A theoretical and experimental study of geometric networks

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Farshi, M. (2008). A theoretical and experimental study of geometric networks. Technische Universiteit

Eindhoven. https://doi.org/10.6100/IR630219

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10.6100/IR630219

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A Theoreti al and Experimental

Study of Geometri Networks

PROEFSCHRIFT

terverkrijgingvandegraadvando tor

aan deTe hnis heUniversiteitEindhoven,opgezagvande

Re torMagni us,prof.dr.ir. C.J.vanDuijn,vooreen

ommissieaangewezendoorhetCollegevoor

Promotiesin hetopenbaarte verdedigen

opdinsdag8april2008om16.00uur

door

Mohammad Farshi

Copromotor:

dr. J.Gudmundsson

CIP-DATALIBRARYTECHNISCHEUNIVERSITEITEINDHOVEN

Farshi,Mohammad

Atheoreti alandexperimentalstudyofgeometri networks/doorMohammadFarshi.

-Eindhoven: Te hnis heUniversiteitEindhoven,2008.

Proefs hrift. - ISBN978-90-386-1135-8

NUR993

Subje theadings: omputationalgeometry /datastru tures/algorithms

Kern ommissie:

prof.dr. M.H.Overmars(Utre htUniversity)

prof.dr. M.Smid(CarletonUniversity)

prof.dr. J.J.vanWijk (Te hnis heUniversiteitEindhoven)

Ministry of Science, Research and Technology

Islamic Republic of Iran

The work in this thesis is supported by the Ministry of S ien e, Resear h and

Te hnologyofI.R.Iran unders holarshipno.800.341.

The work in this thesishas been arried outunder the auspi es of the resear h

s hoolIPA(InstituteforProgrammingresear handAlgorithmi s).

IPAdissertationseries2008-12

Mohammad Farshi 2008. All rights are reserved. Reprodu tion in whole or

in partisprohibited withoutthewritten onsentofthe opyrightowner.

Printing: EindhovenUniversityPress

Prefa e v

1 Introdu tion 1

1.1 Geometri networks . . . 1

1.2 t-Spanners. . . 3

1.3 Whyspanners? . . . 6

1.3.1 Approximateminimumspanningtree . . . 7

1.3.2 Metri spa esear hing . . . 7

1.3.3 Broad astingin ommuni ationnetworks . . . 8

1.3.4 Proteinsvisualization . . . 9

1.4 Thesisoverview . . . 10

2 Region-Fault TolerantSpanners 13 2.1 Introdu tion. . . 13

2.2 Constru ting

C

-faulttolerantspannersusingtheWSPD . . . 16

2.3 Well-separatedpairde omposition . . . 17

2.3.1 Constru tinga

C

-faulttolerantspanner . . . 18

2.3.2 Linear-sizespannersforspe ial ases . . . 20

2.3.3

C

-faulttolerantSteinerspanners . . . 22

2.4 Spe ial ases . . . 24

2.4.1

C

-faulttolerantfattriangulations . . . 24

2.4.2 Limitedboundarydire tions . . . 25

2.5

C

-fault tolerantspannersforarbitrarypointsets . . . 26

2.5.1 SSPDsandfault-tolerantspanners . . . 27

2.5.2 ComputinganSSPD . . . 32

2.6 Testingfor

C

-faulttoleran e . . . 39

2.7 Fault-tolerantgeodesi spanners . . . 40

3.2.2 A(1+")-approximationforEu lideangraphs. . . 50

3.3 Addingabottlene kedge . . . 51

3.4 A(2+")-approximationforEu lideangraphs . . . 55

3.4.1 Linearnumberof andidateedges . . . 55

3.4.2 Speedingupalgorithm3.4.1 . . . 61

3.5 Aspe ial ase:

G

has onstantdilation . . . 62

3.6 Con ludingremarks . . . 65

4 Dilation-OptimalEdge Deletion 67 4.1 Introdu tion. . . 67

4.2 Dilation-minimaledgedeletion ina y le. . . 69

4.2.1 Estimatingthedilationofapolygonalpath . . . 70

4.2.2 Thede isionproblem . . . 72

4.2.3 Theoptimizationalgorithm . . . 76

4.3 Dilation-maximaledgedeletion ina y le . . . 77

4.4 (1+")-Approximationalgorithm . . . 81

4.5 Con ludingremarks . . . 84

5 ComputingSpannerDiameter 85 5.1 Introdu tion. . . 85

5.2 Dynami programmingapproa h . . . 86

5.3 Improvingthe omplexitybounds . . . 90

5.4 Anal approa h . . . 91

5.5 Experimentalresults . . . 93

5.6 Con ludingremarks . . . 94

6 ExperimentalStudyofGeometri Spanners 97 6.1 Introdu tion. . . 97

6.1.1 Spannerproperties . . . 98

6.2 Spanner onstru tionalgorithms . . . 98

6.2.1 Theoriginalgreedyalgorithm andanimprovement. . . 99

6.2.2 Theapproximategreedyalgorithm . . . 101

6.2.3 The-graphalgorithm . . . 103

6.2.4 Theordered-graphalgorithm . . . 104

6.2.5 Therandomordered-graphalgorithm . . . 106

6.2.6 Thesink-spanneralgorithm . . . 106

6.2.7 Theskip-listspanneralgorithm . . . 108

6.2.8 TheWSPDalgorithm . . . 109

6.3 Experimentalresults . . . 111

6.3.6 Maximumandaveragedilation . . . 123

6.3.7 Crossings . . . 125

6.3.8 Thehybridalgorithms . . . 125

6.3.9 Numberofshortestpathqueries . . . 128

6.3.10 Runningtime . . . 130

6.4 Con ludingremarks . . . 135

7 Con lusions 137

ฬ΋ΧΧ୍΋Ϟฬଘάන෎Χϛϩ΋



ฬ΋Ϫ৔̶਀Ϋ̳߮ΧΫϩ୏ϩψ৑߮ι



̵ࠖέ΋άඵහ̵Ψϊॣࠗ

ThisthesisistheresultofalmostfouryearsofworkwhereIhavebeen

a om-panied and supported by many people. I now havethe pleasantopportunity to

expressmygratitudetoallofthem.

First of all, I would like to express my deepand sin ere gratitude to my

su-pervisors,MarkdeBergandJoa himGudmundssonforgivingmethepossibility

toworkunder theirsupervision. ThankstoMarkwhotooktheriskof a epting

me, asapersonwith noknowledgein omputationalgeometry, in hisgroupand

to Joa himwho was mydaily supervisorand who helpedme to understand the

on ept of spanners,dis uss problems and read my manus ripts, notonly when

hewasin TU/ebutalsoafter heleftEindhoven. Thisworkwouldnothavebeen

possible without their support and en ouragement, and I am grateful for their

valuablefriendship.

I would alsoliketo thankmydistinguished o-authorsduringmy PhDstudy

Mohammad Ali Abam, Hee-Kap Ahn, Mark de Berg, Joa him Gudmundsson,

PanosGiannopoulos,ChristianKnauer,Mi hiel Smid, and Yajun Wang,the

re-sults in this thesis is the produ t of our joint work. I would like to express my

thanks to the people who made the joint works possible: Alexander Wolf and

XavierGoao forinvitingmeto theKoreanWorkshop onComputational

Geom-etryand theorganizers oftheworkshop ongeometri networksandmetri spa e

embedding atS hlossDagstuhl. Ialso thankJoa himGudmundssonforinviting

metoNICTAandhishospitalityduring myvisit.

Themembersofmythesis ommitteearegratefully a knowledgedforreading

thethesis,providinguseful ommentsandbeingpresentatmydefensesession. It

wasmyprivilegetohaveMarkdeBerg,Joa himGudmundsson,MarkOvermars,

Mi hiel Smid and Ja k van Wijk in thethesis ommittee and Rolf Klein in the

ofIranianstudentsinEuropefortheirhelp.

I was verylu ky to work with absolutely fantasti people in the Algorithms

group. I thank them all for making my years at Eindhoven delightful. Spe ial

thanksgotoSonjaJoostenforherhelpatthestartofmystudyandAstridVolkers

forherhelpatthenalstagesofmywork. IalsothankmygreatoÆ ematesYuval

Nir, MarkS hrodersandPeterHa henberger.

Aheart-felt thanksgoestoMohammadAli Abam forhisvaluablefriendship,

enjoyabledis ussions,whi halways omewithni eideas,andteabreaks. Ilearned

alotfrom himand I'mlookingforwardto ontinuetowork withhim.

I must thank friends/families whose ompanymade my and my family'slife

mu hmoreenjoyableandtheweekend/holidaymeetingswereourmostwonderful

timesintheNetherlands. Iwouldliketomentionthefollowingfamilies: Cheema,

Eslami, Ghasemzadeh, Mousavi, Moosavi Nezhad, Nazarpoor, Nikoufard,

Reza-eian,Talebi,Vahedi. IwouldalsoliketoexpressmygratitudetoMohammadAli

Abam, MohammadEslami, Hamed Fatemi, Amir HosseinGhamarian,

Moham-mad Ghasemzadeh,KamyarMalakpoor,MohammadRezaMousavi,Mohammad

MoosaviNezhad,MahmoudNikoufard,RezaRezaeian,MohammadSamimi,Saeid

Talebi,MostafaVahedifortheirkindfriendship.

I annot endwithout thankingmy family, onwhose onstanten ouragement

andloveIhavereliedthroughoutmylife. Iamgratefultomyparentsfortheir

un- onditionalsupport,un in hing ourageand onvi tionduringmystudy. Iwould

liketothankmywifeHamidehand mysonAlirezafortheirunwaveringsupport,

patien e and understanding during this time. It is to them that I dedi ate this

work,withloveandgratitude.

ϞฬΔࡣ঒ϩ΋ΧϪরέ΋̵ΧϪরϦૡঙ



߮ଶ৳ฬϦૡঙඟࢂҤΧΔ॥ϩ΋Ϟ߮ଶ৳

Ϟ



1.1 Geometri networks

A network is, informally speaking, a olle tion of \obje ts" with ertain

\ on-ne tions"between the obje ts. An obvious exampleof a network is a omputer

network. Here the obje tsare omputers and thereis a onne tion betweentwo

omputers ifthere is aphysi al able onne tingthem. Other obvious examples

are road or railway networks. Inthe latter type of network the obje tsare the

stations,andthe onne tionsarethetra ks onne tingthevariousstations.

Thereare alsomanyothertypesofnetworks, however,where the onne tions

donotne essarilyhaveaphysi alrealization. For example,in so ials ien esone

studies so ial networks, where the obje ts ould be people and two people are

onne ted if they have a ertain so ial relationship| see Figure 1.1. Another

exampleisformedbybiologi alnetworkssu hasneuralnetworks,generegulatory

networks orprotein-proteinintera tionnetworks|seeFigure1.2.

Sometimesnetworks anberathersmall|thenetworkinFigure1.1for

exam-pleisquitesmall|butsometimes they an alsobehuge,liketheInternet(whi h

hasmore than 500million hosts) and thewebgraph|agraph whose nodes

or-respond to stati pageson the web and whose ar s orrespond to links between

these pages|whi h hasbillions of pages that are onne ted by billions of links.

For example,in 2003Googlesear hengineindexed1.6 billionsofURLs andthis

in reasedto4.2billionsin 2004.

From theseexamplesitis learthat networks formafundamentalmodelin a

Myriel

Mlle. Baptistine

Mme. Magloire

Countess de Lo

Geborand

Champtercier

Cravatte

Count

Old Man

Napoleon

Valjean

Labarre

Marguerite

Mme. de R

Isabeau

Gervais

Fantine

Thenardier

Cosette

Javert

Fauchelevent

Bamatabois

Simplice

Scaufflaire

Woman 1

Judge

Champmathieu

Brevet

Chenildieu

Cochepaille

Mother Innocent

Mlle. Gillenormand

Marius

Enjolras

Bossuet

Gueulemer

Babet

Claquesous

Montparnasse

Toussaint

Tholomyes

Listolier

Fameuil

Blacheville

Favourite

Dahlia

Zephine

Perpetue

Pontmercy

Eponine

Boulatruelle

Brujon

Lt. Gillenormand

Gillenormand

Gribier

Mme. Pontmercy

Mabeuf

Jondrette

Mme. Burgon

Combeferre

Prouvaire

Feuilly

Bahorel

Joly

Grantaire

Child 1

Child 2

Mme. Hucheloup

Baroness T

Mlle. Vaubois

Mother Plutarch

Anzelma

Mme. Thenardier

Woman 2

Courfeyrac

Gavroche

Magnon

Figure1.1: Thenetworkofintera tionsbetweenmajor hara tersinthenovel

LesMiserablesbyVi torHugo, dividedinto11 ommunitiesrepresentedbydierent

olors[NG04 ℄.

of resear hondesigning,analyzing, andoptimizingnetworks. Themathemati al

on ept orresponding to networks are graphs. (In the sequel, we will use the

termsgraphandnetworkinter hangeably.) AgraphGisapair(V;E)whereV is

a(usuallynite)setofnodesandE

⊂

V

×

V isthesetof onne tionsbetweenthe nodes. Basedonthenetwork,we anmakethegraph(edge/vertex)weighted or

dire ted/undire ted. For examplefora graphwhi h models aroadnetwork,the

weightofanedge anrepresentthelengthoftheroad. Alsobymakingitdire ted

we an showone-wayortwo-wayroads.

In some appli ations it is relevant to assume that the set of verti es of the

graph is asubset of a metri spa e and the weightof ea h edge in thegraph is

the distan e between its endpoints. A metri spa e is dened as aset where a

distan ebetweenelementsofthesetisdened. Thedistan e fun tiond( alleda

metri ) shouldbenon-negative,symmetri ,haved(x;y)=0ifandonlyifx=y,

and satisfythetriangleinequality. Obviouslyanyset ofpointsinthe planewith

the Eu lidean distan e as adistan e fun tion makes ametri spa e. As amore

omplexexample,forea hgraphG(V;E)withpositiveedgeweights,we aneasily

Figure1.2: Ayeastproteinintera tionnetwork[MS02 ℄.

thisend,wedenethedistan ebetweenea hpair(u;v)

∈

V 2

asthelengthofthe

shortestpath betweenuand v in G. We all (V;d) themetri spa eindu ed by

the graphG.

A geometri network omes from adding geometry to a network. More

pre- isely,ifthevertexsetofthenetworkisasubsetofd-dimensionalEu lideanspa e,

andthemetri istheEu lideanmetri ,thenthenetworkisageometri network.

Geometri networks model naturally(atleast approximately)manyreal-life

net-works, su h as road networks, railwaynetworks, and so on. Inthis ase we an

usegeometri propertiesto designoranalyzeanetwork. Inthis thesiswealways

onsiderundire tgeometri networks,unlessexpli itlystatedotherwise.

1.2 t-Spanners

When designing a network for a given set V of points, several riteria an be

taken into a ount. In many appli ations it is important to ensure afast

on-ne tion between every pair of points in V. For this it would be ideal to have

adire t onne tion between everypair of points|the network would then be a

omplete graph|but in most appli ations this is una eptable due to the high

osts. This leadsto the on epts of spanners, as dened below. Spanners were

introdu ed byPelegand S haer[PS89℄ in the ontext ofdistributed omputing

andbyChew[Che86℄inageometri ontext.

Let V be a subset of a metri spa e and for ea h u and v in V, let d(u;v)

denotethedistan e betweenuand v inthemetri spa e. Theaimisto designa

asub-quadrati numberofedges. Notethatthenumberof edgesinthe omplete

graph is quadrati in the numberof verti es. Toobtain a\good" graphwith a

sub-quadrati numberofedgeswehavetoallowa(hopefullyshort)detourbetween

pairs of points. For example, instead of asking for a dire t onne tion between

ea hpairofpoints,weareallowedtohaveasmalldetour. Thelengthofthedetour

is givenbyareal numbert>1. Giventheparametert,a onne tionbetweenu

and v in thegraphGis \good"ifthedistan e betweenuand v in thegraphG,

denoted by d G

(u;v), is at most t times the distan e between uand v. We all

su hapathat-path betweenuandv inthegraphG. Theratiod G

(u;v)=d(u;v)

is alledthedilation betweenuandv inG. AgraphGisat-spannerofitsvertex

set ifthedilation betweenea h pairofverti esin Gisbounded byt. Figure 1.3

isanexampleof1:5-spanneron532US ities.

Figure1.3: A1:5-spannernetworkson532US ities[NS07℄.

Here istheformaldenition oft-spanner.

Denition 1.2.1(t-spannerof apointset) AgraphG(V;E)isat-spannerofV,

for areal numbert>1, iffor ea h pair of pointsu;v

∈

V, wehave that

d G

(u;v)

≤

t

·

d(u;v):

Thedilation,orstret hfa tor,ofanetworkG(V;E)istheminimumtforwhi hG

isat-spannerofV. Whenageometri graphsatisesthet-spanner ondition,we

Intuitively,thet-spannerpropertyisstrongerthanthegraph onne tivity

prop-erty,i.e. weshouldnotonlyhaveapathbetweenea hpairofpointsinthegraph

butalsothelengthofthepathshouldbe losetothedistan ebetweenthepairof

points. The parametert de ideshow losethe t-spannerapproximatesthe

om-plete graph. Inother words, the loser tis to one,the loserthet-spanner is to

the ompletegraph.

We an easily extendDenition 1.2.1to at-spannerof agivengraph. Re all

thatforanytwoverti esuandvinagraphGwithpositiveedgeweights,d G

(u;v)

denotes thedistan e between uand v in the graphV, that is, thelength ofthe

shortestpathbetweenuand vin G.

Denition1.2.2(t-spanner ofa graph) LetG(V;E)beagraphwithpositiveedge

weights. A graphG

′

(V;E

′

) on the samevertex set butwith edge set E

′

⊆

E is a

t-spannerof Giffor ea h pair ofverti esu;v

∈

V wehavethat

d G

′

(u;v)

≤

t

·

d

G (u;v):

By omparingDenition1.2.1andDenition1.2.2one anseetwodieren es.

First,at-spanner ofagivengraphGisasubgraphofG. Therefore omputinga

t-spannerofagraphGissometimes alledapruning ofG. These onddieren e

is that, to he k the t-spanner ondition, the distan e betweenea h pair in the

t-spanneris omparedtothedistan ebetweenthemintheinputgraph. Notethat

in Denition 1.2.2itis notne essarythat theverti esoftheinput graphbelong

toametri spa e. Insteadweusethemetri spa eindu edbytheinputgraphto

measurethedistan ebetweenthem.

At-spanneronapointset V|whi h issubsetofametri spa e| anbeseen

asat-spanner ofthe omplete graphon V,denoted byG

(V),where theweight

ofea hedgein the ompletegraphisthedistan ebetweenitsendpoints.

Themainquestionindesigningt-spannersiswhetherspannersexistthathave

\good"properties. Thedesirablepropertiesarethefollowing:

Size: Thenumberofedgesinthegraph. Thisisthemostimportant

measure-mentand generating spannerswith asmall, ideally near-linear, size is desirable.

The fa t that geometri spanners with anear-linear number of edges and small

dilationexisthasmadethe onstru tionofspannersone ofthefundamentaltools

inthedevelopmentofapproximationalgorithmsforgeometri alproblems.

Degree: The maximumnumber of edges in identto avertex. This property

has been shown to be useful in the development of approximation algorithms

[GLNS02a, Yao82℄ and forthe onstru tionof ad ho networks [ALW +

03, Li03℄

wheresmalldegreeisessentialin tryingtodevelopfastlo alizedalgorithms.

Weight: The weight of a network G(V;E) is the sum of the edge weights.

(Re allthat forgeometri graphs theweightof an edge is simply theEu lidean

times the weight of the minimum spanning tree, denoted wt(MST(V)). Low

weight spanners have found appli ations in areas su h as metri spa e

sear h-ing [NP03, NPC02℄|see Se tion 1.3.2|and broad astingin ommuni ation

net-works[FPZW04℄|seeSe tion 1.3.3. Also,ithasbeenusedin the onstru tionof

severalapproximationalgorithms,see[CL00,RS98℄.

Spannerdiameter: Dened asthesmallestinteger

D

su h thatforanypairof verti esuandv in V, thereis at-path in thegraphbetweenuandv ontaining

at most

D

edges. For wirelessadho networksitis oftendesirableto havesmall spanner diameter sin e it determines the maximum number of times amessage

hasto betransmitted inanetwork. Ifagraphhasspannerdiameter

D

then itis saidtobea

D

-hopnetwork.

Ithasbeenshownthat foranyset V ofnpointsin

R

d

andforanyxedt>1

thereexistsat-spannerwith

O

(n)edges, onstantdegree,andwhosetotalweight is

O

(wt(MST(V))) [DN97, NS07℄. Aryaet al. [AMS99℄ designed arandomized algorithm whi h generatesat-spanner ofexpe tedlinearsize and expe ted

loga-rithmi spannerdiameter.

Note that some of the above properties are ompeting, e.g., a graph with

onstantdegree annot have onstantspannerdiameter,and agraphwith small

spannerdiameter annothavealinearnumberofedges[ADM +

95℄.

In most ases we are interested in t-spanners, where t is lose to 1. More

pre isely, weare interestedin s hemes where we an spe ify anyt >1and then

obtainaspannerofdilation t. Toemphasizethefa t thatt is loseto1,wewill

sometimes speakabout(1+")-spannersinsteadoft-spanners.

1.3 Why spanners?

Spannersfor ompletegraphsaswellasforarbitraryweightedgraphsnd

appli a-tionsin roboti s,networktopologydesign,distributed systems,designofparallel

ma hines,andmanyotherareas. Re entlyspannersfoundinterestingpra ti al

ap-pli ationsinareassu hasmetri spa esear hing[NP03,NPC02℄andbroad asting

in ommuni ationnetworks[ALW +

03,FPZW04,Li03℄.

Several well-known theoreti alresults alsouse the onstru tionof t-spanners

as abuilding blo k. For example, Rao and Smith [RS98℄ made abreakthrough

byshowinganoptimal

O

(nlogn)-timeapproximations hemeforthewell-known Eu lidean traveling salesperson problem, using t-spanners (or banyans).

Simi-larly,CzumajandLingas[CL00℄showedapproximations hemesforminimum- ost

multi- onne tivity problems in geometri graphs. The problem of onstru ting

geometri spannershasre eived onsiderableattentionfromatheoreti al

perspe -tive,see[ADD +

93,ADM +

95, AMS99,BGM04, DHN93,DN97,DNS95, GLN02,

thebookbyNarasimhanandSmid[NS07℄. Notethatsigni antresear hhasalso

beendoneinthe onstru tionofspannersforgeneralgraphs,seeforexample,the

book by Peleg [Pel00℄ or the re ent work by Elkin and Peleg [EP04℄ and

Tho-rup and Zwi k [TZ01℄. In this se tion we mention some of the appli ations of

t-spanners.

1.3.1 Approximate minimum spanning tree

Theproblem ofndingaminimumweightspanning tree(MST)ofagivengraph

hasre entlyattra tedalotofattention. Therearelineartimealgorithmsfor

om-putingMSTinarandomizedexpe ted asemodel[KKT95℄orwiththeassumption

that edge weightare integers[FW94℄. For deterministi omparison-based

algo-rithms,slightlysuperlinearbounds areknown[GGST86℄.

Inthegeometri ase,foranydimensiond,one an ndaminimumspanning

tree of aset of n pointsin

O

(n 2

)time by onstru ting the omplete graphand

omputing all the edge weights (=pairwise distan es) in

O

(n 2

) time, and then

running any MST algorithm (su h as Prim's algorithm). Most of the faster

al-gorithmsforthegeometri aseuse asimpleidea: nd asparsesubgraph ofthe

ompletegraphwhi h ontainsanMST,andthen omputeanMSTofthissparse

subgraph. Intheplane,theDelaunaygraphistheappropriategraphtousesin e

it only has

O

(n) edges and an be onstru ted in

O

(nlogn) time [SH75℄. It is nothelpfulinhigherdimensionsbe ausetheDelaunaygraph anhaveaquadrati

numberofedges[Eri01℄.

Salowe [Sal91℄ showed that if G

′

is a (geometri ) t-spanner of a graph G

and wt(MST(G)) denotes the weightof the minimum spanning tree of G, then

wt(MST(G

′

))

≤

t

·

wt(MST(G)). Using this result, we an use any (sparse) t-spanner ofa graphto omputean approximate minimum spanning tree ofthe

graph. For moredetailsaboutapproximatingtheminimumspanningtreeseethe

surveybyEppstein[Epp00℄.

1.3.2 Metri spa e sear hing

Theproblemof\approximate"proximitysear hinginmetri spa esistondthe

elementsofasetwhi hare\ lose"toagivenqueryundersomesimilarity riterion.

Similaritysear hinghasbe omeafundamental omputationaltaskinavarietyof

appli ationareas,in ludingmultimediainformationretrieval,patternre ognition,

omputervisionandbiomedi aldatabases. Insu henvironments,anexa tmat h

haslittlemeaning,and proximity/distan e on epts(similarity/dissimilarity)are

typi allymu hmorefruitful forsear hing. Inall ofthese appli ationswehavea

metri whi h shows the similarity between obje ts. The smaller the distan e is

A typi alqueryqis:

•

ndallelementsinthedatabasewhi harewithindistan er fromq.

•

ndthek losestelementstoq inthedatabase.

Ifthedatabase ontainsnelements,we anansweraquerybyperforming

O

(n) distan e omputations. But evaluating distan es is expensive and thegoal is to

redu ethenumberofdistan eevaluations. Ingeneral,thereareseveralmethodsto

redu e thenumberofdistan eevaluations,seethesurvey[CNBYM01℄. Awidely

used te hniqueforthisisAESA[Rui86℄. Themaindrawba kofthiste hniqueis

thatit omputesallpairwisedistan esandstorestheminamatrixwhi hrequires

alotofspa e.

Navarroet al. [NPC02℄ usedat-spanner as adata stru turefor metri spa e

sear hing to redu e the spa e needed for the AESA. The key idea is to regard

thet-spanner asanapproximationofthe omplete graphof distan esamongthe

obje ts,andtouseitasa ompa tdevi etosimulatethelargematrixofdistan es

required by su essful sear h algorithms su h as AESA. The t-spanner property

impliesthatwe anusetheshortestpathinthet-spannertoestimateanydistan e

withboundederrorfa tort.

They propose several t-spanner onstru tion, update, and sear h algorithms

and experimentallyevaluated them. The experimentsshowthat their te hnique

is ompetitive against urrent approa hes, and that it has a great potential for

further improvements.

1.3.3 Broad asting in ommuni ation networks

Wireless networks onsist of a set of wireless devi es ( alled nodes) whi h are

spread overageographi alarea. These nodesare able to performpro essing as

wellas ommuni ating with ea h other. Thenodes an ommuni ate via

multi-hop wireless hannels: anode an rea h all nodes inside its transmission region

while nodes farawayfrom ea h other ommuni ate throughintermediate nodes.

Wireless ommuni ationnetworkshaveappli ationsin various situationssu h as

emergen yrelief,environmental monitoring,andso on.

There are two ommon types of wireless networks: sink-based networks and

ad ho networks. Inasink-basednetwork,like ellularwirelessnetworks, thereis

one or multiple sink nodes whi h arein hargeof olle tingdata from all nodes

and managing thewhole network. On theother hand,in ad ho networks there

are no su h sink nodes and all nodes are equalin terms of ommuni ation and

networkmanagement.

Energy onsumption andnetwork performan eare themost riti al issuesin

wirelessadho networks,be ausewirelessdevi esareusuallypoweredbybatteries

Awirelessadho networkismodeledbyasetV ofnwirelessnodesdistributed

inatwo-dimensionalplane. Ea hnodehasthesamemaximumtransmissionrange

R whi h, by aproper s aling, we an assume that all nodes havethemaximum

transmissionrangetobeequalto1. Thesewirelessnodesdeneaunitdiskgraph,

denotedbyUDG(V),inwhi hthereisanedgebetweentwonodesiftheEu lidean

distan ebetweenthemisat most1. Themost ommonpower-attenuationmodel

in the literature laims that the powerneeded to support alink (u;v) is

k

uv

k



,

where

k

uv

k

is the Eu lidean distan e betweenu and v and  is a real onstant between 2 and 5 depending on the wireless transmission environment. So the

power onsumptionof anetwork isafun tion oftheweightof thenetwork. The

minimum possible weight for a onne ted network is the weight of a minimum

spanningtree,however,this mighthavelowperforman e.

Severalgraphtheoreti almodelsareused todesignadho networkswithlow

energy onsumption and good performan e|see [Wat05℄. Using a low weight

t-spanner of the unit disk graph is one of the ways|see[ALW +

03℄. This gives

us more exibility to onstru ta network whi h has lowenergy onsumption as

wellas goodperforman e,likebounded degree. Notethat in adho networks, a

network withsmallnodedegree,hasabetter han e tohassmallinterferen e.

1.3.4 Proteins visualization

One of the most important open problems in bioinformati s is the problem of

protein folding. A protein is a long hain of mole ules alled amino a ids. In

naturethereexist20dierentaminoa idsandseveralexperimentsshowthatthe

3D-stru tureofaproteinis ompletelydeterminedbythesequen eofaminoa ids.

The protein-folding problem is the problem of determining the 3D-stru ture

ofaproteingivenitsaminoa idsequen e. Various omputationalmethods have

beenapplied tota kletheprotein-foldingproblem,with varyingsu ess. Among

theseareneuralnetworks,approximationalgorithms,metaheuristi s,

bran h-and-bound,distributedsystemsand omputationalgeometry.

Nowadays, using high omputing power and large s ale storage, resear hers

areableto omputationallysimulate theprotein-foldingpro ess in atomisti

de-tails. Su h simulationsoftenprodu ealargenumberoffoldingtraje tories,ea h

onsisting of a series of 3D onformations of the proteinunder study. As a

re-sult,ee tivelymanagingandanalyzingsu htraje toriesisbe omingin reasingly

important.

Re ently, Russel and Guibas [RG05℄ suggested using geometri spanners for

mappingasimulationto amoredis rete ombinatorial representation. They

ap-ply geometri spanners to dis overthe proximity between dierent segments of

aprotein a rossarange of s ales, and tra k the hanges of su h proximityover

motions easier. Using their stru ture it is possible to visualize proteins in

mo-tion whi h none of the ommonly used software pa kages su h as RasMol[Ras℄,

ProteinExplorer[Mar02℄, orSPV[GP97℄havebeenabletoa hieve.

1.4 Thesis overview

In this thesis we onsider several problems relatedto the design and analysis of

geometri networks.

In Chapter 2, weintrodu e the on ept of region-fault tolerant spanners for

planarpointsets,andprovetheexisten eofregion-faulttolerantspannersofsmall

size. Forageometri graphGonapointsetV andaregionF,wedeneG

⊖

F to bewhat remainsofGafter theverti esand edgesofGinterse ting F havebeen

removed. A

C

-fault tolerantt-spanner isageometri graphGonV su hthat for any onvexregion F, thegraphG

⊖

F is at-spannerforG

(V)

⊖

F, whereG

(V)

isthe ompletegeometri graphonV. Fault-tolerantspannersprovidehighlevels

of availabilityand reliabilityin network onne tions. These networks keeptheir

goodproperties,evenaftersomepartofthenetworkisdestroyede.g. byanatural

disaster.

Weprovethatanyset V ofnpointsadmitsa

C

-faulttolerant(1+")-spanner ofsize

O

(nlogn),forany onstant">0;ifaddingSteinerpointsisallowedthen thesizeofthespannerredu esto

O

(n),andforseveralspe ial asesweshowhow toobtainregion-faulttolerantspannersofsize

O

(n)withoutusingSteinerpoints. Wealso onsiderfault-tolerantgeodesi t-spanners: thisisavariantwhere,forany

disk D, the distan e in G

⊖

D betweenany twopoints u;v

∈

V

\

D is at mostt times the geodesi distan e between u and v in

R

2

\

D. We provethat for any point set V we an add

O

(n) Steiner points to obtain a fault-tolerant geodesi (1+")-spannerofsize

O

(n). These resultsarebasedon[AdBFG07℄.

In appli ations|think of road networks, for instan e|a spanner network is

sometimesexpandedbyaddingoneormoreextra onne tions. Themainquestion

isthenhowtodotheexpansionsu hthattheresultingnetworkisasgoodas

pos-sible. InChapter3westudyaproblemofthistype. Inparti ular,we onsider the

problemofaddinganedgetoagivennetworksu hthatthedilationoftheresulting

networkisminimized. Wepresentoneexa talgorithmandseveralapproximation

algorithms. Thebest approximationalgorithm omputesa(2+")-approximation

oftheoptimalsolutionin

O

(nm+n 2

logn)timeusing

O

(n 2

)spa e,wherenisthe

number ofverti esand m is thenumberof edges in the inputnetwork. For the

spe ial ase, when thedilationof theinputnetwork is onstant,we an improve

the approximationfa torto 1+" and therunningtime to

O

(n 2

). These results

Chapter4studiestheproblemofdilationoptimaledgedeletion. Morepre isely

we are given a geometri network in the plane and wewant to nd an edge in

the network su h that its removal minimizes, or maximizes, the dilation in the

network. Anobviousappli ationis whenwewanttoremovesome onne tionsin

anexistingnetwork,e.g. duetobudget onsideration,andwewanttoknowwhi h

edges should be removed to minimize the ee t on the quality of the network.

Wesolvetheproblem in therestri ted asewhen the network is a simple y le.

A randomized algorithm is presented whi h, given a y le on a set of n points,

omputesin

O

(nlog 3

n)expe tedtime,theedgeofthe y lewhoseremovalresults

in apolygonalpath of smallestpossibledilation. It is also shown that theedge

whoseremovalgivesapolygonalpathoflargestpossibledilation anbe omputed

in

O

(nlogn)time. Iftheinput y leis a onvexpolygon,thelatterproblem an besolvedin

O

(n)time. Finally,itisshownthatgivena y leC,forea hedgee ofC,a(1

−

")-approximationtothedilationofthepathC

\ {

e

}

anbe omputed in

O

(nlogn)totaltime. Theseresultsarebasedon[AFK

+ 07℄.

In Chapter 5 we present algorithms for omputing the spanner diameter of

at-spanner. This is the rst algorithm for omputing spanner diameter of a

t-spanner,tothebest ofourknowledge. Thetime omplexityofthemosteÆ ient

algorithmis

O

(

D

·

mn),wherenisthenumberofverti es,misthenumberofedges and

D

isthespannerdiameteroftheinputgraph,anditrequires

O

(n)spa e. We also omparetherunningtimeofthepresentedalgorithmsexperimentally. These

resultsarebasedon[FG06℄.

The empiri alstudy ofalgorithms is arapidly growingresear h area.

Imple-mentingalgorithmsandtestingtheirperforman eshowstheireÆ ien yinpra ti e

andbring thealgorithmsto thepra ti alstage. InChapter6weexperimentally

studytheperforman eandqualityofthemost ommont-spanneralgorithmsfor

points in the Eu lidean plane. The experimentsare dis ussed and ompared to

thetheoreti alresultsand in several ases wesuggestmodi ationsthat are

im-plemented and evaluated. The quality measurements that we onsider are the

numberofedges,theweight,themaximumdegree,thespannerdiameterandthe

numberof rossings. We omparetherunningtimesofthealgorithmsandsuggest

someimprovements. Thisisthersttimeanextensive omparisonhasbeenmade

between the onstru tion algorithms of t-spanners. These results are based on

[FG05℄and[FG07℄.

Spanners

2.1 Introdu tion

As we mentioned before, geometri networks have appli ations in VLSI design,

tele ommuni ations, roboti s and distributed systems. The major issues with

designing su h anetwork are performan e and reliability. The spanner on ept

apturestheperforman e whenshort onne tionsbetweenthe pointsare

impor-tant. Themainquestionisthenwhetherspannersexistthathaveasmalldilation

andasmall, ideallynear-linear,numberofedges. Other desirableproperties ofa

spannerare forexample that the total weightof the edges is small, or that the

maximumdegreeislow. Asdis ussedintheintrodu tion,su hspannersdoindeed

exist: ithas beenshownthat for any setP ofn pointsin theplane andfor any

xed">0there existsa(1+")-spannerwith

O

(n=") edges,

O

(1=")degree,and whosetotalweightis

O

(wt(MST(P))="

4

),wherewt(MST(P))istheweightofa

minimumspanning treeofP [DN97,NS07℄.

Reliability is on erned with the fa t that in many appli ations the nodes

and/orlinks in a network may fail. In a omputernetwork, for instan e, nodes

mayfailbe ause omputers an rash,andinaroadnetworklinksmayfailbe ause

roads an be ome ina essible due to a idents or maintenan e. A network is

reliable when it retains its good properties even after some nodes or links fail.

With respe t to spanners this meansthere should still be ashort path between

Fault-tolerantspannerwereintrodu edbyLev opolousetal.[LNS98℄. Inthis

paperand a follow-up paper[LNS02℄ they showed the existen eof k-vertex (or:

k-edge) fault-tolerantgeometri spanners with

O

(nklogn) edges. This was im-provedbyLukovszki[Luk99℄, whopresentedafault-tolerantspannerwith

O

(nk) edges,whi hisoptimal. LaterCzumajandZhao[CZ03℄showedthatagreedy

ap-proa h produ esak-vertex (or: k-edge) fault-tolerantgeometri (1+")-spanner

withdegree

O

(k)andtotalweight

O

(k 2

·

wt(MST(P)));theseboundsare asymp-toti allyoptimal.

Thepapersonfault-tolerantspannersmentionedaboveall onsiderfaultsthat

an destroyanarbitrary olle tionofkverti esoredges. Forgeometri spanners,

however, it is natural to onsider region faults: faults that do not destroy an

arbitrary olle tionof verti esand edges,but faultsthat destroyallverti esand

edgesinterse tingsomegeometri faultregion. Thisisrelevant,forinstan e,when

thespannermodelsaroadnetworkandanatural(orother)disastermakesallthe

roads insomeregionina essible. Thisisthetopi ofthis hapter: westudythe

existen e of small spannersin the plane 1

that aretolerantagainst region faults.

Beforewepresentourresults,letusdeneregion-fault toleran emorepre isely.

Let

F

beafamilyofregionsintheplane,whi hwe allthefault regions. For afaultregionF

∈ F

andageometri graph

G

onapointsetP,wedene

G⊖

F to bethepartof

G

that remainsafter thepointsfromP insideF andalledgesthat interse tF havebeenremovedfromthegraph|seeFigure2.1. (Forsimpli itywe

assumethataregionfaultF doesnot ontainitsboundary,i.e. onlyverti esand

edgesinterse ting theinteriorofF willbeae ted.)

F

Figure2.1: Theinputgraph

G

andafaultregionF,thegraph

G⊖

F,andthe graph

G

(P)

⊖

F.

Denition 2.1.1 An

F

-faulttolerantt-spanner isageometri graph

G

onP su h that for any region F

∈ F

, the graph

G⊖

F is a t-spanner for

G

(P)

⊖

F, where

G

(P)isthe ompletegeometri graphonP.

1

The on eptsandmanyoftheresults arryovertod-dimensionalEu lideanspa e.However,

wefeelthe on eptismainlyinterestingintheplane,sowe onneourselvestotheplanar ase

Wearemainly interestedin the asewhere

F

is thefamily

C

of onvexsets. 2

Weshallalso onsider the asewhereweareallowedtoadd Steinerpointstothe

graph. Inotherwords,insteadof onstru tingageometri networkforP,weare

allowedto onstru tanetworkforP

∪

QforsomesetQ ofSteinerpoints. Then wesaythat agraph

G

onP

∪

Qis a

C

-fault tolerant Steiner t-spanner for P if, forany F

∈ C

andanytwopointsu;v

∈

P

\

F,thedistan e betweenuand v in

G⊖

F isatmostttimestheirdistan ein

G

(P)

⊖

F.

We also study another variant of region-fault toleran e. In this variant we

requirethat the distan ebetweenanytwopointsu;v in

G⊖

F is atmostt times thegeodesi distan ebetweenuandv in

R

2

\

F. Notethatthegeodesi distan e in

R

2

\

F|thatis,thelengthofashortestpathin

R

2

\

F|isnevermorethanthe distan ebetweenuandv in

G

(P)

⊖

F. We all aspannerwith this propertyan

F

-faulttolerantgeodesi t-spanner. ItisnotdiÆ ulttoshowthatnitesize

F

-fault tolerantgeodesi spannersdonotexistunlessweareallowedtouseSteinerpoints.

Evenin the ase of Steiner points, nite size

F

-fault tolerant geodesi spanners do not exist when

F

is the family

C

of all onvex sets. Hen e, we restri t our attentionto

D

-faulttolerantgeodesi spanners, where

D

isthefamilyofdisks in theplane.

Weobtainthefollowingresults.

•

InSe tion2.2wepresentageneralmethodto onvertawell-separatedpair de omposition (WSPD)[CK93℄forP into a

C

-faulttolerantspannerforP. We use this method to obtain linear-size

C

-fault tolerant (1+")-spanners for points in onvex position and for points distributed uniformly at

ran-dominsidetheunitsquare,andtoobtainlinear-size

C

-faulttolerantSteiner (1+")-spannersforarbitrarypointsets.

•

InSe tion2.4we onsidertwospe ial ases,fattriangulationsandpolygonal region faults with limited number of edge dire tions, for whi h linear-size

C

-faulttolerantspanners anbeobtained.

•

In Se tion 2.5 we study small

C

-fault tolerant (non-Steiner) spanners for arbitrary point sets. By ombining a more relaxed version of the WSPD

with ideasfrom -graphs[Kei88℄, weshowthat any point set P admits a

C

-faulttolerant(1+")-spannerofsize

O

(nlogn).

•

InSe tion 2.6weaddressaslightlydierentproblem. Insteadof designing a

C

-faulttolerantspanner,itisalsointerestingto he kwhether anexisting networkis

C

-faulttolerantornot. InSe tion2.6wegiveanalgorithmwhi h, givenagraph

G

, he kswhetheritisfaulttolerantunder onvexregionfaults.

2

Itiseasytoseethattherearenosmallregion-faulttolerantt-spannerswithrespe tto

non- onvexfaults: if

HH

denotes thefamilyofregionsthataretheunionof twohalf-planes,then

•

InSe tion2.7westudy thegeodesi ase. WeshowthatforanysetP ofn pointsthere exists a

D

-fault tolerant geodesi Steiner(1+")-spannerwith

O

(n)edgesand

O

(n)Steinerpoints.

2.2 Constru ting

C

-fault tolerant spanners using

the WSPD

In this se tion we show a general method to obtain a

C

-fault tolerant spanner from awell-separated-pairde omposition of apointset P. Although in general

thespanner an have(n 2

)edges, weshowthat forsomespe ial asesasmaller

bound an be proven. We also show how to use the approa h to obtain small

Steinerspanners. Before westartweproveagenerallemmashowingthat, when

onstru ting

C

-faulttolerantspanners,we aninfa trestri tourattentionto half-planefaults. This lemmawillalso beused inlaterse tions. Let

H

bethefamily ofhalf-planes intheplane.

Proposition2.2.1 A geometri graph

G

on a set P of points in the plane is a

C

-faulttolerantt-spanner ifandonlyifitisan

H

-fault tolerantt-spanner. Proof. Obviouslya graphis

H

-fault tolerant ifit is

C

-fault tolerant. To prove theotherdire tionassumethat

G

isan

H

-faulttolerantt-spannerandthatF

∈ C

is anarbitrary onvexregionfault. Weneedto provethat betweeneverypairof

pointsu;v

∈

P

\

F thereisapathin

G⊖

F oflengthatmostttimesthelengthof theshortestpathin

G

′

=

G

(P)

⊖

F.

u

v

F

G

c

(P ) ⊖ F

Π(u, v)

(a)

u

v

F

h

(b)

u

v

F

( )

Figure2.2: IllustrationoftheProposition2.2.1

If uand v are not onne ted in

G

′

we are done. Otherwise, let(u;v) bea

shortest pathbetweenuand v in

G

′

, see Figure 2.2(a). We laimthat forevery

edge (p;q) in (u;v) there is a path in

G⊖

F of length at most t

· k

pq

k

. Sin e the edge(p;q) liesoutsideF andF is onvex,theremust beahalf-planeh that

A

B

C

A

C

B

≥

s × ρ

ρ

ρ

Figure2.3: Denitionofwell-separatedpair.

ontainsF but doesnotinterse t (p;q),see Figure 2.2(b). Sin e

G

is an

H

-fault tolerant t-spanner there is a path (p;q) between p and q in

G⊖

h of length at mostt

· k

pq

k

. Furthermore,sin eF

⊂

hthepath(p;q)alsoexists in

G⊖

F, see Figure2.2( ). The laimand, hen e,thelemmafollows.

2.3 Well-separated pair de omposition

Thewell-separatedpair de omposition (WSPD)was developed byCallahan and

Kosaraju [CK95℄. This powerful data stru ture has been used to solve a wide

varietyofgeometri problemslikeN-bodyproblems(usedinastronomy,mole ular

dynami s, uiddynami sandplasmaphysi s),surfa ere onstru tion[FR02℄and

manymore appli ations. We brie y review this de omposition here be ause we

willuseitinthenext hapters.

Denition2.3.1 [CK95℄ Lets>0bearealnumber,andletAandB betwonite

sets of pointsin

R

d

. We say that A and B are well-separatedwith respe t to s,

ifthere aretwo disjoint d-dimensionalballs C A

andC B

, having the sameradius,

su hthat 1. C A ontainsA, 2. C B ontainsB, and

3. the minimum distan e between C A

and C B

is at least s times the radius

of C A

|seeFigure 2.3.

Theparameterswill bereferredtoas theseparation onstant. Thenextlemma

Lemma 2.3.2 [CK95℄ Let A and B be two nite sets of points that are

well-separatedwithrespe ttos,letx andpbepointsofA, andlety andq bepoints

ofB. Then

(i)

k

xy

k ≤

(1+4=s)

· k

pq

k

,and

(ii)

k

px

k ≤

(2=s)

· k

pq

k

.

Intuitively, byLemma2.3.2, ifAandB arewell-separated,then thedistan e

betweenapointinAandapointinBisroughlythesameasthedistan ebetween

thetwosets Aand B. Alsothedistan ebetweenapairof pointswhi hbothlie

in oneofthesetsismu hsmallerthanthedistan e betweenthetwosets.

Denition 2.3.3 [CK95 ℄ LetP beaset ofnpointsin

R

d

, andlets>0beareal

number. A well-separatedpairde omposition (WSPD) for P with respe tto sis

asequen eof pairs ofnon-emptysubsets ofP, (A 1 ;B 1 );:::;(A m ;B m ), su hthat 1. A i andB i

arewell-separated withrespe ttos,for 1

≤

i

≤

m.

2. for any two distin t points pand q of P, there isexa tly one pair (A i

;B i

)

inthe sequen e, su hthat (i)p

∈

A i andq

∈

B i , or(ii) q

∈

A i andp

∈

B i ,

The integerm is alledthe size ofthe WSPD.

Inotherwords,awell-separatedpairde ompositionofapointsetP onsistsofa

set ofwell-separatedpairsthat overallthepairsofdistin tpoints,i.e.,anytwo

distin tpointsbelongtothedierentsetsofsomepair.

Callahanand Kosarajushowed thatfor anypoint setin Eu lideanspa e and

for any onstant s > 0, there always exist a WSPD of size m =

O

(s d

n) and it

an be omputed in

O

(s d

n+nlogn) time. Inthe geometri problems, when we

need all point pairs in a set, we an easily use aWSPD of the point set as an

approximationwithlinearsize.

2.3.1 Constru ting a

C

-fault tolerant spanner

Callahan and Kosaraju[CK93℄ showedthat the WSPD an beused to obtaina

small(1+")-spanner. SimilarideaswereusedearlierbySalowe[Sal91,Sal92℄and

Vaidya[Vai88,Vai89,Vai91℄. Toobtainthe(1+")-spannerone simply omputes

a WSPD

W

with respe t to s :=4+8=", and then forea h well-separatedpair (A;B)

∈ W

oneaddsanarbitraryedge onne tingapointfromAtoapointinB.

Unfortunatelythis onstru tionisnot

C

-faulttolerant,be auseafaultF an destroy the spanner edge that onne ts a pair (A;B), while some other edges

weneedtoaddmorethanasingleedgefor(A;B). LetCH(A)andCH(B)denote

the onvexhullsof AandB, respe tively. Atrstsightitseemsthataddingthe

twoouter tangents of CH(A) and CH (B) to our spanner may lead to a

C

-fault tolerantspanner, but thisis notthe aseeither. Instead,wewill triangulatethe

regionin betweenthe two onvex hulls in an arbitrarymanner, as illustrated in

Figure2.4(a).

A

B

(a) (b)

Figure2.4: (a)Illustratingthe onstru tionoftheWSPD-graph.(b)Pointsin onvex

position.

LetE(A;B)bethesetofedgesinthetriangulationaddedbetweenCH(A)and

CH(B), and let

G

betheobtainedgraphwith edgeset

E

:= P

(A;B)

∈W

E(A;B).

NotethatanytriangulationbetweenCH(A)and CH (B)hasthesamenumberof

edges. Throughoutthe hapterwewillusethenotation

| · |

todenotethenumber ofelementsinaset.

Lemma2.3.4 The graph

G

is a

C

-fault tolerant (1+")-spanner for P of size P

(A;B)

∈W

|

E(A;B)

|

.

Proof. Thesize of thegraphis obviously P

(A;B)

∈W

|

E(A;B)

|

, soit remainsto show that it is a

C

-fault tolerant (1+")-spanner. Now weobserve that for any half-plane h,

{

(A

\

h;B

\

h) : (A;B)

∈ W}

is a WSPD for P

\

h. Hen e, by Proposition 2.2.1 and the properties of the WSPD it is suÆ ient to show the

following: Lethbeahalf-planefault,letu;vbepointsnotinh,andlet(A;B)be

apairwith u

∈

Aandv

∈

B; thenthere isanedge e

∈

E(A;B)betweenCH(A) andCH (B)that isoutsideh.

Tosee this werst provethat, givena point set P and atriangulationT of

P, the graph T

⊖

h is onne ted for any half-plane h. Assume without loss of generality that his belowand bounded by ahorizontal line. Sin e anypoint of

P

\

h noton the onvexhull musthavean edge onne ting it to a point further awayfrom h,we an walkfrom pawayfrom halong edges of T until we rea h

apoint onthe onvex hullof P. Moreover, anytwo onvexhull points in P

\

h an be onne ted by onvexhulledges outsideh. It followsthat T

⊖

his indeed onne ted.

Now onsideranytriangulationT onA

∪

B thatin ludesE(A;B). ThenT

⊖

h mustbe onne ted. Sin eu;v

6∈

h,andu

∈

Aand v

∈

B,this meanstheremust beanedgee

∈

E(A;B)outsideh.

2.3.2 Linear-size spanners for spe ial ases

Themethoddes ribedabove anbeusedtogetsmall

C

-faulttolerantspannersfor severalspe ial ases. For example, ifP is in onvexpositionthen

|

E(A;B)

| ≤

3 foranypair(A;B)inthede omposition,see Figure2.4(b),soweget:

Theorem 2.3.5 ForanysetP ofnpointsin onvexpositionintheplaneandany

">0,thereexistsa

C

-faulttolerant(1+")-spannerofsize

O

(n=" 2

).

Next weshowthatwe analsogeta

C

-faulttolerantspannerwhose expe ted sizeislinearifthepointsetP isgeneratedbypi kingnpointsuniformlyatrandom

in theunitsquare.

Lemma 2.3.6 LetP beasetofnuniformlydistributed pointsintheunit square

andAbeasub-squareoftheunitsquare. Thentheexpe tednumberofpointson

the onvexhullofP

∩

Ais

O

(log (n

·

area (A))).

Proof. Ifnpointsareuniformly distributed intheunit squarethenitis known

thattheexpe tednumberofpointsonthe onvexhullofthepointsis

O

(logn)[HP98, RS63℄.

Now let X be the number of points on the onvex hull of P

∩

A and let Y :=

|

P

∩

A

|

. Clearly EXP[Y℄ =n

·

area (A). Bythe law of totalexpe tation [Ros98,Proposition4.1℄ ifX andY aretworandomvariablesthen

EXP[X℄=EXP[EXP[X

|

Y℄℄;

therefore

EXP [X℄ = EXP[EXP[X

|

Y℄℄ = EXP[

O

(log (Y))℄

≤ O

(log(EXP [Y℄) ) (Jensen's inequality[Ros98, p.418℄) =

O

(log(n

·

area(A))):

Nowwe ombinetheideasfromthepreviousse tionwithLemma2.3.6to onstru t

a(1+")-spanneroftheuniformlydistributedpointset P.

Theorem 2.3.7 Let P be a set of n points uniformly distributed in the unit

square U. For any " >0 there is a

C

-fault tolerant(1+")-spanner of expe ted size

O

(n="

2 )forP.

Proof. Constru taquadtreepartitioningofU intosmallerandsmallersquares,

untilea hsquare hassize (side length) roughly1=

√

roughly1=nwhi hmeanstheexpe ted numberofpointsin aleafregionis

O

(1). Thequadtree has

O

(n)leaves. Level` of the quadtree orrespondsto a regular subdivisionofU intosquaresofsize1=2

`

. One anshowthatthereexistsaWSPD

W

:=

{

(A i ;B i )

}

i ofsize

O

(n=" 2

)forP su hthatforea hi,thepair(A i

;B i

)either

orrespondstotwosquaresatthesamelevel,orA i

andB i

arebothsingletonpoints

thatlieinnearby ells(orthesame ell) ofthenal subdivision. Moreover,ifwe

denote byn `

the numberof pairsof the WSPD at level` of the quadtree, then

n ` =

O

(2 2` =" 2

). The existen eof aWSPD with these properties followsrather

dire tlyfrom theresultsofFis herand Har-Peled[FHP05℄. For ompletenesswe

brie ysket hanargumentforoursetting.

Foranodeofthequadtree,letP()denotethesubsetofpointsfromP inside

thesquare orrespondingto . Consideralevel `of thequadtree. For ea hpair

ofnodes ;

′

atlevel`su hthatthepointsetsP()andP(

′

)arewell-separated

whilethepointsetsof theparentsof and

′

arenotwell-separated,weputthe

pair (P();P(

′

))into the WSPD. In addition, forea h pair of leaf nodes ;

′

su hthatP()andP(

′

)arenotwell-separated,weputapair(

{

p

}

;

{

q

}

)intothe WSPDforeverypairp

∈

P()andq

∈

P(

′

). Itiseasytoverifythatthisindeed

denesaWSPD. Thebound on thenumberofpairsaddedfor ea h levelfollows

fromastandardpa kingargument.

Now onsider asquare atlevel`. ByLemma 2.3.6,be ausetheareaof is

1=2 2`

,theexpe tedsizeofthe onvexhullofthepointsin  is

O

(log (n=2 2`

)).

If (A;B) is an arbitrary pair in

W

whi h appears at level ` of the quadtree then

EXP[

|

E(A;B)

|

℄

≤

EXP[

|

CH(A)

|

+

|

CH(B)

|

℄ = EXP[

|

CH(A)

|

℄+EXP[

|

CH(B)

|

℄

=

O

(log (n=2 2` )): Therefore EXP 2 4 X (A i ;B i )

∈W

|

E(A i ;B i )

|

3 5 = X (A i ;B i )

∈W

EXP[

|

E(A i ;B i )

|

℄ = 1 2 logn X `=1

O

n ` log(n=2 2` )
= 1 2 logn X

O

(2 2` =" 2 )log(n=2 2` )
:

Toboundthis summation,wesetm:= 1 2

lognandweget:

1 2 logn X `=1 2 2` log (n=2 2` ) = m X `=1 2 2` (2m

−

2`) = 2 m X `=1 2 2` (m

−

`) = 2 m

−

1 X k =0 2 2(m

−

k )

·

k (bysettingk=m

−

`) = 2 2m+1 m

−

1 X k =0 k 2 2k

≤

2 2m+1

∞

X k =0 k 2 2k =

O

(n):

Hen e theexpe tedsizeofthegenerated(1+")-spanneris

O

(n=" 2

).

2.3.3

C

-fault tolerant Steiner spanners

AboveweshowedthattheWSPD an beusedto onstru t

C

-faulttolerant span-nersofsmallsizewhenthepointsarein onvexpositionoruniformlydistributed.

For arbitrarypoint sets, however,the sizeof thespanner may be (n 2

). In this

se tion wewill showthat ifweareallowedto addSteiner points,we an always

usetheabovemethod togetalinear-sizespanner:

Theorem 2.3.8 For any set P of n points in the plane and any ">0, one an

onstru t a

C

-fault tolerant Steiner(1+")-spannerof size

O

(n=" 2

)byadding at

most4(n

−

1)Steinerpoints.

Theideaisto addaset Qof SteinerpointstoP su hthat

|

E(A;B)

|

=

O

(1)for any pair(A;B)in theWSPD of P

∪

Q. Then the theorem immediately follows from Lemma2.3.4.

Our method is based on the WSPD onstru tion by Fisher and

Har-Peled [FHP05℄. Their onstru tion uses a ompressed quadtree, whi h is

dened asfollows.

Let

T

(P)bethequadtreeonP. Wedenotethesquare orrespondingtoanode 

∈ T

(P)by(), and the subsetof pointsfrom P inside ()by P(). When someofthepointsarevery losetogether,aquadtree anhavesuperlinearsize. A

ompressedquadtree

T

∗

(P)forP thereforeremovesinternalnodes from

T

(P)for whi hallpointsfromP lieinthesamequadrantof(). A ompressedquadtree

hasatmostn

−

1internalnodes. FisherandHar-Peled[FHP05℄showthatone an obtainaWSPD of size

O

(s

2

n) forP that onsistsof pairs(P( 1 );P( 2 ))where  1 and 2 arenodesin

T

∗

(P).

The set Q of Steinerpointsthat we use is dened as follows. Let

T

∗

(P) be

a ompressed quadtree for P. Without loss of generality, we may assume that

no point from P lies on any of the splitting lines. For ea h internal node  of

T

∗

(P),weaddthefour ornerpointsof()toQ. Toavoiddegenerate ases,we

slightlymoveea hpointintotheinteriorof(). Notethattwo(ormore)squares

( 1

)and ( 2

) mayshare, for instan e, their topright orner. In this asewe

addthe(slightlyshifted) ornerpointonlyon e. Theresultingset Qhassize at

most4(n

−

1). ThenextlemmanishestheproofofTheorem2.3.8.

= point from P

= Steiner point

Figure2.5: IllustrationfortheproofofLemma2.3.9 .

Lemma2.3.9 Let

T

∗

(P) be a ompressed quadtree for P := P

∪

Q, where the initialboundingsquareU isthesameasfor

T

∗

(P),andlet beaninternalnode

of

T

∗

(P). ThenCH(P())hasat mostfourverti es.

Proof. If the square () ontains zeroor one point from P then at most one

Steinerpointhasbeenaddedinside (),andthelemmaistrue. If() ontains

twoormorepointsthentherearetwo ases,bothillustratedinFigure2.5.

Let bethenodeof

T

∗

(P) su h that P()=P()

∩

P. Note that thefour shifted orners of () were added as Steinerpointsto Q. If () =() then

CH(P())is asquare. Otherwise, ()

⊂

(). Inthis aseCH (P()) isformed bythreeofthefour ornersof()togetherwiththeunique ornerof ()that

generatedaSteinerpointatsomean estorof in

T

∗

(P),seeFigure2.5. Hen e,

2.4 Spe ial ases

In this se tion wepresentalgorithms for onstru ting fault tolerant spannersin

twospe ial ases. InSe tion2.4.1, wegiveanalgorithm that onstru ta

C

-fault tolerant spanner for any point set whi h admits a fat triangulation. Then, in

Se tion 2.4.2, we onstru tspannerswhi h arefault tolerant undermorelimited

regionfaults.

2.4.1

C

-fault tolerant fat triangulations

We allatriangulationofapointset-fat ifallitstrianglesare-fator,inother

words,ifallanglesinthetriangulationareatleast. KaravelasandGuibas[KG01℄

showedthatany-fattriangulationT ofapointsetP isa2-spannerforP. To

make the spanner

C

-fault tolerant, we add some extra edges: we add an edge betweeneverypair ofpoints u;v

∈

P su h that there isapathbetween uandv in T onsistingoftwoedges.

Theorem 2.4.1 Let P be a set of n points in the plane and let T be a -fat

triangulation of P. Then we an augment T with aset of

O

(n=) extra edges su hthattheresultinggeometri graphisa

C

-faulttolerant2-spanner.

Proof. We onne t ea h node v to all other nodeswithin twosteps from v. In

otherwordsweaddanedgebetweenea hpairofpoints onne tedbyapathoftwo

edges. LetT

′

betheresult. Obviouslyweaddatmost P

v

D

·

deg(v)edges,where D isthemaximumdegreein the triangulationT and deg(v) isthe degreeofthe

nodevof T. Sin eforea htriangulation P

v

deg(v)

≤

6nweadd

O

(D

·

n)edges to thetriangulationT. NotethatD=

O

(1=)sin eT isa-fat triangulation.

Now the laim is that T

′

is a

C

-fault tolerant 2-spanner. Using Proposi-tion2.2.1,it suÆ estoshowthat T

′

isan

H

-faulttolerant2-spanner. Lethbe an arbitraryhalf-plane andp;q

∈

P

\

hbe two arbitrarypoints. Karavelas and Guibas [KG01, Theorem 2.1℄ proved that there exist a 2-path (p;q) between

p and q in T zig-zagging above and below the line onne ting p to q|see

Fig-ure 2.6(a). Note that allthe edges in thispath interse t the segmentbetweenp

and q.

If all the nodes on (p;q) lies outside h we are done. Otherwise assume

p

′

∈

(p;q)liesinsidehandletq 1

andq 2

betheotherendpointsofthetwoedges

on (p;q) in identto p

′

|see Figure 2.6(b). Sin ethe segmentpq liesoutsideh

andanyedgeon(p;q)interse tpq,thepointsq 1

andq 2

lieoutsideh. Be ausewe

addededgesbetweenpairswithin twosteps|thedashededgesinFigure2.6(b)|

we anrepla ethetwofatedges(p

′

;q 1 )and(p

′

;q 2 )with(q 1 ;q 2

). Thiswaywe an

p

q

Π(p, q)

(a)

p

q

h

p

′

q

₁

q

₂

(b)

Figure2.6: IllustrationfortheproofofTheorem2.4.1

2.4.2 Limited boundary dire tions

Nowweputsomelimitationontheregionfaults. Let

H

′

beafamilyofhalf-planes

withat mostk boundary dire tions. Bythefollowingpro edure,whi h usesthe

WSPD, we an makean

H

′

-fault tolerant(1+")-spanner ofea hpointset P of

npointswith

O

(kn=" 2 )size. Let

{

d i

}

k i=1

bethe setof dire tions wherethe boundaryof ea h half-plane in

H

′

is parallel to one of d i

's. For ea h 1

≤

i

≤

k assume d (1) i and d (2) i are the

twodire tions perpendi ular to d i

. To onstru t a

H

′

-fault tolerantspanner we

omputeaWSPDofthepointsetwithrespe ttos:= 4("+2)

"

. Thenforea hpair

(A;B) in the WSPD and for ea h dire tion d i

, we add two edges, one between

theextremepointofAandBindire tiond (1) i

andtheotherbetweentheextreme

pointsofAandB indire tion d (2) i

. SeeAlgorithm2.4.1formoredetails.

Algorithm2.4.1: Bounded-Boundary-dire tions

Input: P,">0andaset ofdire tions

{

d i

}

k i=1 . Output:

H

′

-faulttolerant(1+")-spanner

G

=(P;

E

).

W

:=WSPDofP w.r.t. s:= 4("+2) " ; 1 forea h (A;B)

∈ W

do 2 fori:=1;2;:::;kdo 3

AddanedgebetweenextremepointsofA andB indire tiond (1) i

; 4

AddanedgebetweenextremepointsofA andB indire tiond (2) i ; 5 end 6 end 7 return

G

; 8

Theorem 2.4.2 Let P beaset of npointstheplane and

H

′

beafamilyof

half-planeswithatmostkboundarydire tions. Thenforea h">0,we an onstru t

an

H

′

-faulttolerant(1+")-spannerofsize

O

(kn=" 2

)in

O

((nlogn+kn)=" 2

Proof. Obviouslyweaddatmost2k edgesforea hpairinWSPDandtherefore

the size of the graph is

O

(kn=" 2

). Also the time omplexity of the algorithm

is straight forward. Therefore to omplete the proof, we show that the graph

generated byAlgorithm2.4.1is

H

′

-faulttolerant. Toshowthis,itis suÆ ientto

showthatforea hh

∈ H

′

andany(A;B)intheWSPDwhi hissituatedpartially

outsideh,wehaveanedgeoutsidehwhi h onne tAtoB.

Sin eA andB are partiallyoutsideh,theextremepointsofthem inat least

one ofthedire tions perpendi ulartotheboundaryofhisoutsideh. Thismeans

thattheedgesbetweentheextremepoints,whi hareaddedbythealgorithm,lies

outsideh.

Remark 2.4.3 Atrstitmayseemthatwe angeneralizetheresultstoanyfamily

of onvexpolygons with bounded number of edge dire tions (for example

axis-parallel polygons). However as you an see in Figure 2.7 this is not the ase.

p

p

₁

p

2

q

q

₁

q

2

F

A

i

B

i

Figure2.7: Counterexampleforaxis-parallelpolygonalfaults.

2.5

C

-fault tolerant spanners for arbitrary point

sets

In this se tionwe onsider the problem of onstru ting asparse

C

-fault tolerant (1+")-spannerforanarbitrarysetPofnpointsintheplanewithoutusingSteiner

points. Themethodthatwasdes ribedinthepreviousse tiondoesnotguarantee

asmallspannerin general. Here wewilldes ribeamethodthat isguaranteedto

resultinaspannerofsize

O

(nlogn).

Throughout this se tion d(

·

;

·

) denotes the (Eu lidean) shortest distan e be-tween two obje ts (points, disks, et .), and radius(D) denotes the radius of a

2.5.1 SSPDs and fault-tolerant spanners

Theproblem withtheWSPD inourappli ationisthat,eventhoughthenumber

ofpairsintheWSPDis

O

(n),thetotalnumberofpointsoverallthepairs anbe (n

2

). Thereforewewillintrodu earelaxedversionoftheWSPD,theSSPD.

Denition2.5.1 LetAandBbetwosetsofpointsintheplane,andlets>0be

a onstant. Wesaythat A and B are semi-separatedwithrespe t to separation

onstantsiftherearetwodisjointdisksD A andD B ,su hthat (i) D A ontainsAandD B ontainsB, (ii) d(D A ;D B )

≥

s

·

min (radius(D A );radius(D B )).

Thus weallowthe ballsD A

and D B

to be of dierentsizes and weonly require

thatthedistan ebetweenthedisksislargerelativetothesmallerdisk. Notethat

usingthesamenotationswe anreformulatethedenition ofwell-separatedwith

respe ttosasd(D A ;D B )

≥

s

·

max(radius(D A );radius(D B )).

WenowdeneourSSPD.

Denition2.5.2 Let P beaset of npointsin theplane and lets>0beareal

number. Asemi-separatedpairde omposition(SSPD)forP withrespe ttosisa

olle tion

{

(A 1 ;B 1 );:::;(A m ;B m

)

}

ofpairsofnon-emptysubsetsofP su h that

1. A i

andB i

aresemi-separatedwithrespe ttos, foralli=1;:::;m.

2. foranytwodistin tpointspandqofP,thereisexa tlyonepair(A i

;B i

)in

the olle tion,su hthat (i)p

∈

A i andq

∈

B i or(ii)q

∈

A i andp

∈

B i .

Theweight of a set A, denoted by

|

A

|

, is dened as the numberof points in A, theweightofasemi-separatedpair(A;B)is thesumoftheweightsofA andB,

andtheweightofanSSPDisthetotalweightofallthepairs. Laterwewillprove

that itis possibleto omputeanSSPD of weight

O

(nlogn). First, however,we willshowhowto usetheSSPD toobtaina

C

-faulttolerantspanner. Theideais to add edges to thespanner for ea h pair in the SSPD. Be ause the pairsin an

SSPDareonly semi-separated,however,addingasingleedgefor everypairdoes

notne essarilyleadtoagoodspanner. Thereforeweuseanideathatisalsoused

inthe onstru tionof-graphs[Cla87,Kei88℄.

Considerapair(A;B)in anSSPDfor P. Thenthereexist twodisjointdisks

D A

andD B

that ontainAandB respe tively,andforwhi h

d(D A ;D B )

≥

s

·

min (radius(D A );radius(D B )):

Assume without loss of generality that radius(D A

)

≤

radius(D B

), and let o A

denote the enter of D A

|see Figure 2.8(a). Theset E(A;B) ofedges added to