Onl i ne Routi ng i n Convex Subdi v i si ons
Prosenjit Bose y
Andrej Brodnik z
Svante Carlss on x
Erik D . D emaine {
Rudolf Fleis her {
Alejand ro Lopez-Ortiz k
Pat Morin y
J. Ian Munro {
Abstra t
We onside r o nline ro ut ing algo rit hms for nding pat hs be tween the verti es of
planeg ra phs. Weshow( 1)theree xistsaroutinga lg orithmfo rarbitrarytria ngula tio ns
that ha s no memo ry a nd use s no rando miza tio n, ( 2) no equiva lent re sult is possible
for o nve xsubdivisio ns,( 3)thereisno ompet itiveonline routing alg orithmunderthe
Eu lidea n distan e metri in a rbitra ry tria ng ula tions,and (4) there is no o mpetitive
online routing algorit hm unde rthe linkdist an emet ri eve nwhe nthe input graph is
re stri t ed to be a Delaunay,gree dy,o rminimum-we ight tria ng ulation.
1 Introdu ti on
Path nd ing, or rou ting, is entral to a numbe r of e lds in luding ge ographi information
sy ste ms , urbanplanning,roboti s, and ommu ni ation ne twork s. In many as es ,k nowle dge
abou tthe environmentin wh i hroutingtakespla e isnot available beforehand ,an dtheve-
hi le/robot/pa ke tmust learn thisinformationthrough exp loration. Algorithms forrouting
in thes e type s of environme nts are referred toas online [2℄routin g algorith ms.
I n this paper we ons ider on line rou ting in th e follow ing abstra t settin g [3℄: The envi-
ronme ntis ap lanegraph, G(i.e., the planar e mbed dingof G)with nverti es. T hesour e s
and des tination t are ve rti es of G, and a pa ket an only travel on e dges of G. I nitially, a
pa ke tonly know sth e oordinates of s, t, and N(s), wh ere N(v)denote s the se tof ve rti e s
adja entto v. When ap a ket vis its a node v, it learns the oordinates of N(v).
Th isresear hwas part lyfu ndedbytheNaturalS ien esandEngineeringResear hCoun ilofCanada.
y
S hool of Computer S ien e, Carleton University, 1125 Colon el By Dr., Ott awa, Ontario, Canada,
K1S5B6,fjit,morings s. arleton. a
z
IM FM,UniversityofLj ublj ana,Jadranska11,SI-1111Ljub ljana,Sloveniaan dD epartmentofComput er
S ien e, LuleaTe hn i alUniversity, SE-97187Lulea,Sweden. Andrej.BrodnikIMFM.Uni-Lj.SI
x
U niversityofKarlskona/Ronneby,37141KARLSKRON A,Sweden,svante. arlssonsm.luth.se
{
D epartment of Compu ter S i en e, University of Waterloo, Waterlo o, Ontario, Canada, N 2L3G1,
feddemain,rudolf,imunroguwaterloo. a
k
Fa ultyofComput erS ien e,U niversityofNewBrunswi k,P.O.Box4400,Fred eri ton,N ewBrunswi k,
Can ada,E3B4A 1,alopez-ounb. a
randomiz ation. A dete rminis ti routing algorithm is memoryless or oblivious if, given a
pa ke t urre ntly atverte xv andde stin ed fornod et, thealgorithm de ides whe retoforward
the p a ket base d only on the oordinates of v, t and N(v). A ran domize d algorithm is
obliv ious if it de ide s w here to move a pa ket base d only on the oord inates of v, t, N(v),
and the outp utof a randomora le. An algorith m A is defeated by a grap hG if theree xis ts
apairofve rti es s;t 2Gsu hthat apa kets tore d atswill neverrea ht w henbeingrou te d
us ingA. Othe rwis e, we s ay that A works for G.
Le tA(G;s;t)de notethele ngthofthewalktakenbyroutingalgorithmAwhe ntrave lling
from ve rtex s to ve rtex t of G, and le t SP(G;s;t) de note the len gth of the sh orte st path
betwe en s and t. Wes ay that A is - ompet itive for a las sof graph s G if
A(G;s;t)
SP(G;s;t)
for all graphs G 2 G an d all s;t 2 G, s 6= t. We s ay th at A is simp ly ompet it ive if A is
- ompetitive forsome on stant .
Re ently,s everalpapers havedealtw ithon lineroutingandrelate d problemsingeome tri
se ttings. Kalyanas undaram and Pruh s [7℄ give a 16- ompe titive algorithm to explore any
unk now n plane graph , i.e. , vis it allof its nod es. This online ex ploration problemmakes th e
same as sumptions as those made he re, but the goalof the problemis to v is it all ve rti es of
G, not just t. This d i eren e leads to in herently di ere nt solutions .
Kranakis et al. [8℄ give a d eterministi obliv ious rou ting algorith m that works for any
Delaunaytriangulation, andgiveadeterminis ti non-obliv iou salgorithm thatwork sfor any
onn e te d plane graph.
Bos eandM orin[3℄alsostud yonlin eroutingingeome tri s ettings,parti ularlytrian gu la-
tions. Th eygivearandomiz edob liviousroutin galgorithm thatwork s forany triangulation,
and ask whe ther there is a determinis ti obliviou s routing algorithm for all trian gu lations .
Th ey also give a ompetitivenon-ob livious routing algorithm for Delaunay triangulations .
C u ka etal.[5℄e xpe rimentally evaluate the performan e of rou tingalgorithmsve ry sim-
ilar to thos e de s ribed by Kranak is et al. [8℄ and Bos e and Morin [3℄. W hen onsid ering
the Eu lidean distan e travelled d uring point-to-point routing, the ir res ults sh ow th at th e
greedyroutin galgorithm[3℄performsbetterthanthe om pa ssroutingalgorithm [3,8℄on
random graphs , bu t doe s not do as well on Delaun ay triangulations of random point s ets.
1
However, wh en one ons ide rs not the Eu lide andistan e,but the numbe rof edge straverse d
(linkdistan e),the nthe omp as sroutingalgorithmisslightlymoree Æ ie ntforbothrandom
grap hs and Delaun ay trian gu lation s.
I nth ispaperwepres entanumberofne wfundame ntaltheoreti alre sultsth athe lpfu rther
the und erstanding of online routing in plane graph s.
1. We give a deterministi ob livious rou ting algorithm for all triangulations, solv ing th e
open problem posed by Bose and Morin[ 3℄.
1
Cu kaetal . allthese algorithms p-df sandd-dfs,respe tively.
w (v)
t v
Figure 1: De nitionof w(v) and w(v).
2. We prove that no dete rminis ti ob livious routing algorithm works for all onve x sub-
div isions , s how ing s ome limitations of de te rminis ti obliviou s routin galgorith ms.
3. We prove that the randomized obliv ious routing algorithm random - o mpass de-
s ribed by Bos e and Morin[3℄ works forany onvex s ubdiv ision.
4. Wesh owthat,un dertheEu lideanmetri ,n oroutingalgorithm ex iststhatis ompe t-
itive for all triangulations, an d unde r the link dis tan e metri , no routin g algorithm
ex iststhat is ompetitive for allDelaun ay, gre edy, orminimum-we ight trian gu lations .
T heremainderofth epaperisorganizedasfollow s: InS e tion2wegiveourdete rminis ti
obliv ious algorith mforroutingintriangulations. S e tion 3p rese ntsou rre sultsforroutingin
onve xsu bdivis ions. Se tion 4des ribesou rimpos sib ilityre sultsfor ompe titivealgorithms .
Finally, Se tion 5summarizes and on lude s withope n prob lems .
2 Obl i vi ous R outi ng i n Tri angul a ti ons
A triangulation T is a plane grap hfor whi h everyfa e is a triangle,e x e pt the outer fa e,
wh i h is th e omple ment of a onve x polygon. In this s e tion we des ribe a dete rminis ti
obliv ious rou ting algorith m that works for all trian gu lations . Th e algorithm is a arefully
de signed ombin ation of two e xis ting algorithms [3℄. T he greedy algorithm alway s move s
a pa ke t toa ne ighbouring node that min imize s the dis tan e to t. T he om pass algorithm
always moves a pa ke t to the n ode th at is most \inline " w itht. Both th es e algorithms are
de fe ate d by ertaintypesof trian gu lations , b ut theway sinwhi hthey arede featedareve ry
di e re nt. By ombiningthem, we ob tainan algorithm that is works for any trian gu lation.
We u se the n otation
6
a;b ; to denote the angle formed by a b and as measu re d in
the ounter lo k wisedire tion . Let w (v) be the vertex in N(v) w hi hminimiz es the angle
6
w(v);v;t an dle t w(v)beth eve rte x inN(v) w hi h minimiz es the angle
6
t ;v; w(v). If
v has an eighbour w onthe line segment(v;t), th en w(v)= w(v)=w. In parti ular, th e
vertex t is ontained in the we dge w(v);v; w(v). Referto Fig. 1foran illu stration .
T he greedy- o mpass algorithm alway s moves to the ve rtex among f w(v); w(v)g
that minimize s the distan e to t. If the two dis tan es are equ al, or if w (v)= w (v), the n
greedy- ompass hoos es one of f w(v); w(v)g arb itrarily.
t v
i
i +1
w(v
i )
C
D v
f
R
1
R
2
Figure 2: T he proof of The ore m 1.
Theorem 1. Algor it hm greedy- o mpassworks for any triangulation.
Proof. S uppose , by wayof ontrad i tionthat a triangulationT and ap airof verti es s and
t ex ist su hthat greedy- ompass doesnot nd apath from s tot.
I n this as e the re mus tbe a y leof ve rti es C =hv
0
;:::;v
k 1
i of T s u hthat greedy-
om passmovesfromv
i tov
i+1
forall0ik,i. e.,greedy- o mpassge tstrapped y ling
through th eve rti es of C (se e alsoLe mma1of [3℄).
2
Furthermore ,itfollow sfrom Lemma2
of [3℄thatthe des tinationt is ontained inthe inte rior ofC.
Cla im 1. Allverti es of C must lie on the bou ndary of a disk D entered at t.
Proof (of laim). Suppose , by way of ontradi tion, that there is no su h d isk D. T hen let
D be the dis k ente red at t an d hav ing the fu rth es t ve rtex of C from t on its boun dary.
Cons ide r a vertex v
i
in the interior of D su h that v
i+1
is on the bou ndary of D. (Refer
to Fig. 2.) As sume, w.l.o.g., that v
i+1
= w(v
i
). T hen it must be th at w(v
i
) is n ot in
the interior ofD, oth erwis e greedy- om pa ss would not havemoved tov
i+1
. But then th e
ed ge ( w(v
i
); w(v
i
)) uts D into two regions, R
1
ontain ingv
i
and R
2
ontain ingt . Sin e
C pass esthroughbothR
1
andR
2
andis ontain ed inDthe nitmustbeth atC e ntersregion
R
1
at w (v
i
) and leaves R
1 at v
i+1
= w(v
i
). Howe ver, this annot h ap pe n be ause both
w( w(v
i
)) and w( w(v
i
)) are ontained in the halfspa e boun ded by the s upporting lin e
of ( w(v
i
); w(v
i
))and ontain ing t, and are the refore n ot ontaine d in R
1 .
T hus , we have e stablis hed that all verti e s of C are on the bou ndary of D. Howe ver,
sin e C ontains t in its inte rior and the triangulation T is on ne te d, it mus t be th at for
some verte x v
j
of C, w(v
j
) or w(v
j
) is in the interior of D. Su ppos e that it is w (v
j ).
But th en we have a ontradi tion, s in e the greedy- ompass algorithm wou ld have gon e
to w (v
j
) rathe rthan v
j+1 .
2
H ere,andin therem ainderofth ispro of, allsubs riptsaretaken modk.
A onvexsubdivision isane mbe ddedplanegraphsu hthatea hfa eofthegraphisa onve x
polygon, ex eptthe outerfa ewh i histh e omple mentofa onvexpolygon. Trian gu lation s
are as pe ial as eof onve xs ubdiv isionsinw hi hea hfa eis atriangle;thusitisnaturalto
ask wh ethe rthe greedy- om passalgorithm an begeneraliz ed to onvexs ubdivisions . In
this se tion, we s how th at th ere is no dete rminis ti obliv ious routin g algorithm for onve x
su bdivis ions. However, the reisarandomized obliv iou sroutin galgorithm thatus es only on e
randombit pe rs tep.
3. 1 De termi nisti Algo rithms
Theorem 2. Every determinist i oblivious rout ing algor it hm is defeated by some onvex
subdivision.
Proof. Weex hibita nite olle tionof onvexsu bdivis ionssu hthatanyd etermin isti obliv-
ious routing algorithm isde fe ate d by at le as t one of the m.
T here are17 ve rti es thatare ommontoallofou rsubdivis ions. Th ede stinationverte x
tislo atedattheorigin. Th eoth er 16verti esV =fv
0
;:::;v
15
gareth everti e s ofaregular
16-gon entere d atthe origin and listed in ounter lo k wis e order.
3
I n all ou r s ubdiv isions ,
the even-nu mbe re d verti es v
0
;v
2
;:::;v
1 4
h ave de gre e 2. Th e de gree of th e other ve rti e s
varies . All of our subdiv ision s ontain the edges of th e re gu lar 16-gon.
As su me, by way of ontradi tion, that the re exists a routing algorithm A that work s
for any onvex subdiv is ion . Sin e th e even-nu mbere d ve rti es in ou r s ubdiv isions alway s
have the s ame two neighbou rs in all su bdivis ions, A alway s makes the s ame de ision at a
parti ulare ven-numberedvertex . Thus ,itmake ss ens etoaskw hatAdoeswh en itv isitsan
even -numbe red ve rte x, withou tkn owinganythinge lseabou tthe p arti ularsu bdivis ionthat
A is routingon .
For ea h ve rte x v
i
2 V, we olor v
i
bla k or white de pendin g on the a tion of A upon
visiting v
i
, s pe i ally, bla k for movin g ounter lo kwise and white for mov ing lo k wis e
arou nd the regular 16-gon. We laim that all eve n-numbe red ve rti es in V mu st have th e
same olor. If not, th en there ex ists two ve rti es v
i and v
i+2
su h that v
i
is bla k and v
i+2
is w hite. Th en, if we takes=v
i
in the onve x su bdivis ion show n inFig.3. a, the algorithm
be omestrappe dononeoftheedges(v
i
;v
i+1 )or(v
i+1
;v
i+2
)an dneve rrea h esth ede stination
t, ontradi tingthe as sump tion that A work s for any onve x subdiv is ion .
T here fore, as sume w .l.o.g. that all e ve n-numbered ve rti es of V are bla k, and onsider
the onvexs ubdiv isions howninFig.3.b. Fromthis gureitis learth at,ifwetakes=v
1 ,A
ann otvisitxafterv
1
,s in eth enitgetstrappedamongtheverti e sfv
12
;v
13
;v
14
;v
15
;v
0
;v
1
;xg
and never rea hes t.
Note that we an rotate Fig.3.b by integral multiple s of =4 while le aving th e verte x
3
Int heremaind eroft hisproof,allsub s riptsareimp li itlyt akenmod16.
v
i v
i+2
t
v
i+1
1
v
0
v
15
v
14
v
13
v
12 x
t v
2
v
3
v
4
v
5
v
6
v
7
v
8
v
9 v
10 v
11
1
v
0
v
15
v
14
v
13
v
12 v
2
v
3
v
4
v
5
v
6
v
7
v
8
v
9 v
10 v
11
(a) (b) ( )
Figure 3: T he proof of The ore m 2.
Figu re 4: Stri tly onvex su bdivis ions that an be us ed in th e proof ofT heorem 2.
labelsin pla eand makesimilarargumentsfor v
3 ,v
5 , v
7 , v
9 , v
11 , v
1 3 and v
15
. However, this
imp lies that A is defeated by the onvex s ubdivision sh own in Fig.3. s in e if it begins at
anyvertex ofthe regular16-gon,itne ve rentersthe interiorofthe 16-gon . We on lud e that
nooblivious onlin e routin galgorithm works for all onvex s ubdiv isions .
We note that, although our proof u ses su bd ivis ions in whi h s ome of the fa e s are not
stri tly onvex(i.e ., haveve rti es w ithinteriorangle), it is pos sib letomodifythe proofto
us e only s tri tly onve x subdivis ions, but d oin gso leads tomore lutte re dd iagrams . T hes e
diagrams are s how n inFig.4. We leavethe details to the interes ted reade r.
3. 2 R ando mized Al gori thms
Bose and Morin [3℄ des ribe the rando m- ompass algorithm and show that it works for
any trian gu lation. Forap a ketstoredatnodev,therando m- ompassalgorithms ele tsa
vertex from f w(v); w(v)g uniformly at ran dom and move s toit. In this s e tion we show
that ran dom- om pass works for any onvexs ubdiv ision.
Alth ou gh it is we ll kn own that a random walk on any graph G w ill eventually v isit all
Th e rs t advantage is that th e rand om- om pa ss algorith m is more eÆ ient in its us e of
randomiz ation than a random walk . It req uires only one rand om bit pe r s te p, w here as a
random walk req uires logk random bits for a ve rtex of degree k. Th e s e ond advantage is
that the random - o mpass algorithm makes us e of geometry to guide it, and th e res ult
is th at rand om- ompass ge nerally arrives at t mu h more qu i k ly than a random walk .
Neve rth eles s,it anbehe lpfultothinkof ran dom- om passasarandomwalkonadire te d
grap h inw hi he ve ry nod eh asout-degree 1 or 2e x e pt for t w hi his a sink .
Be forewe an makestateme nts abou t whi h graph s d efeat rand om- ompa ss, wemu st
de ne w hat it means fora graph to de fe atarandomiz ed algorithm. We say that agraph G
de fe ats a (randomized) routing algorithm if the re ex istsa pair of ve rti es s and t of Gs u h
that ap a ketoriginatingats with de stinationt has probability 0of re a hin g t inany n ite
nu mbe r of s teps . Note that, for obliv ious algorith ms, provingthat a graph doe s not defeat
analgorithm implies that the algorithm willrea h its d es tination with probability 1.
Theorem 3. Algor it hm random - o mpass works for any onvex subdivision.
Proof. A ssu me, by way of ontradi tion, that th ere is a onve x s ubdiv ision G w ith two
verti e s s andt s u hth at thep rob ab ility ofrea hingsfromt us ing rand om- om passis0.
Th enthereisasu bgrap hH of G ontainings,b ut n ot ontain ingt ,s u hth atforallve rti e s
v 2H, w(v)2H and w (v)2H.
T he ve rtex t is ontained insome fa ef of H. We laimthat this fa e mus tbe onvex .
For the sake of ontrad i tion, ass ume otherw ise . T hen the re is a re ex vertex v on th e
boundary of f su hthat the line s egme nt (t;v) doe s not interse t any edge of H. Howe ver,
this an not happen , s in e w(v) and w(v)are in H, and he n e v would not be re e x.
S in e G is onne ted, it mus tbe that for some vertex u on the bound ary of f, w(u)or
w(u) is ontain ed in the interior of f. B ut this vertex in th e interior of f is als o in H,
ontradi ting the fa t that f is a onvex fa e of H. We on lude that there is no onve x
su bdivis ionth at defe ats rand om- om pass.
4 Competi ti ve Routi ng Al gori thms
Ifweare willingtoa e pt more sophis ti atedrou ting algorithms that makeus e of me mory,
thenitiss ometimesposs ibleto nd ompe titiveroutin galgorithms. BoseandM orin[3℄give
a ompe titive algorithm for Delaunay triangulations under the E u lidean distan e me tri .
Twoq ues tionsaris efromth is: (1)Canthisresu ltbege neralize dtoarbitrarytrian gu lations ?
and (2) Can this res ult be dup li ated for the link d istan e me tri ? I n this se tion we show
that the ans werto both thes e que stions is n egative .
s
(a)
s t
v
b
(b)
Figure 5: (a) T he triangulation T with th e path foun d by A indi ate d. (b) T he res ulting
triangulationT 0
withthe \almost-ve rti al" path sh own in bold .
4. 1 E u li dean Dist an e
Inthiss e tionweshowth at,unde rtheE u lidean metri ,nodete rminis ti rou tingalgorithm
is o(
p
n )- ompe titive for all triangulations . Ou r proof is a modi ation of that use d by
Pap ad imitriouan dYann akak is[9℄toshowth atnoonline algorithmfor ndingade stination
point amongn axis -oriented re tangular ob sta le sin th e p lan e is o(
p
n )- ompe titive.
Theorem 4. Under t he Eu lidean dist an e metri , no determinist i rout ing algorithm is
o(
p
n) ompetit ive for allt riangulations.
Proof. C onsid er annn he xagon allatti e withthefollow ingmodi ations. T he latti e has
had its x- oord inates s aled so that ea h ed ge is of le ngth (n). The latti e als o has two
additionalverti es ,sand t , entere dhorizontally,atoneu nitbelowthebottomrowand on e
unitabovethe toprow,respe tively. Fin ally,allverti e sof thelatti eands andt havebee n
omp leted toa trian gulationT. Se e Fig. 5. afor anillustration.
Le t A be any de terministi rou ting algorithm and obs erve th e a tions of A as it route s
from s to t. In parti ular, on side r the rs t n+1 ste ps take n by A as it rou tes from s
to t. The n A v isits at most n+1 verti e s of T, an d thes e verti e s in du e a s ubgraph T
vis
ons istin gof allve rti es v isite d by A and alle dge sadja entto the se verti es.
For any ve rte x v of T not equ al to s ort, de n ethe x-span of v asth e interval betwee n
the rightmos tandleftmost x- oordin ateof N(v). Thelen gthofany x-sp anis(n),and th e
width of the originaltriangulation T is (n 2
). T his imp liesthat the re is somevertex v
b on
the bottom row of T wh os e x- oordin ate is at mos t n p
n from the x- oordinate of s and is
ontain ed in O(
p
n) x -s pans of the verti es vis ited in the rst n+1step s of A.
We n ow reate the triangulation T th at ontains all verti es and ed ge s of T
vis
. Addi-
tionally, T 0
ontains the s et of e dges forming an \almos t ve rti al" p ath from v
b
to the top
row of T 0
. This almos t ve rti al path is a path that is verti al w here ver poss ible, but u se s
min imal de tou rs to avoid e dge s of T
v is
. Sin e only O(
p
n) detours are req uired , the length
of th is path is O(n p
n). Finally, we omple te T 0
to a triangulation in some arb itrary way
that does n ot in rease the d egrees of verti es on the rs t n+1 ste ps of A. S ee Fig.5.b for
ane xamp le.
Now, sin e A isdete rminis ti , the rst n+1ste ps take nby A onT 0
will beth e s ameas
the rstn+1step staken byA onT,and w illthe re foretrave ladis tan e of(n 2
). Howe ver,
there is a path in T 0
from s to t that rs t vis its v
b
(at a ost of O(n p
n)), then us es th e
\almost-ve rti al" path toth e top row of T 0
(at a ost of O(n p
n)) and the n travels dire tly
to t (at a os t of O(n p
n)). Thus ,the total ost of this p ath, and he n e th e shorte st path,
froms tot is O(n p
n ).
We on lud e that A is not o(
p
n )- ompe titive for T 0
. Sin e the hoi e of A is arbitary,
and T 0
ontainsO(n)verti e s, th isimplie s thatn odete rminis ti routingalgorithm iso(
p
n)
ompe titivefor alltriangulations with n ve rti es .
4. 2 Li nk Dis tan e
Th e link dis tan e metri s imply measure s the number of edges travers ed by a rou ting algo-
rithm. For many networking appli ations, this metri is more me aningfu l th an E u lid ean
dis tan e. I nthiss e tionwes howthat ompe titivealgorithmsund erthe link distan eme tri
are harde r to ome by than und er the Eu lidean d istan e me tri . T hroughout this se tion
weass umethatthe re ad erisfamiliarwiththede nition sofDelaunay,greedyand minimum-
weight triangulation s ( f. Pre parataand Sh amos [ 10℄).
We obtain this res ult by ons tru ting a \bad" family of point se ts as follows . Le t C
i
be the se t of p
n points f(i p
n ;1);(i p
n;2);:::;(i p
n ; p
n )g. We all C
i
th e it h olu mn.
Let D
i
= f(i p
n ;1);(i p
n;
p
n)g, an d d e ne a family of point s ets S = S
1
j=1 fS
j
2g whe re
S
n
=fS
n;1
;:::;S
n;
p
n g and
S
n;i
= i 1
[
j=1 C
j [D
i [
p
n
[
j=i+1 C
j [f(
p
n=2;0);( p
n=2;
p
n+1)g (1)
Twomembers of the se tS
4 9
are s hown in Fig.6.
Theorem 5. Under the link dist an e met ri , no routingalgorit hm iso(
p
n)- ompetit ive for
all Delaunay triangulations.
Proof. Weuseth e notationDT(S
n;i
)tod enotetheDelaunay trian gu lationofS
n;i
. Although
the De launay triangulation of S
n;i
is not unique , we will ass ume DT(S
n;i
) is trian gu late d
as in Fig. 6. Note that, in DT(S
n;i
), the shortes t path betwee n the topmost vertex s and
bottom-mos t ve rtext is of le ngth 3, inde pe nde ntof n and i. Furthe rmore, any path from s
Figu re 6: The pointse ts (a) S
49;2
and (b )S
49 ;5
along w iththeir Delau nay triangulation s.
to t w hose length is le ss than p
n must v isit verti e s from on e of the olu mns C
i 1 , C
i , or
C
i+1 .
T he re st of the proof is bas ed on the follow ing ob servation: I f we hoos e an ele ment i
uniformlyatrandomfromf1;:::; p
ng, the nth e probabilitythat aroutingalgorithm A has
visited ave rtex of C
i 1 , C
i , orC
i+1
after k s teps is at mos t3k=
p
n . Letting k = p
n=6, we
se e th at th e probability that A v isits a vertex of C
i 1 , C
i , or C
i+1 afte r
p
n=6 ste ps is at
mos t1=2.
Le ttingd
i
denote the (e xpe ted, in the ase of rand omize d algorithms) numbe rof step s
wh en rou tingfrom s tot in S
n;i
using routingalgorithm A,wehave
1
p
n
p
n
X
i=1 d
i
p
n=12 : (2)
Sin e, for any S
n;i
, the sh orte st path from s to t is 3 there mus t be s ome i for whi h th e
ompe titiveratio of A forS
n;i
is atle as t p
n=362( p
n).
Theorem 6. Under the link dist an e met ri , no routingalgorit hm iso(
p
n)- ompetit ive for
all greedy t riangulations.
Proof. T his follow simmed iate lyfromthe obs ervationthat forany S
n;i
,aDe launay trian gu-
lation of S
n;i
isalso agre edy triangulation of S
n;i .
Theorem 7. Under the link dist an e met ri , no routingalgorit hm iso(
p
n)- ompetit ive for
all minimu m-weight triangulations.
Proof. We laim that formembe rsof S, any greed y trian gu lation isals oa minimum-we ight
triangulation. To prove this, we use a re sult on minimum-we ight triangulations due to
Ai hholze r et al. [1℄ . Let K
n;i
be the omplete grap h on S
n;i
. The n an edge e of K
n;i is
said tobe a light edge if eve ry edge of K
n;i
th at ros ses e is not s horte r th an e. Ai hholz er
et al. prove that if the se t of light ed ge s ontains the e dges of a triangulation the n that
triangulationis a minimum-weight triangulation .
T here are only 5 di e rent ty pe s of edges in the gree dy triangulation of S
n;i
; (1) ve rti al
ed ge s w ithin a olumn, (2) horizonal edge s be twe en ad ja ent olumns , (3) diagonal edge s
betwe en ad ja ent olumns, (4) e dge s us ed to triangulate olumn i, and (5) edges use d to
joins and t toth e re st of the grap h. It is s traightforwardto verifythat allof the se type s of
ed ge s arein deed lighted ge s.
graph s obliv ious oblivious 4
ompe titive ompetitive
DT Yes [3,8, #℄ Ye s [ ℄ Yes [3℄ N o [h ere℄
GT /M WT Yes [#℄ Ye s [#℄ Yes [4℄ N o [h ere℄
Triangulations Yes [here ℄ Ye s [3, ℄ No [here ℄ N o ["℄
Conv . S ubdv. N o [here ℄ Ye s [h ere℄ No ["℄ N o ["℄
Plane graphs N o [F℄ No [ F℄ No [F℄ N o [ F℄
Table1: A s ummary ofk nown res ults for online routing inplane graph s.
5 Con l usi ons
We have p rese nte d a numbe r of res ults on ernin g onlin e rou tingin plane grap hs. Table 1
su mmarize s w hat is urrently know n about online routing in p lan e graphs . An arrow in a
refere n e in di ates that the resu lt is implie d by the more general res ult pointed to by th e
arrow. An Findi ate sthat the res ult istrivial an d/or folk lore.
We have als o imple mented a s imulation of th e greedy- om pa ss algorithm as well as
the algorithms des ribed by Bose and Morin [3℄ an d omp are d the m u nder th e E u lid ean
dis tan e me tri . Th ese re sults will be pres ented in the full version of the p ape r. H ere we
only s ummarize our main ob servations .
For Delaunay triangulations of rand om point s ets, we fou nd that the performan e of
greedy- ompass is omp arable to that of the o mpass and greedy algorith ms [3, 5,
8℄. For triangulation s obtained by pe rforming Graham's s an [ 6℄ on random point sets ,
the greedy- ompass algorith m d oes s igni antly better than th e o mpass or greedy
algorith ms.
We als o imp leme nte d a variant of greedy - om pass that we all greedy- o mpass-2
that, wh en lo ated at a ve rte x v, move s to th e vertex u 2 f w(v); w(v)g that minimize s
d(v;u) + d(u;t),w hered(a ;b )de note sth eE u lid eandis tan ebetwee naan db . Althoughthe re
are triangulation s thatdefeat greedy- ompass-2,it worked forallour tes t trian gu lations ,
and in fa t se ems to be twi e as eÆ ient as greedy- om pass in terms of the E u lid ean
dis tan e travelle d.
Wen oteth at urre ntly, u nderthe lin k dis tan e metri ,there are no ompetitiverouting
algorith ms for any interes ting las s of geome tri graphs (mes hes donot ou nt). T he re ason
for this se ems to be that the propertie s u sed in de n ing many ge ometri graph s make us e
of prope rties of Eu lide an spa e, and lin k dis tan e in thes e graphs is often ind epend ent of
thes e propertie s. We ons ider it anopen p roblem to nd ompetitivealgorithms , unde r th e
link d istan e metri , foran intere stingan dnaturally o u ring las s of geome tri graphs .
4
In thi s olum n, we onsider only algorit hms th at use a onst ant number of ran dom bits per step.
Otherwise,itiswellkn own thatarandomwal konanygraphGwi llevent uall y visitallverti esofG.
Th is work was initiated at S h loss Dagstu hl S eminar on Data S tru tures , h eld in Wade rn,
Germany, February{M ar h 2000, and o-organized by Sus anne Albers , Ian Mun ro, an d
Peter Widmaye r. Theauthorswouldals oliketothankLars Ja obs enforhelpfuldis us sions .
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