length(FR

s

) ≤

3

2 length(SWR

s

).

Proof: Sin ethe point

v

rea hedby path

FR

s

(s, v, orient ) = SP(s, v)

isaboundarypoint, the result of performing strategy BGO

(v, orient )

, with

orient

being either

cw

or

aw

, is a shortest

wat hmanroutethrough

v

; seeFigure6.

Also the point

v

lies on a shortest path between the horizontal boundary of

Q1(s)

and the

extension

ext(f

u

)

. Hen e, there is a route

SWR

s

that passes through

v

and by Lemma 4.1

length(SWR

v

) ≤ length(SWR

s

)

.

Assume now for the rst ase that DGO-1 performs Steps 4.1 and 4.2, i.e., that when

FR

s

rea hesthepoint

q

,then

||s, q||

y

≤ ||s, q||

x

; seeFigure6(a). Thetourfollowedequals

FR

s

= SP (s, v) ∪ SWR

v

(v, q, aw ) ∪ SP (q, v) ∪ SWR

v

(v, r, cw ) ∪ SP(r, s)

andsin ebyourassumption,that

SWR

v

ontains

s

initsinterior,we anassumewithoutlossof generalitythatthepoint

r

isthelastinterse tionpointof

SWR

v

with theverti alboundaryaxis

of

Q1(s)

;seeFigure6(a). This anbeprovedin thesamewayasin Lemma4.3. Wehave

length(FR

s

) =

||s, q|| + length(SWR

v

) − ||r, q|| + ||r, s||

≤

length(SWR

s

) + ||s, q|| + ||r, s|| − ||r, q||

=

length(SWR

s

) + ||s, q||

x

+ ||s, q||

y

+ ||r, s||

x

+ ||r, s||

y

− ||r, q||

x

− ||r, q||

y

=

length(SWR

s

) + ||s, q||

y

+ ||r, s||

y

− ||r, q||

y

=

length(SWR

s

) + 2||s, q||

y

≤

3

2 length(SWR

s

),

(11)

f

r

s

v

||s, q||

y

> ||s, q||

x

(b)

SWR

v

q

FR

s

f

r

FR

s

v

r

SWR

v

q

||s, q||

y

≤ ||s, q||

x

(a)

p

sin e

length(SWR

s

) ≥ 2||s, q||

byLemma4.2and

||s, q||

y

≤ ||s, q||

x

.

Assume forthese ond asethatDGO-1performsStep4.3,i.e.,on e

FR

s

rea hesthepoint

q

,

then

||s, q||

y

> ||s, q||

x

;see Figure6(b). Thetourfollowedequals

FR

s

= SP(s, v) ∪ SWR

v

(v, r, aw ) ∪ SP (r, s),

with

r

beingthelast interse tionpointof

SWR

v

withthehorizontalboundaryaxisof

Q1(s)

;see Figure6(b). Inthesamewayasbeforewe annowobtain

length(FR

s

) ≤

length(SWR

s

) + 2||s, q||

x

≤

3

2 length(SWR

s

).

This on ludes theproof.

2

4.3 Strategy DGO-2

We ontinuetheanalysisbyrstshowingthesubstrategyDGO-2andthenprovingits ompetitive

(12)

1 Let

e

betheleftmostverti aledgein

Q1(s)

ontheboundarypartseparating

ext

(f

u

)

and

ext

(f

r

)

. Similarly let

e

′

bethe bottommosthorizontal edgein

Q1(s)

onthe boundarypart separating

ext

(f

u

)

and

ext

(f

r

)

2 Let

v

bethetopmostvertexon

e

andlet

v

′

betherightmostvertexon

e

′

3 if

||s, v||

y

< ||s, v

′

_||

x

then

3.1 Mirror

P

atthediagonal of

Q1(s)

/∗ v

and

v

′

arenowswapped

∗/

endif 4 Goto

v

5

q := PBGO(v, aw , Q1(v))

6 if

||s, v|| ≤ ||s, q||

y

then 6.1

r := PIGO(q, north, aw , P)

else

/∗

if

||s, v|| > ||s, q||

y

then

∗/

6.2 Goba kto

v

6.3

r := PBGO(v, cw , P)

endif

7 Goba kto

s

andhalt

End DGO-2

length(FR

s

) ≤

5

3 length(SWR

s

).

Proof: Before we begin thea tual proofwehaveto ensure that thepoints

v

and

v

′

dened in

substrategyDGO-2 are su h that in all ases it holds that

length(SWR

v

) ≤ length(SWR

s

)

and

length(SWR

v

′

) ≤ length(SWR

s

)

. In ases

AA

and

AR

,thepoint

v

liesonsometour

SWR

s

sin e

v

liesonashortestpathfrom

s

to

ext

(f

u

)

andonashortestpathfrom

ext(f

r

)

to

ext

(f

u

)

. Hen e, forthese two asesitfollowsbyLemma4.1that

length(SWR

v

) ≤ length(SWR

s

)

;seeFigures7(a) and (b). Furthermore, sin e

v

′

lies on some tour

SWR

v

sin e

v

′

lies on a shortest path from

ext(f

r

)

to

ext

(f

u

)

,wehavebythesametokenthat

length(SWR

v

′

) ≤ length(SWR

v

)

,thus proving

these ondinequality.

The ase

RR

followsbya ompletelysymmetri argumentifweex hangethepoints

v

and

v

′

forea hother;see Figure7( ).

The nal ase

RA

is a little more involved sin e neither

v

nor

v

′

an be ensured to lie on

a tour

SWR

s

. To prove the inequalities onsider an interse tion point

u

between some tour

SWR

s

and

ext(f

r

)

. The point

u

must exist otherwise

SWR

s

violates the visibility property of

awat hmanroute. Now, byLemma 4.1, wehave that

length(SWR

u

) ≤ length(SWR

s

)

. Sin e

v

and

v

′

lie onsome shortestpath from

u

to

ext(f

u

)

we additionallyhavethat there is sometour

SWR

u

passingthroughboththesepoints. ApplyingLemma4.1againgivesusthetwoinequalities

length(SWR

v

) ≤ length(SWR

u

)

and

length(SWR

v

′

) ≤ length(SWR

u

)

,thusprovingour laimalso

inthis ase;seeFigure7(d).

Assume fortherst asethat DGO-2performsStep6.1,i.e., that

||s, v||

y

≤ ||s, v

′

||

x

andthat

when

FR

s

rea hesthepoint

q

,then

||s, v|| ≤ ||s, q||

y

;seeFigure 8(a). Thetourfollowedequals

(13)

s

ext

_(f

_u

₎

v

′

ext

(f

r

)

s

ext

_(f

_r

₎

ext

_(f

_u

₎

v

′

(b) ( ) (a) (d)

s

ext

_(f

_r

₎

v

′

ext

(f

u

)

s

v

ext

(f

u

)

v

′

_ext

_(f

r

)

p

Figure7: Illustratingthestru tural aseshandledbysubstrategyDGO-2.

Hen e,wehavethatthelengthof

FR

s

isboundedby

length(FR

s

) =

||s, v|| + length(SWR

v

(v, r, aw )) + ||r, s||

≤

length(SWR

v

) + 2||s, v|| ≤ length(SWR

s

) +

4

3 ||s, v|| +

2

3 ||s, v||

≤

length(SWR

s

) +

4

3 ||s, q||

y

+

2

3 ||s, v||

x

+

2

3 ||s, v||

y

≤

length(SWR

s

) +

4

3 ||s, q||

y

+

2

3 ||s, v

′

||

x

+

2

3 ||s, v

′

||

x

=

length(SWR

s

) +

4

3 ||s, q||

y

+

4

3 ||s, v

′

||

x

≤

length(SWR

s

) +

2

3 length(SWR

s

) =

5

3 length(SWR

s

),

sin e

length(SWR

s

) ≥ 2||s, q||

y

+ 2||s, v

′

_||

x

byLemma4.2.

Forthese ond aseassumethatDGO-2performsSteps6.2and6.3,i.e.,that

||s, v||

y

≤ ||s, v

′

_||

x

andthatwhen

FR

s

rea hesthepoint

q

,then

||s, v|| > ||s, q||

y

;seeFigure8(b). Thetourfollowed equals

FR

s

= SP(s, v) ∪ SWR

v

(v, q, aw ) ∪ SP(q, v) ∪ SWR

v

(v, r, cw ) ∪ SP (r, s).

Withoutlossofgeneralitywe anassumethat

r

isthelastinterse tionpointbetween

SWR

v

and theverti alboundaryaxisof

Q1

(s)

. Thelengthof

FR

s

isboundedby

length(FR

s

) =

||s, q|| + length(SWR

v

(v, q, aw )) + length(SWR

v

(v, r, cw )) + ||r, s||

=

||s, q|| + length(SWR

v

) − ||r, q|| + ||r, s||

(14)

s

v

′

SWR

v

||s, v||

y

≤ ||s, v

′

||

x

||s, v|| > ||s, q||

y

(b)

q

r

s

q

v

′

SWR

v

r

||s, v||

y

≤ ||s, v

′

||

x

||s, v|| ≤ ||s, q||

y

p

(a)

=

length(SWR

s

) +

4

3 ||s, q||

y

+

2

3 ||s, q||

y

< length(SWR

s

) +

4

3 ||s, q||

y

+

2

3 ||s, v||

=

length(SWR

s

) +

4

3 ||s, q||

y

+

2

3 ||s, v||

x

+

2

3 ||s, v||

y

≤

length(SWR

s

) +

4

3 ||s, q||

y

+

2

3 ||s, v

′

||

x

+

2

3 ||s, v

′

||

x

=

length(SWR

s

) +

4

3 ||s, q||

y

+

4

3 ||s, v

′

||

x

≤

length(SWR

s

) +

2

3 length(SWR

s

) =

5

3 length(SWR

s

),

sin e

length(SWR

s

) ≥ 2||s, q||

y

+ 2||s, v

′

||

x

byLemma4.2.

2

4.4 Strategy DGO-3

(15)

1 if

(ext (f

u

), ext (f

r

)) ∈ LR ∪ LA

then Mirror

P

atthediagonalof

Q1(s)

endif

2 Let

e

betheleftmostverti aledgein

Q1(s)

ontheboundarypartseparating

ext

(f

u

)

and

ext

(f

r

)

andlet

v

bethetopmostvertexon

e

.Let

v

′

betheverteximmediatelytotherightofthe urrent

point 3 if

||s, v

′

_||

x

≤ ||s, v||

x

then 3.1 Goto

v

′

3.2

q := PBGO(v

′

_{, aw , Q1(v}

′

₎₎

3.3 if

||s, q||

y

≤ ||s, q||

x

then 3.3.1 Goba kto

v

′

3.3.2

r := PBGO(v

′

_{, cw , P)}

else

/∗

if

||s, q||

y

> ||s, q||

x

then

∗/

3.3.3

r := PIGO(q, north , aw , P)

endif else

/∗

if

||s, v

′

_||

x

> ||s, v||

x

then

∗/

3.4 if

||s, v||

y

≤ ||s, v

′

_||

x

then 3.4.1 Goto

v

3.4.2

q := PBGO(v, aw , Q1(v))

3.4.3 if

||s, v|| ≤ ||s, q||

y

then 3.4.3.1

r := PIGO(q, north, aw , P)

else

/∗

if

||s, v|| > ||s, q||

y

then

∗/

3.4.3.2 Goba kto

v

3.4.3.3

r := PBGO(v, cw , P)

endif else

/∗

if

||s, v||

y

> ||s, v

′

_||

x

then

∗/

3.5 Goto

v

′

3.6

r := PBGO(v

′

_{, aw , P)}

endif endif

4 Goba kto

s

andhalt

End DGO-3

length(FR

s

) ≤

5

3 length(SWR

s

).

Proof: Sin ethepoint

v

′

rea hedbypath

FR

s

(s, v

′

, orient ) = SP(s, v

′

)

isaboundarypointthe result of performing strategy BGO

(v

′

, orient )

, with

orient

being either

cw

or

aw

, is a shortest

wat hmanroutethrough

v

′

;seeFigure9.

Also, thepoint

v

′

lieson ashortest pathbetween thehorizontal boundaryof

Q1(s)

and the

extension

ext

(f

r

)

SWR

s

that passes through

v

′

and by Lemma 4.1

length(SWR

v

′

) ≤ length(SWR

s

)

. The point

v

lies on a shortest path from

ext(f

r

)

to

ext(f

u

)

and therefore we havethat there is aroute

SWR

s

that passes through

v

and

length(SWR

v

) ≤

(16)

s

q

r

v

′

SWR

_v

′

s

||s, v

′

_||

x

≤ ||s, v||

x

v

′

q

r

||s, q||

y

> ||s, q||

x

p

SWR

_v

′

||s, v

′

_||

x

≤ ||s, v||

x

||s, q||

y

≤ ||s, q||

x

(a) (b)

s

v

′

v

q

r

SWR

v

||s, v||

y

≤ ||s, v

′

||

x

||s, v|| ≤ ||s, q||

y

s

||s, v

′

_||

x

> ||s, v||

x

v

′

SWR

v

r

q

||s, v||

y

≤ ||s, v

′

||

x

||s, v|| > ||s, q||

y

p

( ) (d)

||s, v

′

_||

x

> ||s, v||

x

||s, v

′

_||

x

> ||s, v||

x

v

′

q

v

s

r

p

SWR

_v

′

||s, v||

y

> ||s, v

′

||

x

(e)

(17)

Assumefortherst asethatDGO-3performsSteps3.3.1and3.3.2,i.e.,that

||s, v

′

||

x

≤ ||s, v||

x

and

||s, q||

y

≤ ||s, q||

x

; seeFigure9(a). Thetourfollowedequals

FR

s

= SP(s, v

′

) ∪ SWR

v

′

(v

′

, q, aw ) ∪ SP(q, v

′

) ∪ SWR

v

′

(v

′

, r, cw ) ∪ SP (r, s),

with

r

beingthelastinterse tionpointof

SWR

v

′

Q1(s)

. Inthe asethat thestrategyperformsStep 3.3.3i.e., that

||s, v

′

||

x

≤ ||s, v||

x

and

||s, q||

y

> ||s, q||

x

; see

Figure9(b). Thetourfollowedequals

FR

s

= SP (s, v

′

) ∪ SWR

v

′

(v

′

, r, aw ) ∪ SP(r, s),

with

r

beingthelast interse tionpointof

SWR

v

′

withthehorizontalboundaryaxisof

Q1

(s)

. In thesamewayasintheproofofLemma4.4we anobtaina ompetitiveratioof

3/2

inthis ase.

Assume for the se ond ase that DGO-3 performs Step 3.4.3.1, i.e., that

||s, v

′

_||

x

> ||s, v||

x

,

||s, v||

y

≤ ||s, v

′

||

x

and

||s, v|| ≤ ||s, q||

y

;see Figure9( ). Thetourfollowedequals

FR

s

= SP(s, v) ∪ SWR

v

(v, r, aw ) ∪ SP (r, s),

with

r

being the last interse tion point of

SWR

v

with the horizontal boundary axis of

Q1(s)

. In the ase that the strategy performs Steps 3.4.3.2 and 3.4.3.3, i.e., that

||s, v

′

_||

x

> ||s, v||

x

,

||s, v||

y

≤ ||s, v

′

||

x

and

||s, v|| > ||s, q||

y

;see Figure9(d). Thetourfollowedequals

FR

s

= SP (s, v) ∪ SWR

v

(v, q, aw ) ∪ SP (q, v) ∪ SWR

v

(v, r, cw ) ∪ SP(r, s)

with

r

beingthelastinterse tionpointof

SWR

v

Q1(s)

. Inthe samewayasintheproofofLemma4.5we anobtaina ompetitiveratioof

5/3

inthis ase.

Assumeforthethird asethatDGO-3performsSteps3.5and3.6,i.e.,that

||s, v

′

_||

x

> ||s, v||

x

and

||s, v||

y

> ||s, v

′

_||

x

;seeFigure9(e). Thetourfollowedequals

FR

s

= SP (s, v

′

) ∪ SWR

v

′

(v

′

, r, aw ) ∪ SP(r, s),

length(FR

s

) =

||s, v

′

|| + length(SWR

v

′

(v

′

, r, aw )) + ||r, s||

≤

||s, v

′

|| + length(SWR

v

′

) − ||r, v

′

|| + ||r, s|| ≤ length(SWR

s

) + 2||s, v

′

||

x

<

length(SWR

s

) + ||s, v

′

||

x

+ ||s, v||

y

≤

3

2 length(SWR

s

),

sin e

length(SWR

s

) ≥ 2||s, v

′

||

x

+ 2||s, v||

y

by Lemma 4.2. From our assumption, that

SWR

v

′

ontains

s

in its interior, we an assume without loss of generality that the point

r

is the last

interse tionpointof

SWR

v

′

Q1(s)

. This anbeprovedin the samewayasinLemma 4.3.

2

4.5 Strategy DGO-4

(18)

1 Let

v

betheverteximmediatelyabove

p

andlet

v

′

betheverteximmediatelytotherightof

p

. 2 if

||s, v||

y

≤ ||s, v

′

_||

x

then 2.1 Goto

v

2.2

r := PBGO(v, cw , P)

else

/∗

if

||s, v||

y

> ||s, v

′

_||

x

then

∗/

2.3 Goto

v

′

2.4

r := PBGO(v

′

_{, aw , P)}

endif

3 Goba kto

s

andhalt

End DGO-4

length(FR

s

) ≤

3

2 length(SWR

s

).

Proof: Consider the point

v

. It lies on the shortest path between the horizontal boundaryof

Q1(s)

and the extension

ext(f

u

)

SWR

s

that passes through

v

and by

Lemma4.1

length(SWR

v

) ≤ length(SWR

s

)

. Bya ompletely symmetri argumentitalsofollows thatthereisaroute

SWR

s

thatpassesthrough

v

′

,andhen e,

length(SWR

v

′

) ≤ length(SWR

s

)

. Assume rstthat DGO-4performsSteps2.1and2.2,i.e.,

||s, v||

y

≤ ||s, v

′

||

x

;seeFigure10(a). Thetourfollowedequals

FR

s

= SP(s, v) ∪ SWR

v

(v, r, cw ) ∪ SP (r, s).

Withoutlossofgeneralitywe anassumethat

r

SWR

v

and theverti alboundaryaxisof

Q1(s)

. This anbeprovedinthesamewayasinLemma 4.3. The

lengthof

FR

s

isboundedby

length(FR

s

) =

||s, v|| + length(SWR

v

(v, r, cw )) + ||r, s||

≤

||s, v|| + length(SWR

v

) − ||r, v|| + ||r, s||

≤

length(SWR

s

) − ||r, v|| + ||s, v|| + ||r, s||

≤

length(SWR

s

) + 2||s, v||

y

≤

length(SWR

s

) + ||s, v||

y

+ ||s, v

′

||

x

≤

3

2 length(SWR

s

),

sin e

length(SWR

s

) ≤ 2||s, v||

y

+ 2||s, v

′

_||

x

byLemma4.2. IfDGO-4performsSteps2.3and2.4,i.e.,

||s, v||

y

> ||s, v

′

_||

x

;seeFigure10(b);thetourfollowed equals

FR

s

= SP(s, v

′

) ∪ SWR

v

′

(v

′

, r, aw ) ∪ SP (r, s)

andsimilarlywe anassumethat

r

SWR

v

′

andthehorizontal boundaryaxisof

Q1(s)

asprovedin Lemma4.3. Inthesamewayasbefore weobtainthat

length(FR

s

) ≤ length(SWR

s

) + 2||s, v

′

||

x

≤

3

(19)

s

||s, v||

y

> ||s, v

′

||

x

(b)

v

′

r

s

||s, v||

y

≤ ||s, v

′

||

x

v

′

v

SWR

v

r

(a)

p

Figure10: Illustratingthe asesintheproofofLemma4.7.

2

Inea hoftheproofsofLemmas4.34.7weestablishedakeypoint,fromnowondenoted

t

,in

Q1(s)

that ispassed by sometour

SWR

s

. If

SWR

t

ontainsthestarting point

s

in the interior

orontheboundarythentheresultsofLemmas4.34.7holdaswehaveshown. Now onsiderthe

asewhen

s

liesoutside

SWR

t

. Weneedto showthat theresultsofthelemmasstillhold inthis ase.

Lemma4.8 Usingthenotationabove,theresultsofLemmas4.34.7stillholdif

s

isnot ontained

in

SWR

t

.

Proof: Ofallthepossibletours

SWR

t

onsideronethathasapoint

s

′

visiblefrom

s

andas lose

to

s

aspossible. Wewillshowthat

length(SWR

s

) ≥ 2||s, s

′

|| + length(SWR

s

′

)

andthat

length(FR

s

) ≤ 2||s, s

′

|| + length(FR

s

′

).

Thus,wehavethat

length(FR

s

) ≤ 2||s, s

′

|| + length(FR

s

′

) ≤ 2||s, s

′

|| + c · length(SWR

s

′

)

≤ 2c||s, s

′

|| + c · length(SWR

s

′

) ≤ c · length(SWR

s

)

forany

c ≥ 1

su hthat

length(FR

s

′

) ≤ c · length(SWR

s

′

)

. Hen e,we anapply

c = 3/2

or

c = 5/3

onea h ofthe asesDGO-0 toDGO-4 appropriatelysin ethepoint

s

′

isbydenition ontained

ontheboundaryof

SWR

t

.

Toprovethe twoinequalities,werst assumethat

SWR

t

is ontainedin

Q1

(s)

and possibly

(20)

s

q

s

′

t

SWR

t

(a)

s

q

t

(b)

SWR

t

s

′

Figure11: IllustratingtheproofofLemma4.8.

symmetri albyamirroringoperationofthepolygon

P

atthediagonalof

Q1(s)

andtherefore an

behandledin thesameway.

Next,notethatthepoint

s

′

liesonan

x

-

y

-monotonepathfrom

s

to

t

. Thisisbe ause

s

′

annot

lieabovethe

y

- oordinateof

t

andit annotlietotherightofthe

x

- oordinateof

t

. If

s

′

did,then

itwouldbe possibleto moveit loserto

s

ontradi tingourinitialassumption. (Indeed if

SWR

t

liesinboth

Q1(s)

and

Q4(s)

,then

s

′

liesonthehorizontalboundaryof

Q1(s)

.)

Considera tour

SWR

s

. Sin e

SWR

t

has no pointsin

Q2(s

′

₎

or

Q3(s

′

₎

it followsthat these

quadrants annot ontain any verti alleft extensions with respe t to

s

′

. Hen e, itis possibleto

moveany verti aledge of

SWR

s

(partially) visiblefrom

s

and tothe leftof

s

′

towardstheright

so that the new tour onstru ted is a wat hman route, passes through

s

′

, has the same length

as

SWR

s

, and has no verti al edge in

Q2

(s

′

)

or

Q3

(s

′

)

; see Figures 11(a) and (b). The tour onstru ted in this way onsists of a wat hman route passing through

s

′

and an

x

-

y

-monotone

path onne ting

s

and

s

′

. Ex hangingthewat hmanroute passingthrough

s

′

foratour

SWR

s

′

possiblyshortensthetotallengthofthetourandthisprovestherstinequality.

Toprovethese ondinequalitywenotethat

FR

s

onsistsofan

x

-

y

-monotonepathfrom

s

to

t

followedbyapartof

SWR

t

andapathba kto

s

. Hen ewe an(forthisproofonly)assumethat theinitialpathfrom

s

to

t

passesthrough

s

′

. Furthermorewenotethat whi heversubstrategyis

appliedfor

FR

s

thesamesubstrategywillbeappliedfor

FR

s

′

. Howeverwithinasubstrategynot ne essarilythesamestepswillbeperformedfor

FR

s

and

FR

s

′

. Wedenethisdieren eformally asfollows.

Considerthetour

FR

p

foranypoint

p

. IfDGOon

p

appliesasubstrategythatdoesoneofthe following:

1. itsrst appli ationof PIGOorPBGO iswith anti- lo kwise orientationand the(possible)

subsequentappli ationofPIGOorPBGOisalsowithanti- lo kwiseorientation,

2. its rst appli ation of PIGO or PBGO is with lo kwise orientation and the subsequent

appli ationofPIGOorPBGO iswithanti- lo kwiseorientation,

(21)

urrentpointto

t

,applyoneofPIGOorPBGOuntilwerea hapoint

q

whereade isionismade

depending on whether

||s, q||

y

≤ ||s, q||

x

or

||s, q||

y

> ||s, q||

x

. This de isionis pre isely the one thatdetermineswhetherthetourwillbeforwardorientedorba kwardoriented.

If both

FR

s

and

FR

s

′

are forwardorientedorba kwardoriented,then these ond inequality follows dire tly. Hen e, we have a problem if either

FR

s

is forward oriented whereas

FR

s

′

is ba kwardorientedorthereverse aseo urs. Wedoa aseanalysistoprovethese ondinequality.

Assume rstthat

FR

s

is ba kward orientedand

FR

s

′

is forwardoriented. Let

o

denote the orientationthat therstappli ation ofPIGO orPBGO in DGOon

s

performs inasubstrategy.

Let

¯

o

betheotherorientation,i.e.,if

o = aw

,then

o = cw

¯

and onversely.

Wehavethat

length(FR

s

) =

||s, s

′

|| + length(FR

s

(s

′

, q, o)) + ||q, t|| + ||t, r|| +

+ length(FR

s

(r, s

′

, ¯

o)) + ||s

′

, s||

=

length(FR

s

′

(s

′

, q, o)) + ||q, s

′

|| + length(FR

s

′

(s

′

, r, o)) + ||r, s

′

|| +

+ 2||s, s

′

|| + ||t, r|| + ||q, t|| − ||q, s

′

|| − ||r, s

′

||

=

length(FR

s

′

) + 2||s, s

′

|| + ||t, r|| + ||q, t|| − ||q, s

′

|| − ||r, s

′

||

≤

length(FR

s

′

) + 2||s, s

′

||,

sin e

length(FR

s

(r, s

′

_{, ¯}

_{o)) = length(FR}

s

′

(s

′

, r, o))

,

length(FR

s

(s

′

_{, q, o)) = length(FR}

s

′

(s

′

, q, o))

, and

||t, r|| + ||q, t|| − ||q, s

′

|| − ||r, s

′

|| =

||t, r||

x

+ ||t, r||

y

+ ||q, t||

x

+ ||q, t||

y

− ||q, s

′

||

x

− ||q, s

′

||

y

− ||r, s

′

||

x

− ||r, s

′

||

y

=

||t, r||

y

+ ||q, t||

y

− ||q, s

′

||

x

− ||q, s

′

||

y

− ||r, s

′

||

x

− ||r, s

′

||

y

=

||r, q||

y

− ||q, s

′

||

x

− ||q, s

′

||

y

− ||r, s

′

||

x

− ||r, s

′

||

y

≤

−||q, s

′

||

x

− ||r, s

′

||

x

− ||r, s

′

||

y

≤

0,

if

o = aw

;seeFigure12(a). Thepenultimateinequalityfollowssin e

||s

′

, q||

y

≥ ||r, q||

y

. If

o = cw

,

thenasimilar al ulation aneasilybemade.

Assume nowthat

FR

s

isforwardorientedand

FR

s

′

isba kwardoriented. If

SWR

t

interse ts both

Q1

(s)

and

Q4

(s)

(thismeansthat

s

′

liesonthehorizontalboundaryof

Q1

(s)

),weargueas

follows. Sin e

||s, q||

y

= ||s

′

_{, q||}

y

inthis ase,thismeansthat if

||s, q||

x

< ||s, q||

y

,then

||s

′

_{, q||}

x

<

||s

′

_{, q||}

y

, and hen e, it is notpossible for

FR

s

to be forward oriented and

FR

s

′

to be ba kward oriented. Therefore

SWR

t

mustbe ompletely ontainedin

Q1(s)

.

We an furthermoreassume that

s

′

is sele ted sothat

SWR

t

lies ompletely in

Q1

(s

′

)

. We

have,ifthesubstrategyusedisnotDGO-4,that

length(FR

s

)

= ||s, s

′

|| + length(FR

s

(s

′

, q, o)) + ||q, t|| + ||t, r

′

|| +

+ length(FR

s

(r

′

, r, o)) + ||r, s

′

|| + ||s

′

, s||

= length(FR

s

′

(s

′

, q, o)) + ||q, t|| + ||t, r|| + length(FR

s

′

(r, r

′

, ¯

o)) + ||r

′

, s

′

|| +

+ 2||s

′

, s|| + ||r, s

′

|| + ||t, r

′

|| − ||r

′

, s

′

|| − ||t, r||

(22)

s

q

FR

_s

′

s

q

s

′

s

′

t

r

t

r

′

FR

_s

′

s

FR

_s

′

t

′

s

′

( ) (b) (a)

Figure12: IllustratingtheproofofLemma4.8.

≤ length(FR

s

′

) + 2||s, s

′

||,

sin e

length(FR

s

′

(r, r

′

_{, ¯}

_{o)) = length(FR}

s

(r

′

, r, o))

,

length(FR

s

(s

′

_{, q, o)) = length(FR}

s

′

(s

′

, q, o))

, and

||r, s

′

_{|| + ||t, r}

′

_{|| − ||r}

′

_{, s}

′

_{|| − ||t, r|| ≤ 0}

by a similar al ulation as above; see Figure 12(b)

for an example of this ase when

o = aw

. The keyinsight is that on e the point

q

is rea hed,

thenallof

Q1(s)

that alsoliesin

Q1(t)

,

Q2(t)

,and

Q3(t)

hasalreadybeenseen. Hen eit only

remainstoexplorethepartin

Q4(t)

.

Finally, ifthe substrategyused isDGO-4, then wehave two dierentkeypoints, thepoint

t

rea hedby

FR

s

andthepoint

t

′

rea hedby

FR

s

′

,giving

length(FR

s

) =

||s, s

′

|| + ||s

′

, t|| + length(FR

s

(t, t

′

, aw )) + ||t

′

, s

′

|| + ||s

′

, s||

=

||s

′

, t

′

|| + length(FR

s

′

(t

′

, t, cw )) + ||t, s

′

|| + 2||s

′

, s|| +

+ ||s

′

, t|| + ||t

′

, s

′

|| − ||s

′

, t

′

|| − ||t, s

′

||

=

length(FR

s

′

) + 2||s

′

, s|| + ||s

′

, t|| + ||t

′

, s

′

|| − ||s

′

, t

′

|| − ||t, s

′

||

=

length(FR

s

′

) + 2||s, s

′

||,

sin e

length(FR

s

(t, t

′

_{, aw )) = length(FR}

s

′

(t

′

, t, cw ))

and

||s

′

_{, t|| + ||t}

′

_{, s}

′

_{|| − ||s}

′

_{, t}

′

_{|| − ||t, s}

′

_{|| = 0}

;see Figure12( ).

2

Next,weprovethe orre tnessof ourstrategy.

Lemma4.9 IfthestrategyDGOdivertsfroman

x

-

y

-monotonepaththenoneofthesubstrategies

DGO-0toDGO-4isperformed.

Proof: Theobje tiveofthisproofistoensurethatall asesaretaken areof. Weenumerateall

thepossible asesandshowthatunlessoneofthesubstrategiesDGO-0toDGO-4isenteredthen

strategywill ontinuealongan

x

-

y

-monotonepathinside

P

.

Webeginbyassumingthat the urrentpointoftherobotisapoint

p

in

Q1(s)

. Furthermore,

(23)

p

has been seenso far. All these assumptions are true for the starting point, i.e., when

p = s

.

Let

ext(f

u

)

and

ext(f

r

)

betheupperandrightfrontierestablishedfromthepoint

p

havingtheir

issuingedgesinside

Q1(s)

. Thisgivesriseto threepossible ases.

Therst aseisthat neither

ext(f

u

)

nor

ext(f

r

)

exist. Thismeansthat thepathfrom

s

to

p

sees allof

Q1

(s)

and substrategyDGO-0 guaranteesa ompetitive ratioof

3/2

in this ase; see

Lemma4.3.

The se ond aseis that the extensions

ext(f

u

)

and

ext(f

r

)

exist but

ext(f

u

) = ext(f

r

)

, i.e., in essen e we only have one extension. If the extension is in

B

or

L

, then substrategyDGO-1

guarantees a ompetitive ratio of

3/2

in this ase; see Lemma 4.4. On the other hand, if the

extensionis in

A

or

R

, thenthe robot movestothe losestpointon theextension. This anbe

done with an

x

-

y

-monotone path moving upwardsand to the rightfrom the urrentpoint thus

showingthatthepathfrom

s

tothenew urrentpointis

x

-

y

-monotone. Furthermore,everything

in

Q1(s)

to theleftandbelowthenew urrentpointhasbeenseen.

The third ase is that the extensions

ext(f

u

)

and

ext(f

r

)

exist and are dierent. This ase has a number of sub ases. If

ext(f

u

)

is in

B

, then

ext(f

r

)

must also lie in

B

, and hen e, one extension dominatesthe other. Sin e

ext(f

u

)

lies in

B

is symmetri with respe t to amirroring operationof

P

along thediagonal of

Q1(s)

to

ext(f

r

)

liesin

L

, wehaveby thesame argument thatoneextension dominatestheotherin this ase. So,ifweassumethat thetwoextensionsdo

notdominateea hother andthat theydo notinterse t,wehavethat the pairof extensions an

bein oneofthefollowingnine ases:

(ext(f

u

), ext (f

r

)) ∈ AA ∪ AB ∪ AR ∪ LA ∪ LB ∪ LR ∪ RA ∪ RB ∪ RR.

If

(ext(f

u

), ext(f

r

)) ∈ AA ∪ AR ∪ RR ∪ RA

,thensubstrategyDGO-2guaranteesa ompetitive

ratioof

5/3

inthis ase;seeLemma 4.5.

If

(ext(f

u

), ext (f

r

)) ∈ LA ∪ LR ∪ RB ∪ AR

,thensubstrategyDGO-3guaranteesa ompetitive

ratioof

5/3

If

(ext(f

u

), ext(f

r

)) ∈ LB

, then substrategyDGO-4 guarantees a ompetitive ratio of

3/2

in

this ase;seeLemma4.7.

Next, we assume that the two extensions interse t, hen e they are orthogonal. The pair of

extensions anbein oneofthefollowingve ases:

(ext (f

u

), ext(f

r

)) ∈ AR ∪ LA ∪ LB ∪ RA ∪ RB.

The ase

(ext (f

u

), ext (f

r

)) ∈ RA

annoto urwiththetwoextensionsinterse tingleavinguswith

thefourremaining ases.

If

(ext(f

u

), ext (f

r

)) ∈ LA ∪ RB

,thensubstrategyDGO-1guaranteesa ompetitiveratioof

3/2

If

(ext(f

u

), ext(f

r

)) ∈ LB

, then substrategyDGO-4 guarantees a ompetitive ratio of

3/2

in

this ase;seeLemma4.7.

If

(ext(f

u

), ext(f

r

)) ∈ AR

, thenby ourindu tive assumption we anmovethe urrent point

withan

x

-

y

-monotonepathtotheinterse tionpointoftheextensions. Everythingtotheleftand

belowtheinterse tionpointisseenbyan

x

-

y

-monotonepathfrom

s

to theinterse tionpoint.

(24)

hasbeenseensofar.

This on ludes the aseanalysisandtheproof.

2

Lemmas4.34.9leadustoestablishthetotal ompetitiveratioofDGO.Wehavethetheorem.

Theorem 1

length(FR

s

) ≤

5

3 length(SWR

s

).

Proof: It only remains to prove the orre tness of our strategy, i.e., that on e the strategy

terminatesthe omplete polygonhas been explored. But this followsfrom Lemma 2.1sin e the

strategyensuresthat

FR

s

hasatleastonepointto therightofeveryextension.

2

5 Con lusions

We haveproved a

5/3

- ompetitivedeterministi strategy alled DGO for exploring a re tilinear

polygonin the

L

1

metri . Weanti ipatethatwithasimilarmethodthe ompetitiveratioshould bepossibletobeimprovedto

3/2

althoughthedetailsareyettobeironedout. Thisnewstrategy

makesextensiveuseofthestrategyDGOanditssubstrategiesDGO-0to DGO-4.

Closing thegapto theknown lowerboundof

5/4

is stillanopenproblem. Wehopethat our

methodwillgivenewinsightsothatthegap anbenarrowedfurtheroreven losedaltogether.

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