Approximate Shortest Path and Distance - Christian Sommer

188 Pages·2010·3.25 MB·English

Approximate Shortest Path and
Distance Queries in Networks
Christian Sommer
Bornon8March1983inZurich,Switzerland
CitizenofWinterthur(ZH)andSumiswald(BE),Switzerland
MasterofScienceETHZurich(2006)
Submittedinpartialfulfilmentoftherequirements
forthedegreeofDoctorofPhilosophy
January2010
DepartmentofComputerScience
GraduateSchoolofInformationScienceandTechnology
TheUniversityofTokyo

Abstract
Computingshortestpathsingraphsisoneofthemostfundamentalandwell-studiedproblemsin
combinatorial optimization. Numerous real-world applications have stimulated research investi-
gations for more than 50 years. Finding routes in road and public transportation networks is a
classical application motivating the study of the shortest path problem. Shortest paths are also
soughtbyroutingschemesforcomputernetworks: thetransmissiontimeofmessagesislesswhen
theyaresentthroughashortsequenceofrouters. Theproblemisalsorelevantforsocialnetworks:
onemaymorelikelyobtainafavorfromastrangerbyestablishingcontactthroughpersonalcon-
nections.
This thesis investigates the problem of efficiently computing exact and approximate shortest
pathsingraphs,withthemainfocusbeingonshortestpathqueryprocessing. Strategiesforcom-
putinganswerstoshortestpathqueriesmayinvolvetheuseofpre-computeddatastructures(often
calleddistanceoracles)inordertoimprovethequerytime. Designingashortestpathquerypro-
cessing method raises questions such as: How can these data structures be computed efficiently?
What amount of storage is necessary? How much improvement of the query time is possible?
How good is the approximation quality of the query result? What are the tradeoffs between pre-
computationtime,storage,querytime,andapproximationquality?
Fordistanceoraclesapplicabletogeneralgraphs,thequantitativetradeoffbetweenthestorage
requirement and the approximation quality is known up to constant factors. For distance oracles
that take advantage of the properties of certain classes of graphs, however, the tradeoff is less
wellunderstood: forsomeclassesofsparsegraphssuchasplanargraphs,therearedatastructures
that enable query algorithms to efficiently compute distance estimates of much higher precision
thanwhatthetradeoffforgeneralgraphswouldpredict. Thefirstmaincontributionofthisthesis
is a proof that such data structures cannot exist for all sparse graphs. We prove a space lower
bound implying that distance oracles with good precision and very low query costs require large
amounts of space. A second contribution consists of space- and time-efficient data structures for
a large family of complex networks. We prove that exploiting well-connected nodes yields effi-
cientdistanceoraclesforscale-freegraphs. Athirdcontributionisapracticalmethodtocompute
approximateshortestpaths. BymeansofrandomsamplingandgraphVoronoiduals, ourmethod
successfullyaccommodatesbothhighlystructuredgraphsstemmingfromtransportationnetworks
andlessstructuredgraphsstemmingfromcomplexnetworkssuchassocialnetworks.
iii

Acknowledgements
IthankmyadvisorandpatronShinichiHonidenforthewonderfulJapaneseenvironmenthepro-
vided,inwhichIlearnttheessentialsofJapanesestudentlifeandculture,forthefreedomhegave
me, andforhisverygenerousfinancialsupport, whichallowedmetotraveltomanyconferences
allaroundtheworld,andwhichalsoallowedmetoworkwithhigh-qualityequipment.
I thank my co-advisor and mentor Michael E. Houle for the many fruitful and interesting
discussionsaboutalgorithmsandeverythingelse,forannouncingtheshortestpathproject,forhis
advice,guidance,andintelligentquestions,forhiseffortsasacollaborator,forteachingmehowto
improve my writing, and, honestly, for him patiently insisting on important things I did not want
tohearandbelieve.
IamveryhonoredtohaveHiroshiImaiaschairandDavidM.Avis,TakeoIgarashi,Kunihiko
Sadakane,andTetsuoShibuyaasmembersofthecommitteeformyPhDthesis. Manythanksfor
investingtheirtimeinstudyingmyworkandforexaminingthisthesis.
I profited and learned enormously from my co-authors I worked with on the results of this
thesis. I am indebted to Wei Chen, who first told me about the connection between path queries
and compact routing, Shang-Hua Teng, for empowering and inspiring me, Elad Verbin, for his
fascinating questions and his great intuition (and the invitation to Tsinghua), Yajun Wang, for
insistingandworkinghardonthesubtledifferencesbetweenrandomgraphmodels,MartinWolff,
forhisprecioushelponbootstrappingmythesisandforhisimpressiveskillsasanon-nativeeditor,
andWeiYu,forworkingonallthetediousandnastycalculations.
I also profited from collaborating with Cyrille Artho, Hristo Djidjev, Stephan Eidenbenz,
DaisukeFukuchi,NicolasW.Hengartner,Pierre-Lo¨ıcGaroche,ShivaKasiviswanathan,Ken-ichi
Kawarabayashi, Yusuke Kobayashi, Martin Mevissen, Johan Nystro¨m, Yoshio Okamoto, David
Roberts,SunilThulasidasan,andTakeakiUno. IgotpreciousadvicefromIttaiAbraham,ErikD.
Demaine, Jittat Fakcharoenphol, Cyril Gavoille, Stephan Langermann, Xiang-Yang Li, Mikkel
Thorup,andUriZwick.
This thesis has improved substantially by the valuable comments of those who read parts of
preliminaryversionsofit. IwouldliketothankMichaelE.Houle,CyrilleArtho,JacopoGrazzini,
and Johan Nystro¨m for their helpful suggestions and proofreading. The remaining errors and
omissionsareentirelytheauthorsresponsibility.
I owe a big thanks to countless individuals who I was fortunate to meet and to spend time
with in one way or the other — o-sewa ni nari mashita! in the Honiden laboratories: Miki Nak-
agawa (and Family Nishida in Takatsuki), Kyoko Oda, Akiko Shimazu, Shuko Yamada, Ai To-
bimatsu, Mizuki Inoue, Kenji Taguchi, Yasuyuki Tahara, Nobukazu Yoshioka, Fuyuki Ishikawa,
Kenji Tei, Rihoko Inoue, Nik Nailah Binti Abdullah, Rey Abe, Hikari Aikawa, Yukino Baba,
Valentina Baljak, Takuo Doi , Katsushige Hino, Satoshi Kataoka, Yojiro Kawamata, Kazutaka
Matsuzaki, Hirotaka Moriguchi, Mohammad Reza Motallebi, Hiroyuki Nakagawa, Yoshiyuki
Nakamura, Hikotoshi Nakazato, Eric Platon, Jose´ Ghislain Quenum, Yuichi Sei, Ryota Seike,
Shunichiro Suenaga, Ryuichi Takahashi, Ryu Tatsumi, Susumu Toriumi, Eric Tschetter, Kayoko
Yamamoto,AdrianKlein,MaximMakatchev,DanieleQuercia,andMartinRehak;atNII:Shigeko
Tokuda, Michael Nett, Weihuan Shu, Nizar Grira, Sebastien Louis, Se´bastien Duval, Takeshi
Ozawa, Yuzuru Sawato and all the other nice and friendly guards; at Microsoft Research Asia
andinBeijing: KunChen,YukiArase,YasuyukiMatsushita,‘Tommy’,YuanZhou,JialinZhang,
LolanSong,Peter&UrsiZu¨rcher,GabrielSchweizer,andJaimieHwang;attheLosAlamosNa-
tionalLaboratoriesandinLosAlamos: UlrikeCampbell&Glenn,FamilyEidenbenz,DouglasD.
Kautz, Jacopo Grazzini, Nicolas Jegou, Leticia Cuellar, Vishwanath Venkatesan, Guanhua Yan,
Keren Tan, Lukas Kroc, Leonid Gurvits, Nandakishore Santhi, and Carrie Manore; on my trips
v
totheUSandaroundtheworld: AliciaAponte,JittatFakcharoenphol,DanieleQuercia,Thomas,
Kati&LarsSpirig,andAndrewYao;aspecialthankstomyrunningmatesfromTokyo’sInterna-
tionalRunningClub“NambanRengo”formanyenjoyablehoursoftraining,racing,andtravelling
(andpartying): thanksGuillaumeBouvet,Gary&MamiChandler,KazuoChiba,ChadClark,Ed-
ward Clease, Joachim Dirks, Jon Holmes, Jay Johannesen, Chika Kanai, Gordon Kanki Knight,
DanielKershner,StephenLacey,ChristianeLange,BrettLarner,JasonLawrence,OmarMinami,
Teruyuki Minegishi, Keren Miers, Satohi Numasawa, Paddy O’Connor, Rie Onodera, Bob Poul-
son,GarethPughe,FabrizioRaponi,GerardRobb,DavidMotozoRubenstein,PhilipRyan,Daniel
Seite, Yuka Shigihara, Mika Tokairin, Martin Verdier, Ju¨rgen Wittstock, Kiyonari Yoshida, and
others; from the Swiss club in Japan: Jo¨rg Aschwanden, Daniel & Monika Hagemeier, Felix
Mo¨sner, Hans Prisi, Christoph Saxer, Uwe Sievers, Martin Wenk, and Hermann Werner; Japan
friends (tea ceremony & BBQ) Mitsuhiko & Yuko Kusuyama, Chikako, Tomoaki, & Noriaki
Sawada, Machiko & Yoshinobu Nagura, Junichi ‘Andy’ Kimura, Hitomi Sakamoto, Tomoko
Sawada, Toshiro ‘Joe’ Suzuki, Akemi Okado, Masae Ono, John & Yuki Mettraux; more friends
in Japan: Ryudo Tsukizaki, Koichi Matsumoto, Filiz Gencer, Jumi Klaus, Cedric S. Rutishauser,
JerryRay, myextremelypatientandlovelyJapaneseteacherChiekoOkamoto, KatsuroIshimasa
ofIdaten,andKenYamagataofIrohaformakingthebesteelsushiintown.
It was great to have friends visiting me in Japan; thanks to Samuel Burri, Adrian Doswald,
Martin Halter, Roger Herzog, Julia Imhof, Tanja Isker, Moritz Isler, Hannes Schneebeli, Georg
Troxler,andStefanWolf.
Last but definitely not least, I wish to thank my wonderful family. None of this would ever
havebeenpossiblewithouttheloveandthefantasticsupportofHansruedi,Hermine,andStefan.
Theytaughtmethatshortestpathsarenotalwaysoptimalanddesirable,andthat“derWegistdas
Ziel”. And I wish to thank my Japanese family in Miyazaki, Kayashima, and Tottori, especially
mybelovedRie,foreverything.
ChristianSommer,December2009
vi
Table of Contents
1 Introduction 1
1.1 NetworksandGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 TransportationNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 ComplexNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 ShortestPaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 ClassicalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Point-to-PointShortestPathQueries . . . . . . . . . . . . . . . . . . . . 9
1.3 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Preliminaries 17
2.1 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 GraphProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.2 GraphClasses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.3 SyntheticGraphModels . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 GraphAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 ComputationalModels . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2 ApproximationAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 CommonTechniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.1 SpannersandEmulators . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.2 DistanceLabelingsandMetricEmbeddings . . . . . . . . . . . . . . . . 29
2.3.3 PlanarGraphTechniques . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.4 Well-SeparatedPairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 ShortestPaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.1 SingleSourceShortestPath(SSSP)Algorithms . . . . . . . . . . . . . . 32
2.4.2 AllPairsShortestPath(APSP)Algorithms . . . . . . . . . . . . . . . . 35
2.4.3 ManyPairsShortestPath(MPSP)Algorithms . . . . . . . . . . . . . . . 37
2.4.4 ShortestPathandDistanceQueries . . . . . . . . . . . . . . . . . . . . 38
3 ReviewofShortPathQueryProcessing 39
3.1 Theoretical—DistanceOracles . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.1 LowerBounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.2 GeneralGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.3 RestrictedGraphClasses . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Practical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.1 HierarchicalApproaches . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.2 GraphAnnotationApproaches . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.3 RoadNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.4 ComplexNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 LowerBoundsforSparseGraphs 59
4.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 CommunicationComplexity . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.2 RegularGraphswithLargeGirth. . . . . . . . . . . . . . . . . . . . . . 64
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Table of Contents
4.2.3 CountingPermutations . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 ReductionfromLopsidedSetDisjointness . . . . . . . . . . . . . . . . . . . . . 69
4.3.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.2 Reductionfromadatastructuretoacommunicationprotocol . . . . . . . 71
4.3.3 Communicationcomplexityimpliesspacecomplexity . . . . . . . . . . 73
4.3.4 CountingPaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.5 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 ConclusionandOpenProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 DistanceOraclesforPower-lawGraphs 81
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1.1 OverviewoftheResult . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.1.2 RelatedWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2.1 DistanceOracleofThorupandZwick . . . . . . . . . . . . . . . . . . . 83
5.2.2 PropertiesofRandomPower-lawGraphs . . . . . . . . . . . . . . . . . 84
5.3 TheAdaptedDistanceOracle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.4 TimeandSpaceComplexities . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4.1 CoreSize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.4.2 BallSizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.4.3 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.5 ConclusionandOpenProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6 ApproximatingShortestPathsUsingVoronoiDuals 97
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.2.1 GraphVoronoiDiagram . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.3 TheVoronoiMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.4 ComputationalComplexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.5 StretchAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.6 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.6.1 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.6.2 DataSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.6.3 ExperimentalSetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.7 ConclusionandOpenProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7 Conclusion 121
viii
1
MILLEVIAEDUCUNTHOMINEM
PERSAECULAROMAM.
(AllroadsleadtoRome.)
(Esfu¨hrenvieleWegenachRom.)
AlanusdeInsulis
intheLiberParabolarum
12thcenturyAD
Introduction
Imagine youwantedto travel fromcentralTokyoto visittheplacewhereIgrew up, whichisthe
villageofOttenbachinSwitzerland. Whatisthefastestwaytogetthere?
ImagineyouwantedtogetintouchwithNelsonMandelausingasequenceofpersonalintro-
ductionsthroughfriendsandfriendsoffriends. Whatistheshortestsuchsequence?
ImagineyouwantedtoaccessawebpageontheInternet. Whichroutersshouldbeusedsuch
thatthenecessaryinformationisdownloadedtoyourcomputerfastest?
These questions have something in common, in that their (optimal) solution is the shortest
path between two points of a network: a transportation network, a social network, and a router
network. AllroadsmayleadtoRome, butwewishtoarriveassoonaspossible. Theaimofthis
thesisistoprovidemeanstoefficientlycomputeshortestpathsinnetworks.
1.1 Networks and Graphs
Figure 1.1: The Ko¨nigsberg bridges as depicted by Leonhard Euler in his article “Solutio Prob-
lematis ad Geometriam Situs Pertinentis” on page 129 in volume 8 of Commentarii Academiae
ScientiarumPetropolitanaein1741.
YoumayhaveheardofthesevenKo¨nigsbergbridgesandthequestionastowhetheronecan,
in one walk, cross each bridge exactly once. Leonhard Euler resolved this question in 1735 by
proving that there is no such walk. His proof works as follows. Consider the island denoted by
the letter on the illustration in Figure 1.1. Euler made the following important observation:
A
although the island contains many buildings, streets, and paths, the relevant information is the
1
CHAPTER 1. INTRODUCTION
island’sconnections(thebridges)totheotherpartsofthecity. Thisobservationledhimtocreate
an abstract discrete structure, later termed a graph. He identified each landmass with a node
and each bridge with an edge connecting the two corresponding nodes in the graph. A walk in
Ko¨nigsberg corresponds to a walk in the graph; crossing a bridge is represented by traversing an
edge. Euler then noted that the number of edges adjacent to a node is essential. Except for the
endpointsofthewalk, allintermediatenodesmusthavean evennumberofadjacentedges, since
any walk must leave the node exactly once for every time entering it. Since all of the four nodes
haveanoddnumberofedges,therecannotbeanywalkthattraverseseachedgeexactlyonce.
Modeling the bridges by the edges of a graph helped Euler to solve the problem of the
Ko¨nigsberg bridges. Ever since, graphs have been used as an abstraction of structures in the
realworld:
Graphs are, of course, one of the prime objects of study in Discrete Mathemat-
ics. However, graphs are among the most ubiquitous models of both natural and
human-madestructures. Inthenaturalandsocialsciencestheymodelrelationsamong
species, societies, companies, etc. In computer science, they represent networks
of communication, data organization, computational devices as well as the flow of
computation, and more. In mathematics, Cayley graphs are useful in Group Theory.
Graphs carry a natural metric and are therefore useful in Geometry, and though they
are “just” one-dimensional complexes, they are useful in certain parts of Topology,
e.g. Knot Theory. In statistical physics, graphs can represent local connections be-
tweeninteractingpartsofasystem,aswellasthedynamicsofaphysicalprocesson
suchsystems. [HLW06]
In a graph representing a road network, intersections and streets can be modeled by nodes
and edges, respectively. Two nodes have an edge in between if there is a street connecting the
two corresponding intersections. For a computer network, routers and the connecting network
cables are mapped to nodes and edges, respectively. In a social network, the connections are not
physical. Individuals can be modeled by nodes; two nodes are connected by an edge whenever
the corresponding individuals are friends. In other social networks, an edge may also indicate a
privateorprofessionalrelationshipotherthanfriendship.
Different streets of a road network have different lengths. This is modeled by assigning each
edge a number, called its edge weight. This cost can reflect real-world values such as distance,
traveltime,transmissiontime,andlatency. Inagraphmodelingasocialnetwork,theedgeweight
can also reflect the quality of a friendship, although this is arguably difficult to capture appropri-
atelybyasinglenumber[XNR10].
In the remainder of this introduction, we first review various example structures, for which a
graph serves as a suitable model. We then consider one central problem that can be solved using
graphs: we investigate applications of the shortest path problem (Section 1.2), in particular con-
temporary applications of the point-to-point shortest path problem (Section 1.2.2). We conclude
thechapterbystatingthecontributionofthisthesis(Section1.3).
1.1.1 Transportation Networks
Transportation networks are an integral part of the infrastructure in many countries. For this
reason, the study of transportation networks is an important field of research. We give three
examplesoftransportationnetworks: road,railway,andairlinenetworks. Amorerealisticmodel
oftransportationwouldideallyintegratenetworksofallthreetypes[Fra08,DPW09,DPWZ09].
2