Alex Slivkins

133 Pages·2009·1 MB·English

EMBEDDING, DISTANCE ESTIMATION AND OBJECT LOCATION IN
NETWORKS
ADissertation
PresentedtotheFacultyoftheGraduateSchool
ofCornellUniversity
inPartialFulfillmentoftheRequirementsfortheDegreeof
DoctorofPhilosophy
by
AleksandrsSlivkins
August2006
(cid:176)c 2006AleksandrsSlivkins
ALLRIGHTSRESERVED
EMBEDDING,DISTANCEESTIMATIONANDOBJECTLOCATIONINNETWORKS
AleksandrsSlivkins,Ph.D.
CornellUniversity2006
Concurrentwithnumeroustheoreticalresultsonmetricembeddings,agrowingbodyofresearchinthenet-
workingcommunityhasstudiedthedistancematrixdefinedbynode-to-nodelatenciesintheInternet,result-
ing in a number of recent approaches that approximately embed this distance matrix into low-dimensional
Euclideanspace. Afundamentaldistinctionbetweenthetheoreticalapproachestoembeddingsandthisre-
centInternet-relatedworkisthatthelatteroperatesundertheadditionalconstraintthatitisonlyfeasibleto
measurealinearnumberofnodepairs,andtypicallyinahighlystructuredway. Indeed,themostcommon
frameworkhereisabeacon-basedapproach: onerandomlychoosesasmallnumberofnodes(’beacons’)in
thenetwork,andeachnodemeasuresitsdistancetothesebeaconsonly. Moreover,beacon-basedalgorithms
are also designed for the more basic problem of triangulation, in which one uses the triangle inequality to
inferthedistancesthathavenotbeenmeasured.
We give algorithms with provable performance guarantees for triangulation and embedding. We show
thatinadditiontomultiplicativeerrorinthedistances,performanceguaranteesforbeacon-basedalgorithms
typicallymustincludeanotionof”slack”–acertainfractionofalldistancesmaybearbitrarilydistorted.
For arbitrary metrics, we give a beacon-based embedding algorithm that achieves constant distortion
on a (1−(cid:178))-fraction of distances; this provides some theoretical justification for the success of the recent
networking algorithms, and forms an interesting contrast with lower bounds showing that it is not possible
to embed all distances with constant distortion. For doubling metrics (which have been proposed as a
reasonableabstractionofInternetlatencies),weshowthattriangulationwithaconstantnumberofbeacons
canachievemultiplicativeerror1+δ ona(1−(cid:178))-fractionofdistances,forarbitrarilysmallconstants(cid:178),δ.
We extend these results in a number of directions: embeddings with slack that work for all (cid:178) at once;
distributed algorithms for triangulation and embedding with low overhead on all participating nodes; dis-
tributed triangulation with guarantees for all node pairs; node-labeling problems for graphs and metrics;
systemsprojectonlocation-awarenodeselectioninalarge-scaledistributednetwork.
Biographical Sketch
AleksandrsSlivkinswasbornDecember1,1978inRiga,Latvia(thenSovietUnion). HelivedinRigauntil
he finished high school in August 1996. Then he went to California Institute of Technology; he graduated
in June 2000 with B.S. in Mathematics. From August 2000 till present Alex has been a graduate student
withtheComputerSciencedepartmentofCornellUniversity. Followinghiscandidacyexam,hereceiveda
M.S. in Computer Science in 2004. He expects to graduate with a Ph.D. in August 2006. After Cornell he
isgoingforaone-yearpostdocatBrownUniversity. HehasacceptedaresearchstaffpositionatMicrosoft
Research,SiliconValleyCenterstartingfromJuly2007.
iii
Acknowledgements
Firstofall,IwouldliketothankmythesisadvisorJonKleinbergforhisinvaluablesupportandmentorship
in both research- and career-related issues. I have benefited tremendously from interacting with my other
twothesiscommitteemembers,EvaTardosandEminGu¨nSirer. IadditionallythankJon,EvaandGunfor
theiradviceduringthejobsearchprocess.
Throughout my graduate career I have been fortunate to collaborate with many wonderful researchers.
Itismypleasuretoacknowledgemyfacultycoauthors(JonKleinbergandGunSirerfromCornell,Matthew
AndrewsfromBellLabs,ShukiBruckfromCaltech,andAnupamGuptafromCarnegieMellon),andthank
them for their guidance and collegiality. Furthermore, I would like to thank Shuki Bruck for being my
undergraduateresearchmentor.
I thank graduate students at Cornell Computer Science department for creating a fruitful academic en-
vironment. Iwilltrytoavoidlistingtoomanynameshere, butIwouldliketomentionElliotAnshelevich,
AnirbanDasgupta,AraHayrapetyan,MartinPal,MarkSandler,ZoyaSvitkina,andTomWexler,aswellas
severalolderstudents: AaronArcher,DavidKempe,TimRoughgarden,andChaitanyaSwamy.
Finally, I thank my parents and grandparents for supporting and motivating me throughout my entire
life.
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Table of Contents
1 Introduction 1
1.1 Overviewofresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Relatedwork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Definitionsandtheorems: embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Definitionsandtheorems: distributedalgorithms. . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Bibliographicnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Backgroundandpreliminaries 11
2.1 ExpandergraphsandProbability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Metricembeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Relationsbetweendifferent(cid:96) norms . . . . . . . . . . . . . . . . . . . . . . . . . 14
p
2.2.2 Embeddingsoffinitemetricsinto(cid:96) spaces . . . . . . . . . . . . . . . . . . . . . . 14
p
2.2.3 Embeddingsintotreemetrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3 Lowdimensionalityinmetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Growth-constrainedmetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Doublingmetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.3 Decomposablemetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 TriangulationandEmbeddingusingSmallSetsofBeacons 20
3.1 Beacon-basedtriangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Beacon-basedembeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Beacon-basedapproaches: furtherresults . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.1 Black-boxGNP-styleembedding . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.2 Strongtriangulationwithaconstantnumberofbeacons . . . . . . . . . . . . . . . . 29
3.3.3 Infinitemetricsandarbitrarymeasures . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Fullydistributedapproaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 Improvedembeddingsforgrowth-constrainedmetrics . . . . . . . . . . . . . . . . . . . . . 35
3.6 Lowerboundsonembeddingswithslack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6.1 Generallower-boundingtechnique . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6.2 Lowerboundsforcontractingembeddings . . . . . . . . . . . . . . . . . . . . . . . 42
4 GracefullyDegradingDistortionforDecomposableMetrics 44
4.1 Distancescalesandscalebundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Theembeddingalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Analysis: proofofLemma4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
v
4.5 Analysis: toolsfromProbability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.6 Analysis: mapsf andg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
ij (i,j,0)
4.7 ABourgain-styleproofofLemma4.2fordoublingmetrics. . . . . . . . . . . . . . . . . . . 52
4.8 Anextensiontoarbitrarymetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 NetworkTriangulationviaRingsofNeighbors 55
5.1 Frameworkandresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Tools: distributedrandomwalks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 RandomizedRingsofNeighbors: ProofofTheorem5.3 . . . . . . . . . . . . . . . . . . . . 60
5.4 NetworkTriangulation: ProofofTheorem5.5 . . . . . . . . . . . . . . . . . . . . . . . . . 62
6 Location-awarenodeselectionviaRingsofNeighbors 66
6.1 Meridian: aframeworkforlocation-awarenodeselection . . . . . . . . . . . . . . . . . . . 66
6.2 Analysisofscalability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.2.1 FormaldescriptionoftheMeridianframework . . . . . . . . . . . . . . . . . . . . 73
6.2.2 QualityoftheMeridianrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.2.3 Nearestneighborsandcentralleaders . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2.4 Extensions: exactnearestneighbors . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2.5 Extensions: load-balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2.6 Fine-tunedversionsoftheresults . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3 FullproofofTheorem6.6oncentralleaderelection . . . . . . . . . . . . . . . . . . . . . . 79
6.4 FullproofofTheorem6.7onexactnearestneighbors . . . . . . . . . . . . . . . . . . . . . 81
6.5 FullproofofTheorem6.9onload-balancing . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.5.1 Setup: Meridianringsandthesearchalgorithm . . . . . . . . . . . . . . . . . . . . 83
6.5.2 Setup: randomizationandrandomvariables . . . . . . . . . . . . . . . . . . . . . . 84
6.5.3 Theactualproof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7 DistanceEstimationandObjectLocationviaRingsofNeighbors 88
7.1 Thefourproblemsandrelevantbackground . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.2 Alow-stretchroutingschemefordoublingmetrics . . . . . . . . . . . . . . . . . . . . . . 94
7.3 Triangulationanddistancelabelingschemes . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.4 Low-stretchroutingschemes,revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.4.1 Routingschemesonmetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.5 Searchablesmall-worldnetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.5.1 FullproofofTheorem7.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7.5.2 ComparisonwithKleinberg’ssmallworlds . . . . . . . . . . . . . . . . . . . . . . 106
7.5.3 Comparisonwiththesingle-link-per-nodemodel . . . . . . . . . . . . . . . . . . . 107
7.6 FullproofofTheorem7.12onroutingschemes . . . . . . . . . . . . . . . . . . . . . . . . 108
8 Conclusionsandfurtherdirections 114
Bibliography 117
vi
List of Tables
1.1 Lowerboundsforembeddingswithslack . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1 Lowerboundsforembeddingswithslack . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7.1 Low-stretchroutingschemesfordoublinggraphs . . . . . . . . . . . . . . . . . . . . . . . 90
7.2 Low-stretchroutingschemesfordoublingmetrics . . . . . . . . . . . . . . . . . . . . . . 102
7.3 Theorem7.12: spacerequirements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
vii
List of Figures
3.1 Triangulationindoublingmetrics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Perfecttriangulationfordensepointsets . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.1 Meridian: multi-resolutionrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2 Meridian: closestnodediscovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.3 Meridian: multi-constraintqueries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.4 Meridian:progressateachhop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.5 Meridian:trade-offbetweenperformanceandaccuracy . . . . . . . . . . . . . . . . . . . . 76
6.6 Meridian:thein-degreeratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.1 Interconnectionsbetweenvariousresultsinthischapter . . . . . . . . . . . . . . . . . . . 93
7.2 ProofofTheorem7.1: Aroutingschemeondoublingmetrics . . . . . . . . . . . . . . . . 96
viii
Chapter 1
Introduction
Thepastdecadehasseenmanysignificantandelegantresultsinthetheoryofmetricembeddings(forrecent
surveys, see [Ind01, Lin02, Mat02a, IM04]). Embedding techniques have been valuable in the design and
analysis of algorithms that operate on an underlying metric; many optimization problems become more
tractablewhenthegivenmetricisembeddedintoonethatisstructurallysimpler.
Meanwhile,anactivelineofresearchinthenetworkingcommunityhasstudiedthedistancematrixde-
finedbynode-to-nodelatenciesintheInternet [FJJ+01,GSG02,GS95,HFP+02,KSB01,VPSV02],result-
ing in a number of recent approaches that approximately embed this distance matrix into low-dimensional
Euclideanspace [DCKM04,NZ02,PCW+03,ST03].1 However,thereisafundamentaldistinctionbetween
thisInternet-relatedworkandthelargebodyoftheoreticalworkonembedding,duetothefollowingintrin-
sic problem: in any analysis of the distance matrix of the Internet, most distances are not available. The
costofmeasuringallnode-to-nodedistancesissimplytooexpensive;instead,wehaveasettingwhereitis
generally feasible to measure the distances among only a linear (or near-linear) number of node pairs, and
typicallyinahighlystructuredway. Indeed,themostcommonframeworkforInternetmeasurementsofthis
typeisabeacon-basedapproach: onechoosesuniformlyatrandomaconstantnumberofnodes(‘beacons’)
inthenetwork,eachnodemeasuresitsdistancetoallbeacons, andonethenhasaccesstoonlytheseO(n)
measurementsfortheremainderofthealgorithm. (Forexample,thedatacanbesharedamongthebeacons,
whothenperformcomputationsonthedatalocally.)
This inability to measure most distances is the inherent obstacle that stands in the way of applying
algorithms developed from the theory of metric embeddings, which assume (and use) access to the full
distance matrix. Thus, to obtain insight at a theoretical level into recent Internet measurement studies, we
needtoconsiderproblemsinfollowingtwogenres.
(i) What performance guarantees can be achieved by metric embedding algorithms when only a sparse
(beacon-based)subsetofthedistancescanbemeasured?
(ii) At an even more fundamental level, many Internet measurement algorithms are seeking not to em-
bed but simply to reconstruct the unobserved distances with reasonable accuracy (see e.g. [FJJ+01,
GSG02,GS95,KSB01]). Canwegiveprovableguaranteesforthistypeofreconstructiontask?
Reconstruction via triangulation. Within this framework, we discuss the reconstruction problem (ii)
first, as it is a more basic concern. Motivated by the research of Francis et al. on IDMaps [FJJ+01], and
1WespeakofInternetlatenciesasdefiningasa“distancematrix”ratherthanametric,sincethetriangleinequalityisnotalways
observed; however, one can view the recent networking research as indicating that severe triangle inequality violations are not
widespreadenoughtopreventthematrixofnode-to-nodelatenciesfrombeingusefullymodeledusingnotionsfrommetricspaces.
1