302nd Airlift Wing, U.S. Air Force Reserve Command Peterson Air

12 Pages·2006·0.31 MB·English

Adaptive Location Constraint Processing
Zhengdao Xu and Hans-Arno Jacobsen
UniversityofToronto
[email protected], [email protected]
ABSTRACT andpreviouslyunthinkablepotentialsfortracking,correlating,and
filtering of informationabout moving objects. Forexample, it is
Animportantproblemformanylocation-basedapplicationsisthe
possibletotrackthelocationofmobileusersinawirelessnetwork,
continuous evaluation of proximity relations among moving ob-
evenacrosswirelesscoverageareas,andinbuildingsacrosscam-
jects. These relations express whether a given set of objects is
pus[28,19],totrackthediscretelocationofvehiclesorpackages
in a spatial constellation or in a spatial constellation relative to
indelivery[1],andeventrackthemovementoflivestockorfishfor
a given point of demarcation in the environment. We represent
environmentalpurposes[2].
proximityrelationsaslocationconstraints,whichresemblestand-
Animportantprobleminthecontextofapplicationsthatleverage
ingqueriesovercontinuouslychanginglocationpositioninforma-
thistrackingpotentialishowtoefficientlydeterminewhetherfor
tion.Thechallengeliesinthecontinuousprocessingoflargenum-
anygivennumberofsetsofmovingobjects,theobjectspersetare
bersoflocationconstraintsasthelocationofobjectsandthecon-
closeto,nocloserthan,orfurther-away-thanaspecifieddistance
straint load change. In this paper, we propose an adaptive loca-
toeachotherortoagivenpointofdemarcationintheenvironment.
tion constraint indexing approach which adapts as the constraint
Werefertothisproblemasthelocationconstraintmatchingprob-
load and movement pattern of the objects change. The approach
lem.Agivensetofobjectsspecifyingaproximityrelationthatmust
takes correlations between constraints into account to further re-
beenforcedisreferredtoasalocationconstraint. Therearemany
duce processing time. We also introduce a new location update
applicationsthatrequireasolutiontothisproblem. Thechallenge
policy that detects constraint matcheswith fewer location update
istodevicealgorithmsthatcanefficientlymonitorlargenumbers
requests. Ourapproachstabilizessystemperformance,avoidsos-
oflocationconstraintsoverlargernumbersofcontinuouslymoving
cillation,reducesconstraintmatchingtimeby70%forin-memory
objects.
processing, and reduces secondary storage accesses by 80% for
Inacellularnetwork,forexample,anoperatormaywanttoof-
I/O-incurringenvironments.
feralertingservicesthatnotifymembersofagroup(e.g.,agroup
CategoriesandSubjectDescriptors
of friends or family), if they are close to each other (e.g., friend
H.[InformationSystems];H.2.8[DatabaseApplications]: Spa- finderandbuddytrackingapplications)orareclosetoadesignated
tialdatabasesandGIS;H.3.3[InformationSearchandRetrieval]: pointintheenvironment(e.g.,theCNTowerinTorontooraroad
Information Filtering; H.4[Information SystemsApplications]: congestiononHighway407.) Inmetropolitanareasthenumberof
Location-basedServices objectstrackedcaneasilyreachhundredsofthousands,specifying
GeneralTerms asimilarnumberoflocationconstraints.
Otherexamplesdrawfromprotectingsecuritysensitiveareasin
Algorithms,Design,Experimentation,Performance citiesbyspecifyingthatnoairplaneistocomecloserthanaspec-
Keywords ifieddistancefromdesignatedpoints,suchasnuclearplants,tow-
ers,stadiumsetcetera.Airtrafficcontrollersandpilotsmaybenefit
Location-basedServices,LocationConstraintProcessing,Moving
fromalertingservicesthatwarnifdesignatedplanescometooclose
ObjectIndexing,AdaptiveIndexing,LocationUpdatePolicy,Lo-
toeachother.Similarly,majorproductionplantsmayspecifypoli-
cation Query, Standing Query, Continuous Location Query, and
ciesthatrestrictpersonnelfromenteringrestrictedareasorenforce
ConstraintMatching.
constraintsthataspecificareamustbeguardedbyasetnumberof
personnel. Again, these scenarios may involve large numbers of
1. INTRODUCTION continuously moving objects with many associated location con-
Thepervasivepresenceofwirelessnetworkscombinedwithad- straints. Altogether different applications can be found in multi-
vances in location positioning technology [28] gives rise to new playeronlinegames, wherefriendorfoemaywanttosetupcon-
straintstonotifyandwarnofeachothers’presenceandproximity
inthegameworldorpartsofthevirtualspace.Multi-playeronline
gamesareplayedby thousandsof playersatthe sametime. The
Permissiontomake digital orhardcopies ofallorpartofthisworkfor efficienttrackingofconstraintsamongplayersandamongplayers
personalorclassroomuseisgrantedwithoutfeeprovidedthatcopiesare andthegameworldisofcriticalimportancetothesuccessofthe
notmadeordistributedforprofitorcommercialadvantageandthatcopies game.
bearthisnoticeandthefullcitationonthefirstpage.Tocopyotherwise,to Tomodelthesescenariosandtoexpressvarioustypesofprox-
republish,topostonserversortoredistributetolists,requirespriorspecific
imity relations, we introduce two classes of location constraints,
permissionand/orafee.
the n-body constraint and the n-body staticconstraint. Werefer
SIGMOD’07,June11–14,2007,Beijing,China.
Copyright2007ACM978-1-59593-686-8/07/0006...$5.00.
to these constraints as location constraints, as they are evaluated itors whether a specific set of objects are in a given spatial con-
overthelocationdataoftheassociatedobjects.Theconstraintsare stellation to each other or to a given point of demarcation in the
definedasfollows: environment.
Then-bodyconstraintisoftheform p t;p t;:::;p t <(or> Inthispaperweproposeanadaptivespacepartitioningscheme
1 2 n
j j
)d,andalsoabbreviatedasP <(or>)dforobjectsetP(= p ;p ; andalgorithmstosolvethelocationconstraintmatchingproblem.
1 2
f
:::;p ). Theconstraintwithoperator<issatisfiedifthenmov- Weexperimentwithtwospacepartitioningschemes,oneisatree-
n
g
ingobjects,identifiedby,p ;p ;:::;p ,canbeenclosedbyacircle basedorganizationofthespaceandtheotheroneagridfile-based
1 2 n
withadiametersmallerthandattimet.Theconstraintwithopera- organization.Oursolutionsupportsthecontinuoustrackingoflarge
tor>issatisfiedifthediameterofthesmallestcircleenclosingthe numbers of location constraints for large populations of moving
objectsisgreaterthandattimet. p (1 i n)istheidentifier objects. Ouralgorithmsaredesignedtoadapttochangesinloca-
i
ofobjecti. Inournotationp t isinterpr(cid:20)eted(cid:20)asthecoordinateof tionconstraintloadsandchangesinmovementpatternsofobjects.
i
objectiattimet.disreferredtoasthealertingdistance. In particular, our solution adapts to clusters of objects changing
Then-bodystaticconstraintisoftheform A;p t;p t;:::;p t overtimeinsize,skewnessandtheformationofnewclusters.Fur-
1 2 n
j j
< (or >)d, alsoabbreviated as A;P < (or >)dforobject thermore,oursolutionconstitutesaframeworkcustomizablefora
j j
setP (= p ;p ;:::;p ). Aisthecoordinateofsomestaticpoint. wide-rangeofquerytypesexpressingdifferentgeometricrelations
1 2 n
f g
The constraint is satisfied if the n moving objects, identified by, among moving objects. We demonstrate and evaluate alternative
p ;p ;:::;p ,arewithin(foroperator<)oroutside(foroperator query types such as the continuous reverse range query and the
1 2 n
>)thecircledefinedbydaroundthestaticpointAattimet. continuous reverse k nearest neighbor query. We also introduce
Theaboveconstraintdefinitionsdonotinclude=, and ,as thenotionofconflictingandharmoniousconstraintstocapturede-
(cid:20) (cid:21)
theresultingconstraintscanbeeasilyexpressedwiththenegation pendencyrelationsamongconstraintsthathelpinconstraintprun-
oftheconstraintitself.Forexample,P dissimply (P >d) ing.Wedevelopadetailedanalyticalmodeltoassesstheconstraint
ci (cid:20) : ci
andP = disequivalent to (P > d) (P < d),where evaluationcostforin-memoryandI/O-incurringenvironments.Fi-
ci : ci ^: ci
P denotesthemovingobjectsassociatedwithconstraintc . This nally, weexperimentallyvalidatetheefficiencyofouralgorithms
ci i
simplifiesthedesignoftheconstraintmatcher. based on various movement patterns, compare to alternative ap-
Thelocationconstraintmatchingproblemcanthenbeformally proaches,anddemonstratescalability.
statedasfollows:Givenasetoflocationconstraints,C = c ;c ; InSection2,wedescribetheadaptivelocationconstraintmatch-
1 2
f
:::;c , which designate the location relationshipamong a set of ing algorithm. In Section 3, we develop the analytical model to
k
g
m possibly moving objects, P = p ;p ;:::;p , continuously assesstheconstraintevaluationcost,determinesystemparameters,
1 2 m
f g
determineallconstraintsc inCthataresatisfied. andestimatesecondarystorageaccesscost. Thesystemarchitec-
i
Location constraints arelikecontinuousqueries thatoncesub- tureofafullyimplementedlocationconstraintprocessingsystemis
mittedtothesystemremainactiveuntilexplicitlyrevoked. Fig.1 reviewedinSection4.Section5presentstheexperimentalevalua-
showsthedataflowofatypicallocation-basedserviceemploying tionofouralgorithms.InSection6,weputourworkinperspective
location constraint matching, as defined in this paper. Location torelatedapproaches.
updates of mobile objects are streamed into the system and trig- 2. MATCHINGALGORITHM
ger the evaluation of constraints and the constraint satisfaction is
Oursolutiontothelocationconstraintmatchingproblemisbased
communicatedbacktotheinterestedsubscribersvianotifications.
onspacepartitioning. Theintuitionisthatspacepartitioningcan
Existing data management and indexing techniques for moving
approximatethelocationofthemovingobjects,withouttheneed
fortheexactlocationofobjectsallthetime,toresolvemanycon-
3-body constraint (matched)(cid:13)
straintswithcertainty. Inthissectionwedescribeourapproachin
<d(cid:13) Location Constraint(cid:13) detail.
Location Updates(cid:13) Matcher(cid:13)
<obj_id, x, y>(cid:13) 2.1 Background &BasicAlgorithms
constraint 1(cid:13)
constraint 2(cid:13)
static point(cid:13) Location based(cid:13) constraint 3(cid:13) Asolutiontothelocationconstraintmatchingprobleminprior
Comm. Tower(cid:13) Applications(cid:13) ...(cid:13) work[23]dividesthewholespaceintopartitionsthatareindexed
with a k-d-tree [5] or a multi-layered grid file structure [13]. To
2-<bdod(cid:13)y static constraint (matched)(cid:13) NoStiefircvaictieo(cid:13)nWo/r(cid:13)kstation(cid:13) MRaetscuhlitnsg(cid:13)(cid:13) cavhoanidgethpeaorvtietirohne,aadboafcakttorapc-kdionwgnaslgcoanritihnmthiesku-sde-dtretoeqwuhiceknloybajsescots-
ciate theobject with the new partition. In amulti-layer gridfile,
Figure1:DataFlowforLocationConstraintProcessing thespaceispartitionedwithdifferentlayersofgrids, thegridsin
thesamelevelareequal-sizedcellswhicharefurtherindexedwith
objects[11,3,16,17,14]aremainlyfocusingontheefficientpro- hashtables.
cessing of a single spatio-temporal query, such as a range query WeassumethatthepositionofeachobjectisretrievedwithGPS,
and the k nearest neighbor query (kNN). However, the location ground-basedsensorsorotherlocationpositioningtechnology. We
constraintmatchingproblemdealswiththeconcurrentandcontin- calleachpositionretrievalalocationupdateandwecallthemove-
uousevaluationofmanyconstraintsatthesametime(i.e.,hundreds mentofanobjectfromonepartitiontoanotherapartitionupdate.
of thousand of location constraints for the applications we advo- Constraintevaluation consistsoftwoparts,theconstrainteval-
cate.) Due to the continuous and frequent movement of objects, uationbasedonexactpositioninformationandthepartition-based
anunderlyingdatabasewouldbemostlybusyupdatingtheobjects’ constraint evaluation given partition information. Fortheevalua-
locationwithoutbeingabletoevaluatetheconstraints. Moreover, tionbasedontheexactposition,Welzl’salgorithm[20]isadopted,
thelocationconstraintmatchingproblemisdifferentfromthekNN whichcomputesthesmallestcirclethatenclosesnpointsinO(n)
problem[10,27]inthatthekNNproblemdeterminestheknear- time. Below, weintroducepartition-based evaluation thatis trig-
est object(s) among all the objects in the space to a given point, gered when an object updates its partition or when the partition
whilethelocationconstraintmatchingproblemcontinuouslymon- schemeisadjustedbytheadaptiveadjustmentalgorithm. Thepar-
titioninformationisusedasaroughapproximationfortheposition Sri(cid:13)oj(cid:13):Pwa (cid:13)jr(cid:13)tthit icoonl uinm (cid:13)ni(cid:13)th(cid:13)(cid:13) O(cid:13)1(cid:13)((cid:13)d=81 / 2)(cid:13)(cid:13)
ofthemovingobjectsandformanyconstraints,thisapproximation j(cid:13)i(cid:13) 1(cid:13) 2(cid:13) 3(cid:13) 4(cid:13) 5(cid:13) 6(cid:13) n-(cid:13)body constraint:(cid:13)
isePnaorutgithiotno-dBeatseermdiEnevaalureastuioltn. for n-body Constraints: For n- 1(cid:13) p1(cid:13)(cid:13) O(cid:13)5(cid:13)((cid:13)r=1.5)(cid:13)A(cid:13) (c(cid:13)s1(cid:13)(cid:13):a |tpis1 f(cid:13),ipe 2d,(cid:13)p(cid:13)) 3(cid:13)|(cid:13) < 3, c(cid:13)2:(cid:13) |p2 (cid:13),p 3,(cid:13)p5 (cid:13)| > 3(cid:13)
p(cid:13)
bothonedtychoecnopsnatrsrattriitaniiotnsnt,iitinfifssooarmmsseaotocioibanjte.ecdStupwpiiptmhoosnveeeteshdainttotcojbaeisnretewh-eepvcaaorlutniastittoreandinSbtiat,shaealdlt 23(cid:13)(cid:13) p2(cid:13)(cid:13) 3(cid:13) p4(cid:13)(cid:13) p(cid:13) (c(c(cid:13)(cid:13)uu53(cid:13)(cid:13)(cid:13)(cid:13)::nn ||csppea2 5(cid:13),,(cid:13)rtppitsa 36f,,(cid:13)(cid:13)iippne 5(cid:13)7)d||(cid:13)(cid:13)(cid:13) (cid:13)<<)(cid:13) 33,, cc(cid:13)(cid:13)64::(cid:13)(cid:13) ||pp15 (cid:13)(cid:13),,pp2 6(cid:13),,(cid:13)pp3 7(cid:13)||(cid:13) >> 33(cid:13)(cid:13)
p isassociatedwithandP = p ;p ;:::;p isthesetofbodies 7(cid:13) n-(cid:13)body static constraint:(cid:13)
inivolved incj (pi2Pcj). Acjfterfthe1pa2rtitionnsgthatcontainobjects 4(cid:13) O(cid:13)2(cid:13)((cid:13)d=101 / 2)(cid:13)(cid:13) p5(cid:13)(cid:13) (c(cid:13)s7(cid:13)(cid:13):a |tAis,f ipe 4d|(cid:13) (cid:13))<(cid:13) 1.5, c(cid:13)8(cid:13): |A, p5 (cid:13)| > 1.5(cid:13)
itcbniootimuoPnnpcdsjutoaeanfdrde.thitTedhheesenisztmsiefiizaeoeldfle,otshftthetcehaiercsscmtelueaatllwilnecotsiertcrcsiclreeicrclcttehilnseagtaerenaenlclcltohltohsesieneusgsepptaphelaelrrtatoihntbeidojsenelcsotpswaaeirrne-r 5(cid:13) M1(cid:13) (cid:13)u(cid:13)n(cid:13)i(cid:13)t(cid:13)obile poinOt(cid:13)(cid:13)3(cid:13)((cid:13)d=S1ta0ti1 c/ 2 )(cid:13)p(cid:13)Ooi(cid:13)4n(cid:13)(td(cid:13) =1)(cid:13) p6(cid:13)(cid:13) cc(((cid:13)(cid:13)uu19(cid:13)(cid:13)1(cid:13):nn(cid:13): |scA|Aae,tr, i ptspa5 f7(cid:13) |iin(cid:13) |e <)<d(cid:13) 1(cid:13))1(cid:13)..55,, cc(cid:13)1(cid:13)01(cid:13):2 (cid:13):| A|A, ,p p4 (cid:13)|7 (cid:13)|> > 1 1.5.5(cid:13)(cid:13)
P .Basedonthisupperandlowerbound,theresultofmanycon-
stcrajintcanbedetermined. Figure2:IllustrationofPartition-basedEvaluation
For instance, the Partition Based NB Algorithm for the con-
straints with the ”<” operator (p t;p t;:::;p t < d) is shown
1 2 n
below. If the diameter of the sjmallest intersectjing circle (lower unsatisfied(Line4); ifallobjectsareintheinternalpartition,the
bound)islargerthand(checkedbyfunctionIntersectinLine3), constraintissatisfied(Line6);iftheconstraintdoesnotbelongto
c isunsatisfied.Ifthesmallestenclosingcircle(upperbound)has theabovetwocases,theconstraintisuncertain(Line8). Theso-
j
adiametersmallerthand(checkedbyfunctionEncloseinLine lutiontotheconstraintwiththe”>”operatorcanbederivedina
5),c issatisfied. Ifdisbetweenthislowerandupperbound,the similarfashionduetoitssymmetrywith”<”.
j InFig.2,the1-bodystaticconstraintc andc aresatisfiedbe-
constraint is uncertain (Line 8). Due to the symmetry, the con- 7 8
causew.r.t.circleO ,partitionS (containsp )isaninternalpar-
straint p1t;p2t;:::;pnt > dcanbesolvedbyexchangingLine4 titionandS (cont5ainsp )isan25externalpart4ition. Similarly,we
j j 45 5
withLine6. canverifiedthatc andc areunsatisfied. However, constraints
9 10
The result of an evaluation is valid as long as the objects re- c andc areuncertainbecauseS (containsp )isabounding
11 12 36 7
mainintheirpartitionsandthepartitionschemedoesnotchange. partitionsw.r.t.O .
5
Fig.2illustratespartition-basedevaluation. The3-bodyconstraint
Algorithm Partition Based NBS(MobileObjectp)
c (p ;p ;p <3,rightsideofthefigure)issatisfiedbecausethe
d1iamj e1ter2of3thjecircleO whichenclosespartitionsS , S and 1. foreachConstraintcthatpisassociated
1 11 21 2. letP = p ;p ;:::;p bethesetofbodiesinc;
Sun2s2a1tis(cfioendtabiencianugspe1t,hpe2dainadmpet3e)riosfpth8e(c<irc3l)e.OTh2ewchoincshtrianitnetrsce3citss 34.. if(9cp:ir2efsPu1l;tp=i22unesxattenirsgnfiaedl;partition)
S21, S22 and S45 is p10 (> 3). However, the constraint c5 is 5. else if( pi P;pi internalpartition)
uncertainbecausethediameterofthecircleO3enclosingS36,S45 6. 8c:r2esult=2satisfied;
andS isp10andthediameterofthecircleO intersectingthese 7. else //inboundingpartition
56 4
partitionsis1,butthealertingdistance3isbetweenthelowerand 8. c:result=uncertain;
upperbound(p10 > 3 > 1). Furthercomputationbasedonthe
precise object position reveals that c5 is actually satisfied (circle ConstraintEvaluationAlgorithm:Alocationupdatetriggerscon-
notshownforclarity). Butthiscannotbeestablishedwithparti-
straintevaluation(i.e.,Evaluation).Wedistinguishthecaseswhere
tioninformationalone. Similarly,onecanverifytheevaluationof
thelocationchangeleadstoapartitionupdateandwherenosuch
constraintsc ,c andc .
2 4 6
partitionupdateisincurred.
Algorithm Partition Based NB(MobileObjectp) Forapartitionupdate,allconstraintsanobjectisassociatedwith
1. foreachConstraintcthatpisassociatedwith havetobeevaluatedusingthepartition-basedevaluationfunction
2. letP = p1;p2;:::;pn bethesetofbodiesinc; (Lines4,5)andtheuncertainconstraintsinvolvingthatobjecthave
3. if(Intefrsect((cid:0) ni=1gpi:current partition)>c:d)) tobeexplicitlyevaluated basedontheexactlocation positionin-
4. c:result=unsatisfied; formationafterwards(withMatch Uncertain Constraints
5. else if(Enclose((cid:0) ni=1pi:current partition)<c:d) in Line 6). The number of uncertain constraints (expressed as
6. c:result=satisfied; Uparti tionid) in the previous and current partitions are updated
7. else afterwards(Line7).Ontheotherhand,ifthelocationupdatedoes
8. c:result=uncertain; notincurapartitionupdate,onlyuncertainconstraintsneedtobe
evaluated(Line9).Partition-basedevaluationisinvokedeverytime
Partition-BasedEvaluationforn-bodyStaticConstraints:The anobjectchangesapartitionorwhenthepartitionisadjusted,but
n-body static constraint, A;p1t;p2t;:::;pnt < d, is matchedif notforeverylocationupdate.
j j
and onlyifallpi (i 1::n) areinthecircleO withstaticpointA Algorithm Evaluation(MobileObjectp)
2
as center and das radius. Dependingon whether the partition is 1. p:previous partition=p:current partition;
inside,intersecting,oroutsidetheboundaryofO,weidentifythe 2. p:current partition=Find New Partition(p);
internal,bounding,andexternalpartitionsw.r.t. O(e.g.,theparti- 3. ifp:previous partition=p:current partition
tioncompletelyinsideOisdefinedasaninternalpartitionandso 4. Partition Based6 NB(p);
5. Partition Based NBS(p);
on.)ThedistancebetweenthemovingobjectandthestaticpointA
6. Match Uncertain Constraints(p);
canbetrackedaccordingtothetypeofthepartitiontheobjectisin
7. updateU andU ;
p:currentpartition p:previouspartition
(i.e.,”internal”,”external”,or”bounding”). 8. else *ifp:previous partition=p:current partition*
Thepartition-basedevaluation(Partition Based NBS)forcon- 9. Match Uncertain Constraints(p);
straintswiththe”<”operatorworksasfollows: ifsomeobjectin
2.2 Adaptive SpacePartitioning
P = p ;p ;:::;p isintheexternalpartition, theconstraintis
1 2 n
f g
Thestaticspacepartitioningdescribedabovesuffersinenviron-
1S representsthepartitionintheithrowandjthcolumn mentswhereboththeconstraintsandthemovementpatternsofthe
ij
objects are continuously changing. For many applications, con- wheretheyleadtothegreatestreductioninthenumberofuncertain
straints are continuously updated based on changing user prefer- constraints(recordedbyset(cid:0)U inLine4)andthesmallestincrease
ences(i.e.,insertsanddeletes.)Thus,apartitionschemeoptimized in partition update rate (recorded in (cid:0)PUR in Line 5). Only the
tentativesplittinglinesinsidepartitions(cid:0) (cid:0) areconfirmed
foronesetofconstraintsandmovementpatternmaynotbeopti- (Line6).ThefinalnumberofpartitionsspUlit\depPeUnRdsonthesizeof
malforanothersetofconstraintsandadifferentmovementpattern.
(cid:0) and(cid:0) thatarecontrolledbythewindowsizew,whichis
U PUR
Also,partitionupdatesincuroverheadwhenalargenumberofob- themainindicatoroftheadaptationspeedandisdirectlycoupled
jects are moving between partitions, which is to be expected for with, and therefore adjustedaccording to, the linear combination
applications with varying movement patterns. These characteris- oftheaveragespeedofmovementandtheconstraintupdaterate,
tics arethe motivations to develop an adaptive spacepartitioning a1v+a2r.Thereductioninthenumberofuncertainconstraintsis
computedinLine3.Fig.3illustratesthesplitadjustment.
schemethatevolveswithchange.
The AKDT and AMLG Index: We propose the adaptive k- Algorithm Split Adjust(intw)
1. foreachcandidateleafnodeS
d-tree (AKDT) index and the adaptive multi-layer grid (AMLG) 0 i
2. S =S splitbytentativesplittinglines;
indextoaccommodatethissituation. AKDTisfundamentallyak- i i
0
d-tree,exceptthatitpartitionsthespaceratherthantheobjectsin 34.. (cid:0) =4tUopSiw=praerdtiuticotniosnwoifthunhcigehrteasitncoUnstr;aintfr.SitoSi ;
the space. In AKDT, eachleafnode representsa single partition 5. (cid:0)U =topwpartitionswithlowe4stPSUiR ;
inthespace. Eachinternalnodestoresinformationaboutasplit- PUR 0 Si
6. replaceS withS ifS (cid:0) (cid:0) ;
tinglineandrepresentstheunionofpartitionsofthe(leaf)nodes i i i U PUR
2 \
underneathit. Similarly, inAMLG, eachgrid atthelowestlayer
presentsasinglepartitioninspaceandthehigherlayergridstores
top w(cid:13)
the informationof thegrid layout and representsthe union ofall Partition i(cid:13)1(cid:13)Partition i(cid:13)2(cid:13)Partition i(cid:13)3(cid:13)
partitionsunderneaththislayer. AKDTandAMLGadaptivelyset U(cid:13)i1(cid:13)(cid:13) U(cid:13)i2(cid:13)(cid:13) U(cid:13)i3(cid:13)(cid:13) >(cid:13)
andrelinquishthesplittinglines. Candidates(cid:13)
(sorted)(cid:13)
The adaptive space partition algorithm consists of two stages, Partition j(cid:13)1(cid:13)Partition j(cid:13)2(cid:13) Partition j(cid:13)3(cid:13)
namely,theinitialpartitioningstageandtheadjustmentstage.The PUR(cid:13)j1(cid:13)(cid:13) PUR(cid:13)j2(cid:13)(cid:13) PUR(cid:13)j3(cid:13)(cid:13) <(cid:13)
initialpartitioningsimplydividesthewholespaceintosmallcells
top w(cid:13)
with arbitrary granularity. The adjustment stage tunes the parti-
Partitions for Adjustment(cid:13)
tionschemewithanadjustmentfrequencyproportionaltothelin-
ear combination of the average velocity over all moving objects
Figure3:SplitAdjustment
tracked,v,andtheaveragerateofconstraintupdates,r,i.e.,a v+
1
a r,forconstantsa anda . Theaveragevelocityisderivedfrom
2 1 2 TheMerge AdjustAlgorithmperformsaninverseoperationto
the location updates recorded over time. In this way, the parti-
thesplitalgorithm.Itremovesthecurrentsplittinglinesandmerges
tionschemeevolves asthemovement patternand constraintload theleafnodepartitions. Themergealgorithmaimsatreducingthe
change. We formally show with Lemma 2 in Section 3 that the partitionupdaterateatthecostofaminorincreaseinthenumber
splitting of a partition can reduce the number of uncertain con- ofuncertainconstraints.
straintsandcanincreasethepartitionupdaterate,whilethemerg- Algorithm Merge Adjust(intw)
ing of partitions can reduce the partition update rate and can in- 1. foreachcandidatesecondlevelnodeS
i
0
crease the numberof uncertain constraints. The objective is to 2. S =S withsplittinglinestentativelyremoved;
i i
balancethetrade-offbetweensplittingandmergingsuchthatboth 3. U =increaseofuncertainconstraintfr.S toS 0;
the number of uncertain constraints and the partition update rate 4. (cid:0) =4topSiwpartitionswithlowest U ; i i
decreasetothegreatestextentpossible. 5. (cid:0)UPUR=topwpar0titionswithhigh4estPSiURSi;
Fortheefficientadjustment,eachpartitionSi (iisthepartition 6. replaceSiwithSi ifSi (cid:0)U (cid:0)PUR;
2 \
ID)maintainsthefollowingadditionalinformation:1.PUR ,the
Si OverheadReductionforSplit&MergeCountingthechange
estimatedpartitionupdaterateinsideS .Itequals(cid:26) v L ,
i Si(cid:3) Si(cid:3) Si inthenumberofuncertainconstraintsbyrecomputingallthecon-
where(cid:26)Si,vSi andLSi aretheobjectdensity,theaverageveloc- straints associated with the partition (Line 3 in Split Adjust and
ityandthetotallengthofthesplittinglinesinsideSi,respectively. Merge Adjust)istoocostly.Lemma2inSection3showsthatthe
2. U : Thenumberofuncertainconstraints associatedwith the
Si uncertain constraints after the split are a subset of the uncertain
objectsinpartitionS . Theadjustmentprocessiscomposedofthe
i constraints before splitting, therefore, only the original uncertain
splitSplit AdjustAlgorithmandthemergeMerge AdjustAlgo-
constraintsareconsideredwhencomputingthenumberofuncertain
rithm. The split algorithm creates new smaller partitions from a
constraints after splitting. A similar property holds for merging.
leaf node partition by adding tentative splitting lines(Line 2) in-
Thisreducestheoverheadofthesplitandmergeprocess,however,
sidethecandidateleafnode.ForAKDT,thetentativesplittingline
therearebetterwaystofurtherreducethiscost.
generatedinthesplitprocessisrandomintermsofitsorientation
Anuncertainconstraintmaybeassociatedwithmultipleobjects
(horizontalorvertical)anditspositionisstrict(e.g.,bisectsthepar-
indifferentpartitionsandaddingsplittinglinesintothesepartitions
tition)orfollowssomegivendistribution(e.g.,uniformornormal
mayhavedifferenteffectsonthisconstraint.Forexample,inFig.4,
distribution). Therandomizationofthesplittinglinesreducesthe
c (p , p , p < 6) isassociatedwithpartitionsS , S andS
1 1 2 3 1 2 4
potentialforoscillationofthealgorithm. ForAMLG,thesplitting j j
(where p , p and p are located). c is uncertain because the
1 2 3 1
lines do not have thisdegree offreedom, they have todivide the circle, O , enclosing these partitions has a diameter p52 (> 6)
1
spaceintoequal-sizedpartitions.However,theobjectsineachpar- andthecircle,O ,intersectingthesepartitionshasadiameterp10
2
tition are in the same grid file indexed by a hashtable. The split
(< 6). After splittingS , byaddingtwolinesegmentsl andl
1 1 2
processreducesthenumberofuncertainconstraints,butitalsoin- 0
(dashedlines),p fallsinsidepartitionS andtheenclosingcircle
ducesmore partitionupdateoverhead. Weevaluate thistrade-off 1 1
becomesO .O ’sdiameterisp34(<6),thereforec issatisfied.
lateron. 3 3 1
IfwefurtherpartitionS ,whichcontainsp ,c remainsuncertain
The algorithm only commits the splitting lines in the partition 2 2 1
becauseS isboundingneitherO norO . Mergingalltheparti-
2 1 2
tionsinS rendersc uncertainagain. However,mergingS and bodystaticconstraintcanbederivedinasimilarfashion[25]).The
1 1 2
S doesnothaveanyeffectonc . Thepartitionsplittingprocess firstandsecondcolumninthetablearethedescriptionoftwocon-
3 1
may affect the result of the constraint when the partition bounds straints, the third column are the conditions based on which the
intersecteithertheenclosingcircleortheintersectingcircle(i.e., conflictingandharmoniousrelationisidentified. Thelastcolumn
S boundsbothcircles). Thepartitionmergingprocessmayaffect specifiesthetransitconditionbetweenthetwoconstraints. Forin-
1
theresultoftheconstraintwhenthepartitionafteramergeisin- stance, the relationamong c and c (if c is satisfied, c is also
3 4 3 4
tersectedbytheenclosingorintersectingcircle(i.e.,S intersects satisfied) is identified with the rule in Line 1, because c and c
1 3 4
withO ).Wecallaconstraintsplitaffectedw.r.t.acertainpartition havethesamealertingdistanceandtheobjectsetforc isasuper-
3 3
iftheconstraintisuncertain,andhassomebodiesinvolvedinside setoftheobjectsetforc .
4
thepartition,andthepartitionboundseithertheintersectingcircle Table1:ConflictingandHarmoniousRules(ExplanationinText)
ortheenclosingcircle. Aconstraintismergeaffectedw.r.t. acer- c1 c2 conditions transitcondition
tainpartitioniftheconstraintissatisfiedorunsatisfied, hassome P1<d1 P2<d2 d1(cid:20)d2,P1(cid:19)P2 c1issatisfied!
bodiesinvolved insidethepartition thatis beingmerged, and the P1>d1 P2>d2 d1(cid:21)d2,P1(cid:18)P2 c2issatisfied
mergedpartition(asecondlevelnode)isintersectedbyeitherthe P1<d1 P2>d2 d1(cid:20)d2,P1(cid:19)P2 c1issatisfied!
enclosingortheintersectingcircle. P1>d1 P2<d2 d1(cid:21)d2,P1(cid:18)P2 c2isunsatisfied
Intheaboveexample,c issplitaffectedw.r.t.S andmergeaf- P1>d1 P2<d2 d1<d2,P1(cid:19)P2 c1isunsatisfied
fectedw.r.t.S10.Sincethes1plitaffectedconstraintsa1retheonlycon- PP11<<dd11 PP22><dd22 dd11>(cid:21)dd22,,PP11(cid:18)(cid:18)PP22 !c1ics2uinsssaatitsisfifieedd
straintswhoseresultsmightchangeinthesplitadjustment, when P1>d1 P2>d2 d1(cid:20)d2,P1(cid:19)P2 !c2isunsatisfied
computingthereductionoftheuncertainconstraintsafterthesplit-
ting (Line 3 in Split Adjust), only the split affected constraints
Fortheconditioncolumn,itmustholdthattheobjectsetofone
are considered because no other constraint changes its result due
constraintisthesubsetorthesupersetoftheobjectsetofanother
tothesplit. Similarly, onlymerge affectedconstraintshavetobe
constraint.IfP isnotasubsetorasupersetofP andyettheirin-
1 2
considered(Line3inMerge Adjust)forcountingtheincreasein
tersectionP (=P P )containsmorethantwoobjects,thereis
1;2 1 2
uncertainconstraintsafteramerge. Thisgreatlyreducestheover- \
stillapotentialdependency betweentheseobjects. Consequently,
headofrecountinginthesplitandmergeprocesses.
theresultofoneconstraintmaybepartiallyvaluablefortheeval-
Also, since the purpose of splitting is to reduce the uncertain
uationofotherconstraints. Tofullyexploitthecorrelationofcon-
constraints, the candidate nodes for a split are selected such that
straints with intersecting object sets, the notion of a virtual con-
theycontainthelargestnumberofsplitaffectedconstraints(Line1
straintisdeveloped.Avirtualconstraint,c ,involvestheintersect-
v
in Split Adjust). This improves the performance of the partition
ingobjectsP (=P P ).Therulesthatdefinethegenerationof
1;2 1 2
adjustmentbecauseonlytheconstraintsthatmaybeaffectedbythe \
c aregiveninTable2.(cid:15)belowdenotesthesmallestpositivevalue
v
splitarecounted.Thepurposeofmergingistoreducethepartition
allowedbythecomputer.Itishardwaredependent.
update rate, therefore the candidate nodes (second level partition
Table2:VirtualConstraintGenerationRules
S )formergeareselectedsuchthatthesenodescontainthelargest
i c1 c2 cv transitcondition
estimatedpartitionupdaterate,PURSi(Line1inMerge Adjust.) P1>d1 P2>d2 P1;2>max(d1;d2) cvissatisfied!c1
andc2aresatisfied
O(cid:13)1(cid:13)(d=521 / 2)(cid:13)(cid:13) P1<d1 P2<d2 P1;2>max(d1;d2) cvissatisfied!c1
2(cid:13) (cid:13)u(cid:13)n(cid:13)i(cid:13)t(cid:13)S(cid:13)1(cid:13) l(cid:13)21(cid:13) (cid:13)u(cid:13)n(cid:13)i(cid:13)t(cid:13)(cid:13) l(cid:13)1(cid:13)p1(cid:13)(cid:13) OS(cid:13)3(cid:13)1(cid:13)((cid:13)’d(cid:13)=3O41 (cid:13)2/ 2(cid:13)()(cid:13)d(cid:13) =101 / 2)(cid:13)(cid:13) M1Sais(cid:13)n2(cid:13) oBu(cid:13)(cid:13)d bwne il(cid:13)lci(cid:13)2felole(cid:13), rnprceto(cid:13)o1a (cid:13) tsiib nnpmet(cid:13).l(cid:13) ciaAtotkifnmeteg etr h SsSi s(cid:13)1(cid:13)s(cid:13)1 (cid:13),a(cid:13) d icstifi(cid:13)1 sf(cid:13):sef pi|repel1id tn(cid:13), (cid:13).pwc 2S e,(cid:13)itp.ph(cid:13)3 l |(cid:13)i(cid:13) tl(cid:13)1<ti(cid:13)n 6g(cid:13)(cid:13) PP11><dd11 PP22><dd22 P+P11(cid:15);;(22(cid:15)<<>mm0aa)xx((dd11;;dd22)) accaannnvvdddiissccc222uunnaaassrrraaeeettiisuussannfifitsseeiaasddfittii!!essdfifieeccdd11
p2(cid:13)(cid:13) P1>d1 P2<d2 P1;2>max(d1;d2) cvissatisfied!c1is
p(cid:13) 2(cid:13) Merging S(cid:13)1(cid:13)' (cid:13)by removing (cid:13)l(cid:13)1(cid:13) and (cid:13)l(cid:13)2(cid:13),(cid:13) satisfied,butc2isnot
3(cid:13) changes c(cid:13)1(cid:13) from (cid:13)satisfied(cid:13) back to(cid:13) P1>d1 P2<d2 P1;2<max(d1;d2) cvissatisfied!c1is
S(cid:13)2(cid:13) S(cid:13)3(cid:13) S(cid:13)4(cid:13) unnoc eefrfteacint (cid:13)o, nb uct(cid:13)1 m(cid:13).(cid:13) erging S(cid:13)2(cid:13) and S(cid:13)3(cid:13) has(cid:13) +(cid:15)((cid:15)>0) satisfied,butc2isnot
Toamortizetheevaluation cost, asecondarygraphstructureis
Figure4:Split/MergeAffectedConstraints usedtoindexconflictingandharmoniousconstraints. Thisgraph
structureisconstructedandincrementallyupdatedwhennewcon-
2.3 Conflicting andHarmonious Constraints
straints are inserted or removed from the system. At the start,
There can be constraints that are in conflict with each other. toconstructthegraphstructure,eachconstraint(includingvirtual
For instance, if c (p t;p t > d) is satisfied, c (p t;p t < constraints) are regarded as nodes in the graph. Each constraint
1 1 2 2 1 2
j j j j
d)must be unsatisfied. Similarly, constraints can alsobe in har- connects to its associated conflicting and harmonious constraints
monywitheachother. Ifaconstraintissatisfied(unsatisfied),this (othernodes)withunidirectionallinks(representingtheasymmet-
could imply that another constraint is satisfied (unsatisfied). For ricrelation).Eachlinkisappropriatelylabeledwithatransitcondi-
instance,ifc (p t;p t;p t < d)issatisfied,thenitfollowsthat tion,forexample,satisfied unsatisfied(shorthandedass u)
3 1 2 3
c (p t;p t <j d) is also sjatisfied. The conflicting and harmo- etcetera.Thenodeswiththe!sameneighborsandsametransit!con-
4 1 2
j j
nious relationis asymmetric. c is unsatisfied mightnotindicate ditionlabelsaremergedintoacompoundnode. Ifaconstraintis
2
that c is satisfied. Also, even if c is satisfied, c might stillbe removed,thecorrespondingnodemayneedtobedeleted.
1 4 3
unsatisfied. Forthediscussionbelow, weassumeconstraintc is In the evaluation stage, after the result for one of the indexed
1
associatedwithobjectsetP andconstraintc is associatedwith constraints is computed, a breadth-first transversal of the graph
1 2
objectsetP etcetera. startingfromthatconstraintisperformed,immediatelygivingrise
2
Conflictingandharmoniousrelationsforn-bodyconstraintsare totheresultsofassociatedconstraints(i.e.,representedbylinked
identified with asetofrulessummarized inTable1(rulesfor n- nodes in the graph.) Virtual constraints are evaluated based on
adaptivespacepartitioning,justasanordinaryconstraint,butthey tionandarecapableofsimplecomputationforcheckingwhether
have higher evaluation priority and will always be evaluated be- coordinatesareinsideacertainpartition(rectangularregion).2
foretheoriginalconstraintsbasedonwhichtheyareconstructed. If the currentresult ofthe constraint basedon the partition in-
Conflictingandharmoniousconstraintmanagementisorthogonal formationissatisfied,wesetacirclewiththealertingdistanceas
to partition-based pruning performed by the indexes. The graph thediametersuchthatitcoversthemaximalnumberofpartitions
index canpruneuncertainconstraintswhichspacepartitioningis includingallthepartitionscontaininginvolvedobjects. Then,the
incapableofpruning. partitionscoveredbythiscircleareregardedasasaferegionand
InFig.5,wegiveanexampleofconflictingorharmoniouscon- returnedtothedevice, whichdoesnotneedtoupdateitslocation
straintsrepresentedbyourgraphindex. Thefiveconstraintsbeing aslongasitismovingwithinthesaferegion,becauseinthiscase
indexedareshowninnodeN ,N ,N N andN .Theconstraints theconstraintsarealwayssatisfied(seeFig.6(A)).
1 3 4 5 6
innodeN andN sharethesameobjectsp andp ,thereforea If the result is unsatisfied, we first identify the objects p and
3 4 2 3 i
virtualnodeN representingconstraint p ;p >disconstructed p (called sentinel objects) that are located inside two partitions
2 2 3 j
j j
(ruleinLine2ofTable2). IfthevirtualnodeN issatisfied,N withadistancegreaterthanthealertingdistance. Then, abisect-
2 3
and N are unsatisfied (unidirectional edge s u from N to ing stripe is set. It is defined as a stripe with width equal to the
4 2
!
both N and N ). Also, if N or N is satisfied, then N must alertingdistance. Itbisectsthedistancebetweenthetwopartitions
3 4 3 4 2
beunsatisfied (unidirectionaledges ufromboth N and N (seeFig. 6(B)).Aslong as eachsentinel object ismoving within
3 4
!
to N ). Since the two unidirectional edges between N and N thepartitionsonitssideofthestripe(regardedassaferegion),the
2 2 3
(N )havethesamelabel(s u),theyarecombinedasasingle constraints are unsatisfied. Therefore, p and p track their own
4 i j
!
bi-directional edge. Observe thatN and N areconnectedwith positionanddonotneedtoupdatetheirlocationunlesstheygobe-
3 4
the edgesofthe samelabel toothernodes, therefore they canbe yondthesaferegionontheirsides. Atthesametime,alltheother
combinedtoformacompoundnode.Alltheothertransitedgesare objectswaitfortheirpositionprobe. Uponnotificationthateither
basedontheruleslistedinTable1. p orp areoutsidethesaferegion,theconstraintmatcherforcesall
i j
Anexampleofconstraintprocessingisthatwhentheobjectp theinvolvedobjectstoupdatetheirlocations,andvalidatesthenew
2
updatesitslocation,thevirtualconstraint p ;p > dinN will sentinelobjectsandthestripe. Itispossiblethatsuchsentinelob-
2 3 2
j j
beevaluatedfirst(givenitisuncertain)becauseavirtualnodehas jectsdonotexist. Inthiscase,thedeviceupdatesitslocationwith
higher evaluation priority. If it is unsatisfied, then the constraint somedefaultrate. Thissaferegionpolicyisverycosteffectivein
in N is satisfiedbecausethe transit conditionfrom N toN is thatitcapturesmostconstraintmatcheswithsignificantreduction
1 2 1
u s. SincethereisnoedgecomingoutfromN labeledwith oftherequiredlocationupdates.
1
!
s , thetransversalterminates. Ontheotherhand,ifthecon-
! (cid:3)
straintsinN2 issatisfied,thenconstraintsinN3,N4 andN5 must Stripe bisecting the distance bw.(cid:13)
beunsatisfied(s u). Alerting distance(cid:13) partitions containting p(cid:13)i(cid:13) and p(cid:13)j(cid:13)
! p(cid:13)
i(cid:13)
|p2(cid:13)N,(cid:13) p(cid:13)13(cid:13)(cid:13)(cid:13)|<du(cid:13)(cid:13)+ (cid:13)(cid:13) (cid:13)s(cid:13) (cid:13) (cid:13)s (cid:13)(cid:13)s(cid:13)u(cid:13) (cid:13) (cid:13)V|upi(cid:13)rt2(cid:13)u,(cid:13)aN lp N3(cid:13)2b(cid:13)(cid:13)(cid:13)(cid:13)i(cid:13)|od>(cid:13)d(cid:13)ir(cid:13)ede(cid:13)(cid:13)c(cid:13)(cid:13)(cid:13)t(cid:13)iso(cid:13)s(cid:13)n (cid:13)(cid:13)(cid:13) (cid:13)a(cid:13) (cid:13)(cid:13)luu(cid:13)(cid:13) (cid:13)e(cid:13)(cid:13)d(cid:13)g(cid:13)e|(cid:13)p2(cid:13)(cid:13),| ppC 13(cid:13)s(cid:13)(cid:13),o, (cid:13) m pp (cid:13)N bu(cid:13)42(cid:13)(cid:13)o,(cid:13) |(cid:13)(cid:13)3 u< (cid:13)p n 3dd(cid:13)(cid:13)| (cid:13)n<oddse(cid:13)(cid:13)s (cid:13)(cid:13)u (cid:13)(cid:13) (cid:13) (cid:13) (cid:13) (cid:13)s(cid:13)s (cid:13)(cid:13)(cid:13)u (cid:13)u(cid:13) (cid:13)(cid:13) u(cid:13)(cid:13)|p1(cid:13)(cid:13),p s(cid:13)p (cid:13)|5 (cid:13) u(cid:13)p2(cid:13)|(cid:13), <4(cid:13) NN, (cid:13) p d 3p(cid:13)(cid:13)56(cid:13)(cid:13) 5,(cid:13)(cid:13) (cid:13) (cid:13)| p> 4(cid:13)(cid:13)d(cid:13) Safe region(cid:13)FpM1(cid:13)i(cid:13)og(bAuil)er(cid:13) peO3(cid:13)p(cid:13)b62(cid:13)j(cid:13)e:ctS(cid:13) afeRSaefeg rieognionC foor mp(cid:13)i(cid:13)puA(cid:13)l(cid:13)te(cid:13)(cid:13)r(cid:13)ti(cid:13)n(cid:13)ag(cid:13)(cid:13) d(cid:13)(i(cid:13)s(cid:13)(cid:13)ta(cid:13)Bn(cid:13)tc(cid:13)e(cid:13))i(cid:13)onSafe prej(cid:13)(cid:13)gion for p(cid:13)j(cid:13)
N(cid:13)4(cid:13)
u u(cid:13) 2.5 ExtendedConstraints
s(cid:13) (cid:13) (cid:13) (cid:13)u(cid:13)
s s(cid:13)
d (cid:13) +>(cid:13) d (cid:13)u(cid:13): (cid:13)unsatisfied(cid:13) s(cid:13): (cid:13)satisfied(cid:13) Inthissubsection weillustrate how thepartition-basedevalua-
tionalgorithmcanbeappliedtootherconstraints. Theillustration
is based on a continuous reverse range query (CRR) and a con-
Figure5:IndexforConflictingandHarmoniousConstraints
tinuousreversek-nearest neighborquery(CRKNN), whichare
definedasfollows:
2.4 Location UpdatePolicy
1. GivenarectangularregionR,aContinuousReverseRange
Thelocationupdatepolicymanagestheconsistencybetweenthe query,CRR(R;p )continuouslycheckswhetheramoving
i
lastpositionassumedbytheconstraintsolverandtheactualposi- objectsp isinsideR.
i
tionofthemovingobject.Eachlocationupdateexertscertainload
ontothesystem(e.g.,network,power,computeresources,andeco- 2. GivenastaticpointA,aContinuousReversek-NearestNeigh-
nomical.Thelocationupdatepolicyistherulethatspecifieswhen borquery,CRKNN (A;p ),continuouslycheckswhether
i
theupdateshouldbescheduled. Thepolicybalancesthetrade-off Aisoneoftheknearestneighborsofp .
i
betweenthe resourcesconsumedand the accuracy represented in
Inthediscussionbelowwerefertothesequeriesasconstraintsto
thesystemandachievableintheresults.
emphasize the continuous character of the problem, i.e., given a
Inpractice,therearemanydifferentwaystoobtainlocationdata
potentiallylargesetofconstraintsandchangingobjectlocationin-
andpowerisaconcernthatrequiresthelocationupdatepolicytobe
formation, wearelookingforthematchedconstraints. Asinthe
adjustedtothecharacteristicsofthemobiledevice.Here,weadopt
previoussubsection,theobjectiveofthepartition-basedevaluation
thenotionofasaferegiontodefineadevice-dependentlocationup-
datepolicy[9].Theobjectivesaretomaintainup-to-dateconstraint 2ThedevicecouldeitherobtainitslocationviaGPSorrequestit
results,andreducewirelesscommunicationandre-evaluationcost. fromthenetworkinginfrastructure.Bothareviablemodelsinprac-
Weassumethatthemobiledevicesareabletotracktheirposi- tice.
is to reduce the number of constraints that have to be evaluated with atotalof omoving objects. Supposethateachconstraint is
againstprecisepositioninformation. associatedonaveragewithoobjectsandamongthecconstraints,
Withoutlossofgenerality,supposethatp islocatedinsidepar- u are uncertain. The model estimates the matching cost for in-
i
titionS . Thepartition-basedevaluationforCRR(R;p )works memory and I/O-incurring environments and formalizes algorith-
pi i
asfollows. IfS isdisjointwithR,theconstraintisunsatisfied. micproperties. Duetospacelimitations,theproofsofallthethe-
pi
IfS iscompletelyenclosedbyR,theconstraintissatisfied.Oth- orems and lemmas below are omitted. They can be found inthe
pi
erwise(S intersectsR),itisuncertain. Fig.7showsthreerange technicalreport[25].
pi
queries, R , R and R . Since partition S , which accommo-
1 2 3 53 3.1 Cost forIn-memory Processing
dates p , is disjoint with R , the constraint CRR(R ;p ) must
1 1 1 1
beunsatisfied. CRR(R ;p )issatisfiedsinceS iscompletely AllcomputationsaredoneinmainmemoryandnodiskI/Oisin-
2 1 53
insideR andCRR(R ;p )isuncertainsinceR partiallyinter- curred. Onaverageeachobjectisassociatedwithco=oconstraints
2 3 1 3
sectsS . anduo=oofthemareuncertain. Sincethefrequencyoftheparti-
53
tionadjustmentprocessesismuchlowerthanthelocationupdate
1(cid:13) 2(cid:13) 3(cid:13) 4(cid:13) 5(cid:13) 6(cid:13) 7(cid:13) Mobile point(cid:13) frequency, thecostformaintainingtheindex data structureis ig-
p(cid:13) nored. Theconstraintevaluationcostiscomprisedoftwofactors:
1(cid:13) 2(cid:13) A(cid:13)2(cid:13) Static point(cid:13) thecostforevaluatingaconstraintbasedonpreciselocationposi-
S(cid:13)15(cid:13)
234(cid:13)(cid:13)(cid:13) SS(cid:13)(cid:13)5111Or(cid:13)(cid:13)(cid:13)(cid:13)r(cid:13) A(cid:13)1(cid:13)Rth(cid:13)(cid:13)2(cid:13)e(cid:13)(cid:13) n(cid:13)S(cid:13)e(cid:13)a(cid:13)1(cid:13)r1e(cid:13)(cid:13) (cid:13)s (cid:13)t (cid:13) d(cid:13) O(cid:13)i(cid:13)st(cid:13)t(cid:13)(cid:13)ar(cid:13)h(cid:13)(cid:13)n(cid:13)e(cid:13)c(cid:13) (cid:13)e(cid:13)n(cid:13) (cid:13)e(cid:13)b(cid:13)a(cid:13)wb(cid:13)r(cid:13).(cid:13)w(cid:13)eR (cid:13)(cid:13)S(cid:13)s(cid:13).(cid:13) (cid:13)(cid:13)13(cid:13)t(cid:13)S1(cid:13)((cid:13)(cid:13)f(cid:13)(cid:13)(cid:13)1(cid:13)ua1(cid:13)(cid:13)r(cid:13)n(cid:13)(cid:13)ta(cid:13)d(cid:13)h(cid:13)n(cid:13) (cid:13)e(cid:13)S(cid:13)d(cid:13)s4(cid:13)(cid:13) (cid:13)t(cid:13)7(cid:13)S)(cid:13)(cid:13) (cid:13)1(cid:13)(cid:13)d(cid:13)5(cid:13)i(cid:13)(cid:13)s(cid:13)t(cid:13)a(cid:13)An(cid:13)c(cid:13)4(cid:13)e(cid:13)(cid:13)S(cid:13)47(cid:13) CCCCCCRRRRRR222RRRNNN(cid:13)(cid:13)(cid:13)(((RRRNNN(cid:13)(cid:13)(cid:13)321(cid:13)(cid:13)(cid:13)((((cid:13)(cid:13)(cid:13) AAA,,,ppp(cid:13)(cid:13)(cid:13)431111 (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)))),,,(((ppp(cid:13)(cid:13)(cid:13)usu222 (cid:13)(cid:13)(cid:13)a)))nn(((tcs(cid:13)(cid:13)(cid:13)uusiseaannfrttisciteisasaedffitrini(cid:13)eit)esa(cid:13)(cid:13)d)df(cid:13)ii(cid:13)n)e(cid:13))(cid:13)(cid:13)(cid:13))d(cid:13)(cid:13))(cid:13) taSrtbihetaouicesonpenenpiidvntocheofsoedoens,rtctmnhpwoeaaswhtttoieffoflnooonwrrca(aanhadtiteoiclionobmhnocjeetaidenctdaditcotuuuaanrprsatdautCpikapoateodrnestsas,itatttieilpt,ouspadn)pruatoauornxriptdtniaidmgtltiahooatoeefntnespl(el.yadlorue(cid:11)Itcnniftaiopitteotiraeoonocdfn-chbeatuaissempssCveidenadaolgteueseqtvatsiutapmiaulaourel)nes-,.
5(cid:13) A(cid:13)3(cid:13) p1(cid:13)S(cid:13) (cid:13)53(cid:13) R(cid:13)1(cid:13) 1(cid:13) (cid:13)u(cid:13)n(cid:13)i(cid:13)t(cid:13) f(u).o=Wto)i(cid:11)thl.thNios,tethtehaatbooltveislothceataiovneruapgdeautepdcaotsetfcraenqubeensciym(pdleifinoedtetdoa:s
Cost =uo(cid:11)f (1)
lu
When the object is moving into a new partition, all the con-
Figure7:CRRQueryandCRKNNQuery straintsitis associatedwitharesubjectedtopartition-basedeval-
uation. Ifpartition-basedevaluationincurs(cid:12) processingtime,the
The partition-based evaluation for CRKNN(A;pi) works as cost of apartition update perunit of time is (co=o)(cid:12)p. p=l is the
follows. SupposeAisinsidethepartitionSA andpi islocatedin probabilityofanobjectupdatingitspartition(denottedasPpu)and
Spi.ThesystemmaintainsacurrentlistofkNNsandthepartitions thepartitionupdatecostcanberewrittenas:
wheretheyarelocated(supposetheyareSlj(1 (cid:20) j (cid:20) k).) Ifthe Costpu =co(cid:12)fPpu (2)
nearest distance between Spi and SA is greater than the furthest Thevalueof(cid:11)and(cid:12) dependonthehardwareusedandcanbe
distance between Spi and Slj, or more formally, infp2Spi;q2SA estimatedexperimentally. FromEq.1andEq.2,itfollowsthatthe
jjp;qjj2 > max1(cid:20)j(cid:20)k supp2Spi;q2Sljjjp;qjj2, then A can not averagecostperunitoftimefortheadaptivealgorithmisgivenby:
beoneofthek nearestneighborsofp , andCRKNN(A;p )is Cost =Cost +Cost =uo(cid:11)f +co(cid:12)fP (3)
i i adapt lu pu pu
unsatisfied.Letr=sup p;A andO isacirclewithras The per unit of time cost for the na¨ıve approach is simply the
theradius.TheniftheMpin2kSopwijsjkisujmj2ofS randO (S O )3 costforevaluatingallconstraints:
pi r pi (cid:8) r Cost =co(cid:11)f (4)
containsatmostkneighbors,CRKNN(A;p )mustbesatisfied. na(cid:127)(cid:16)ve
i InEq.3,ifP issmall,thesecondtermcanbeignored,andthe
Ifaconstraintisneitherunsatisfiednorsatisfied,itisuncertain. pu
costcanbeapproximatedto:
InFig.7,supposethecurrenttwonearestneighborsofmoving u
objectp2 areA1 andA2. Sinceinfp2S11;q2S47(p;q) = p29 > Costadapt(cid:25)uo(cid:11)f = cCostna(cid:127)(cid:16)ve (5)
max(sup p;q )= p26, A cannotbeone Discussion:Ouralgorithmoutperformsthena¨ıveapproachwhen
p2fS22;S15g;q2S11 jj jj2 4
ofp2’s2nearestneighbors,CR2NN(A4;p2)isunsatisfied.How- Costadapt < Costna(cid:127)(cid:16)ve. However,withstaticspacepartitioning,
ever,constraintCR2NN(A ;p )mustbesatisfied,becauseA is thiscannotbeguaranteed.Forinstance,ifthemajorityofthecon-
1 2 1
theonlyneighborinsidetheregionS11(cid:8)Orwherer=supp2S11 straintsareuncertain(largeru)orPpuistoohigh,thentheperfor-
p;A . AnditcanbeverifiedthattheconstraintCR2NN(A ; manceofthealgorithmdegenerates.Givenastaticspacepartition-
1 2 3
jj jj
p2)isuncertain. Infact,ifp2ismovingtothebottom-leftofS11, ing,Ppudependsonlyonthemovementpattern(e.g.,thevelocity
A willreplaceA tobethesecondnearestneighborofp . The andtheposition)oftheobjectswhilethenumberofuncertaincon-
3 2 2
resultofconstraintCR2NN(A ;p )istrulyundeterminedunless straintsdependsonthecorrelationofthepositionofobjects.There
3 2
theprecisepositionofp2isgiven. isnodirectrelationshipbetweenPpuandthenumberofuncertain
Bear in mind that for location change within the partition, ex- constraints.Also,derivingtheoptimalpartitionschemeisdifficult,
plicit computation is only required for the constraints previously becausethereisnodirectdependencybetweenP andtheparti-
pu
classified as uncertain and the constraint classification only hap- tionscheme.Forexample,evenifthespaceispartitionedsparsely,
pensinresponsetoapartitionupdate. theobjectcouldstillgeneratealargenumberofpartitionupdatesby
movingback-and-forthacrossasplittingline.Ontheotherhand,an
3. ANALYTICALMODEL objectcouldmovefastwithoutgeneratinganypartitionupdatesin
adenselypartitionedregionbyremaininginsideasinglepartition.
In this section, we develop an analytical model for the cost of
Weaddressthischallengeinthenextsubsection.
evaluatingn-bodyconstraints. Theanalysisforn-bodystaticcon-
straintsissimilar.Inourmodel,atotalofcconstraintsisassociated 3.2 Determiningthe OptimalPartitionSize
3M M = p+qp M ;q M ,wherep+qisthevector Inthissectionwedevelopamodeltoestimatetheoptimalparti-
1 2 1 2
sumo(cid:8)fthevectforspajnd2q. 2 g tionsize. Theideaistosampleanumberofpartitionschemesand
selectthebestone(i.e.,theonethatleadstoasmallerCost .) constraints. By continuousadjustment, the algorithm adapts toa
adapt
Thisiswhatouralgorithmdoesbytentativelysplittingandmerg- non-uniformenvironmentandevolveswiththechangeofthemove-
ing partitions to determine a partition change that ultimately re- mentpatternsandreachesalocallyoptimalpartitioning.
duces the overall cost. The change aimsto reduceboth P and
pu 3.3 Cost underSecondary StorageAccess
thenumberofuncertainconstraints,andaimstobalancethetrade-
offbetweenthem. Ouranalyticalmodelisbasedonthefollowing For applications that need to manage a large number of loca-
assumptions: tionconstraintswithalargenumberofmovingobjects,secondary
1. Allobjectsfollowtherandommovementpattern.Asaresult, storageaccesscostofthealgorithmmaybecomeaperformancede-
thedistributionoftheobjectsontheplaneisuniform. gradingfactor. Forinstance,objectsmaybeassociatedwithlarge
2. Thepartitionsareequal-sizedsquareswiththelengthofeach profiles, as is common in telecommunicationapplications, or the
sideequaltoa,whichismuchsmallerthantheextent4ofthe applicationmaybecollocatedwithotherapplications,thusnothave
wholespaces(a s). exclusiveaccesstomainmemory. TheI/Ocostforaccessingdata
(cid:28) insecondarystorageis typicallyordersofmagnitudehigherthan
3. Alltheconstraintshaveauniquealertingdistances,denoted
accessing data residing in main memory and it becomes the ma-
asd(d s).
(cid:28) joroverhead. Thereforereducingthenumberofsecondarystorage
First,weestablishthatthesizeofthediameterofthesmallestcircle
accessesbecomesaprimaryconcerninthisenvironment.
enclosingasetofobjectsisuniformlydistributed. Therefore,the
Foreverysecondarystorageaccess,dataisfetchedinpagesrather
probabilityofaconstraintbeinguncertain(satisfiedorunsatisfied)
than in records and each access to secondary storage transfers a
is an expression of the partition size, space size and the alerting
constantamountofdataintomemory. TheI/Ocostismeasuredas
distance.ThisiscapturedinLemma1.
thenumberofaccessestothedataonthesecondarystoragedevice.
LEMMA 1. Under random movement of objects in Euclidean Foroptimizingsecondarystorageaccesses,ouralgorithmaimsat
2Dspace,thedistributionofthesizeofthediameterofthesmallest holdingasmanyuncertainconstraintsaspossibleinmainmemory
circleenclosingasetofobjectsisuniform.Therefore,aconstraint because they are the most frequently accessed data. Constraints
isunsatisfiedwithprobability1 add=ae,issatisfiedwithproba- thatdonotfitintomainmemoryresideondiskandarebroughtin
(cid:0) s
bility abd=ac andisuncertainwithprobability a. ondemand.Thefollowingtheoremquantifiesthecost(thenumber
s s
ofI/Oaccessespertimeunit)ofouralgorithmrelativetothena¨ıve
Basedon Lemma1and theaforementionedassumptions, wecan
approach.Aproofcanbefoundin[25].
estimatetheoptimalsizeofthepartitionthatminimizestheevalu-
ationcost.Thisisstatedinthefollowingtheorem. THEOREM 2. If all the uncertain constraints can be held in
main memory, thecostof thematching algorithm is thepartition
THEOREM 1. UnderrandommovementofobjectsinEuclidean updateprobability(P )timesthecostofthena¨ıveapproach(Co-
pu
2Dspace,thereexistsauniquelocal(thereforealsoglobal)optimal
stI=O =P *CostI=O ).
partitionsize,whichis kv(cid:12)s forsomeconstantk.(recallthat(cid:11)and adapt pu na(cid:127)(cid:16)ve
(cid:12)aretheaveragecostfor(cid:11)evaluationbasedonprecisepositionand This result is somewhat surprising, it means that even with lim-
partitioninformation,respectively.) itedmemory,reducingthepartitionupdateratebysomeproportion
causestheevaluationcosttobereducedwiththesameproportion,
Discussion: If the movement pattern is not random, we can not
giventhatalluncertainconstraintsareheldinmemory. Underthe
compute a globally optimal partition size. However, if we inter-
randommovementpattern,thenumberofuncertainconstraintsis
pret a local distribution as random, then there still exists a local
onlyasmallportionofthetotalnumberofconstraints(a asstated
optimum, which gives us a locally optimal partitioning. We ap- s
inLemma1).Itcanthereforeoftenfitintomainmemory,evenfor
proximate this local optimum by rescheduling the partitioning in
largeconstraintloadsgoingbeyondmainmemorycapacities.Also
theareathatisnotyetoptimizedbasedonthemergingandsplit-
anefficientalgorithmshouldstopfurtherreducingthenumberof
tingalgorithm,introducedintheprevioussection.
uncertainconstraintsbysplittingaslongastheyallfitintomem-
We say a partition scheme, ps, is the ancestor of a partition
ory.Merginginthatcasemayactuallydampenthepartitionupdate
scheme, ps0, (ps0 is the descendantof ps), if ps0 can be reached
rateandhelptolowertheI/Oaccesscostfurther.
frompsbyaddingsplittinglines. Thisdescendantrelationshipis
denotedasps0 ps.InSection2.2,toreducethecomputation,we
(cid:31) 4. SYSTEM IMPLEMENTATION
mentionedthattheuncertainconstraintsafterthesplitting(merg-
ing)processmustbeasubset(superset)oftheuncertainconstraints
Phone Network Carrier(cid:13)
beforethesplitting(merging). Thispropertyisformallycaptured
inLemma2.Proofdetailscanbefoundinthetechnicalreport[25]. App.L Coogngtirnogll(cid:13)er &(cid:13) P&S Staging(cid:13) Publications & Subscriptions(cid:13) PrDoaxyta S Leinrvke(cid:13)r(cid:13) W
(cid:13)A
(cid:13)P
iaTasllhssaeLoolsiEaanoMntsauaMunittniAisuosfinane2ctdib.eserficIthoefaidnipnnssdct0croLaon(cid:31)einsnmtstrptamrisnai,naipntt2hstie0nii;nnspatpasnhns0da;.tuaawnnssiutaahnttiicassefifimreetddaoirccneoocpnnorssenttrcrsaaitsriienantitaniiptnnpinrpposspxsiiiss-0 Database((cid:13)UsBLeLoorooP agItD&dsgti,Sinr naC(cid:13)ggpo(cid:13)(cid:13)(cid:13)nsLtraoMinctaa ItDtci)oh(cid:13)ninP Cg&S oE(cid:13)nnsgtrinaein(cid:13)t(cid:13) MLUoapcndaaaOtgiotpeeen(cid:13)rn(cid:13)(cid:13)wave P&So DrseKipt(cid:13)iPloyno(cid:13) sqLituo(iloeconarny tgSi(cid:13)o.e nlra(cid:13)vte. ra(cid:13)ltN.)(cid:13)(cid:13))(cid:13)(cid:13)0(cid:13).(cid:13)(cid:13)(cid:13)2(cid:13)WAP (cid:13)(cid:13)( (cid:13)r(cid:13)(cid:13)e(cid:13)ws(cid:13)(cid:13)(cid:13)o(cid:13)Br(cid:13)
muainmastaisotianstfi(aewdirtebhdyputscht0ie)o,pnmaoortfriettihoceonn-nbsuatrmsaeibdnetersvcoaaflunuabntiecoerner.tsaoTilnhveecdsopnalssittsrtaianitnigstfisperadotcaetnhsdes ApSpLeliBrcvaSet(cid:13)iro(cid:13)n(cid:13) (URseelra ItDed, SMeSrvGic)(cid:13)e(cid:13) NoEtinfigcianteio(cid:13)n(cid:13) WAP PNuosthif icRaetqioune(cid:13)st for(cid:13) NoStieficrvaetiro(cid:13)n(cid:13) (cid:13)n(cid:13)o(cid:13)i(cid:13)t(cid:13)(cid:13)ac(cid:13)(cid:13)if(cid:13)i(cid:13)(cid:13)t(cid:13)oCell phone(cid:13)
represents data flow(cid:13)
costofaslightincreaseinthepartitionupdaterate. Ontheother
hand,themergingprocessaimsatareductionofthepartitionup- Figure8:End-to-endSystemArchitecture(fullyimplemented)
daterateatthecostofaslightincreaseinthenumberofuncertain
4Theextentofthespaceisdefinedasthemaximumpossibledis- We have developed a complete end-to-end location-based ser-
tancebetweentwopointsinsidethespace. vice prototype incorporating the location constraint matcher de-
scribedinthispaper. Moreinformationaboutanearlierversionof doesnotevolveatallovertimetobetteraccommodatethemove-
theprototypeissummarizedin[23]andisknownastheLocation- ment pattern, the matching time for both AKDT and AMLG is
based Toronto Publish/Subscribe System (L-ToPSS). The system around 250ms and the partition size is 25km2. When the win-
wasdeployedasaproofofconceptonamobilecellularnetwork. dow size (which controls the speed of adjustment) is adjusted to
TheoverallsystemarchitectureisshowninFig.8. 25,AKDTandAMLGactivelyevolvetolowertheevaluationload
Inourimplementation, thesystemusesthecellularnetwork to throughsplitting.Withinsevenminutes,thematchingtimeofboth
obtain location position informationof subscribers. The network indexes is reduced by 60% and stabilizes below 100ms, and the
exportslocationtrackingcapabilitiesviaaWebservice. Location partitionsizeisadjustedtoaround5km2. Thisperformancegain
position data is retrieved through a location position server over comes about through the pruning of large numbers of uncertain
theInternet. Boththeserver andthe subscribercan initiateposi- constraints (by about 60%) at the cost of a moderate increase in
tionrequests.Whenthelocationupdatestreamsintotheconstraint partitionupdates(toabout5%)(graphsomittedduetospacelim-
matching engine, it is evaluated against the location constraints itation). Thisshowsthattheadaptivealgorithmsfindtheoptimal
stored in the system and the matches are communicated back to partitionsizethroughself-adjustment.
thesubscribersviathenotificationserveronthecarrier’sside.Dif- On the other hand, if the initial partition size is set to 1km2
ferentsolutionsforobtaininglocationpositiondataareavailablein (Fig.9(C,D)),partitionadjustment(w =25)evolvestomergethe
practice. Inourcase, theoperatorcombinedGPS,networktrian- partitionstoanaveragesizeofabout5km2.Theaveragematching
gulation,andcellsitelocationtechnologiestopositionsubscribers, timeisreducedfrom350mstobelow100ms(byabout70%).This
aimingtoprovidethemostaccuratepositionforarequiredpreci- performancegainisduetothereductioninthepartitionupdaterate
sion, specifiedaspartofthelocationrequestinput. Abriefeval- (byabout90%)ofthemergeprocess.Thematchingtimeresulting
uationoftheaccuracyofthesystemunderdifferentexperimental from the na¨ıve approach, where all the constraints are evaluated
conditionsissummarizedin[22]. without partition-based pruning, is an order of magnitude higher
(notshowninthegraphsforclarity.)
5. EXPERIMENTS InFig.10(A),wevarythemeanofthevelocity(v = 5;10;15)
andrunthedatasetagainstadaptiveindexing withdifferentwin-
Inthissection,wepresenttheexperimentalresultsthatdemon- dowsizes.(Forclarityofpresentation,weonlyshowresultsforthe
strate the performance of the algorithms. All experiments were AKDT;resultsforAMLGaresimilar.) Thefigureshowsthatthe
conductedonaPentium4with2GHzand2GRAM)runningun- adjustmentcost(theoverheadofsplittingandmerging)increases
derLinux. Intheexperimentswesimulatemobileobjectsmoving linearlyasthewindowsizeincreasesfordifferentvelocities.How-
in a test field of size 40km 40km. We model two movement ever, the adjustment cost is negligible compared to the time for
(cid:2)
patterns,randommovementandclusteredmovement. Therandom constraint evaluation. To obtain the same evaluation time in the
movementpatternmaintainsauniformdistributionoftheobjectsin threescenariosshown,alargerwindowsizeisrequiredforhigher
thetestfield. Theclusteredpatternmaintainsanumberofclusters velocity.
ofobjectsinthetestfield,wherethepositionoftheobjectinthe Giventhedistributionofthevelocity, theoptimalpartitionsize
clusterfollowsanormaldistributionwithmean,(cid:22),asarandomly canbefoundwiththesteepestdescentmethod(SDM[7])bysam-
selectedpointinthetestfield. Wevarythestandarddeviation,(cid:27), plingdifferentpartitionsizesapproachingtheoptimum,whichbased
ofthedistributiontomodeldifferentdegreesofskewness. Inthe onTheorem1isunique.InFig.10(B),wevalidatetherelationship
extremecase,wherethestandarddeviationissufficientlylarge,the between the average velocity of the objects and the optimal par-
clustered movement pattern is reduced to the random movement tition size and show that the optimal partition size obtained with
pattern. Tomodelthemovementofclusters,(cid:22)isdeterminedasa SDMisnearlyproportionaltotheaveragevelocityoftheobjects
functionoftime,t. (asconfirmedbyTheorem1). Moreover,thefigureshowsthatthe
Prior to the experiment, constraints are generated; each con- adaptiveprocessingeventuallyresultsinapartitionschemethatis
straint is associated with n bodies among all the moving objects closetotheoptimum.
inthefield. Thealertingdistancesareuniformlydistributedwith InFig.10(C),weevaluatetheadaptationofthealgorithmwith
acertainmean(e.g.,500m). Bychangingthemean,thematching the evolution of the clustered movement pattern. 50 clusters are
load5canbeadjusted.Thevelocityofthemovingobjectsfollows generated with constant standard deviation (400), the means of
anormaldistributionwithagivenmean(5m/sorasspecified). theseclustersaremovingwithrandomspeed(uniformintherange
Thepartitionadjustmentisperformedonceeveryminutetoop- [20,50])anddirection. Asevidentfromthemovingclusterwork-
timizethepartitionschemeasneeded. Forsakeoffairnessinthe load,thestaticindexcannotkeepupandperformancedeteriorates.
comparison, wemaintain the same adjustmentspeed for both in- The matching time is not only high but also very unstable with
dexesbysplittingandmergingthesamenumberofcandidatepar- many spikes resulting fromlargepartition updateoverhead when
titions. the clusters move across splitting lines. AKDT and AMLG han-
dle this situation well. The matching time is relatively low and
5.1 Effect ofAdaptation
muchmorestable.Whenweadjusttheskewnessofthecluster(by
This experimentisconductedunderrandommovement pattern varyingthestandarddeviation(cid:27)),weobtainsimilarresults,except
wheretheobjectsareuniformlydistributed. ItshowshowAKDT thathigherskewness(smaller(cid:27))exacerbatestheinstabilityforthe
andAMLGadapttotherandommovementpatterns.Theresultsare static indexing. Fig. 10(D) shows the superiority of the adaptive
comparedagainstanon-adaptivesolution(w = 0). With100,000 indexingfortheCRRandCRKNN constraints.Comparedwith
constraintsand 10,000uniformlydistributedobjects, wemeasure thebasicprocessingwithoutconstraintpruning,thematchingtime
thematchingtimeandtheaveragepartitionsizeovertime.Fig.9(A, isgreatlyreduced(to14%forCRKNN andto9%forCRR)for
B)showstheresultsduringthefirst12minutesoftheexperiment theadaptivepartition-basedapproach.
whentheinitialpartitionsizeis25km2. Thestaticindex(w =0) Fig. 11(A) demonstrates the cost of indexing for different lo-
cation update loads (i.e., number of location updates incurred.)
5The matching load is the ratio between the average number of
With static indexing (w = 0), for AKDT with backtracking op-
satisfiedconstraintsandthetotalnumberofconstraints.
(A) (B) (C) (D)
300 25 400 7
matching time (ms)112205050000 AAAAKMKMDDLLTGTG((((wwww====0202)5)5)) 2partition size (km)1120505 AAAAKMKMDDLLTGTG((((wwww====0202)5)5)) matching time (ms)112233050505000000 AAAAKMKMDDLLTGTG((((wwww====0202)5)5)) 2partition size (km)23456 AAAKMKDDLTGT(((www===020)5))
AMLG(w=25)
500 2 4 6 8 10 12 00 2 4 6 8 10 12 500 2 4 6 8 10 12 10 2 4 6 8 10 12
time (minutes) time (minutes) time (minutes) time (minutes)
Figure9:AdaptationtotheUniformMovementPattern
(A) (B) (C) (D)
time (ms)234567890000000000000000 aaaeeedddvvvaaajjjuuullluuussstttaaammmtttiiioooeeennnnnn(((ttt(((vvvvvv======511511)05)05)))) 2partition size (km)111112222024680246 oAApKMtDiLmTGa((wlw s==iz22e55)) matching time (ms)123456000000000000 AAAAKMKMDDLLTGTG((((wwww====0202)5)5)) matching time(ms)11114680246000000000000000000000 CCCCCCRRRRRRRRRKKKNNN AANNNKM DAALTGKM(D(LwwTG==((2w2w55==))2255))
100 8 2000
05 10 15 20 25 30 35 40 45 65 10 15 20 00 20 40 60 80 100 120 01 2 3 4 5 6 7
window size average velocity (m/s) time (minutes) number of constraints x 105
Figure10:AdjustmentCost(A),OptimalPartitionSize(B),AdaptationtoClusteredPattern(C),andCRR/CRKNNqueries(D)
timization,6 theoverheadisreducedby50%ascomparedtonon- 5.2 Effect ofSecondaryStorageAccess
backtracking-basedindexing. Foradaptiveindexing(w=25),the Inthis subsection wemeasurethe number ofdisk accesses re-
costofAKDT(withorwithoutbacktracking)isveryclosetothat quiredforprocessing10,000objectsand100,0002-bodyconstraints,
of AMLG, because adaptive spacepartitioning avoids high parti- withstrictlimitsimposedontheavailablememory.Weassumethat
tion updaterates and themaintenance differencebetweenAKDT theindexingstructure,thepositioninformationoftheobjectsand
andAMLGissmall. Bothcostareabout10%ofthecostofstatic acertainnumberoflocationconstraintsareheldinmemory.Other
indexing. locationconstraintshavetobepagedinandoutofmemoryonde-
WealsoobserveinFig.11(B)thatthematchingtimedecreases mand(weassumetheLRUpagereplacementpolicyandapagesize
linearlyasthepercentageoftheconflictingand harmoniouscon- of4k). Thealgorithmtriestostoreasmanyuncertainconstraints
straints increases. These constraints are pruned directly through inmemoryaspossible,sothatdiskaccessisminimized.
the secondary graph-structured index and do not need to be re- Fig.12(A)plotsthenumberofsecondarystorageaccessesagainst
evaluated. available memory. For this experiment, we test the AKDT and
Fig. 11(C) shows the adaptation of the algorithms to different AMLG indexes against objects with various movement patterns.
constraint sets. Three constraint sets are deliberately generated Fig.12(A)showsthatthenumberofdiskaccessesforbothindexes
such that for static indexing (w = 0), the sets contain different aremuchlowerthanforthena¨ıvealgorithm. Thisisbecausethere
number of uncertain constraints (set 1 < set 2 < set 3), there- arefewer constraints thatneedtobe accesseddue to thepruning
fore,weobserveanincreaseinmatchingtimefromconstraintset1 capabilitiesoftheindexes. Thepartition-basedalgorithmsexhibit
to set 3. With adaptive indexing (w = 25), the matching time a point in memory use after which the number of disk accesses
stabilizesaround100msforallsets,becausethepartitionscheme is nearlyproportionaltothediskaccessesforthena¨ıve approach
evolvestoreducetheuncertainconstraintstothesamelevel. (around80pagesforw=25,120pagesforw=12and160pages
Now, withalltheconstraintmatchesdetected, wecomparethe forw =0). Thisisthepointwheretheavailablememorysuffices
number oflocation updatesissued by the saferegion policy with toloadalluncertainconstraints.Theadaptivealgorithmswithwin-
thefrequency-basedpolicy. Thelocationupdatefrequencyofthe dow size25reducetheuncertainconstraints (by60%), therefore
frequency-based policy is adjusted (to around 1/s) such that all lessmemoryisrequiredtostorethem.Ifthepagenumberexceeds
thematchesaredetected(withlowerfrequency,somematchesare thisamount,thenumberofdiskaccessesequalsP CostI=O ,as
missed). Fig. 11(D) shows that, for all three constraint sets, the pu na(cid:127)(cid:16)ve
predicted by Theorem 2. The graph for the adaptive algorithms
safe region policy is the most cost effective in that it detects the
(w > 0) ismuchflatterthanthenon-adaptive one(w = 0)after
samenumberofmatcheswithonly10%ofthelocationupdatecost
thispoint. BasedonTheorem2,wecandeducethattheadaptive
ofthefrequency-basedpolicy. Thisisasignificantsavingthatcan
algorithmshavepartitionupdateratesthataremuchlower(about
directlytranslatetodollarvalue.
60% lower for w = 25 and 30% lower for w = 12) than the
non-adaptive algorithms. Therefore, the adaptive algorithms out-
performthenon-adaptiveones,alsoinI/Oincurringenvironments.
6Thesearchbacktrackslevel-by-levelupthek-d-treefromtheorig- Furthermore,inFig.12(B),wesuppressthesplittingprocessas
inalleavenodewheretheobjectislocateduntilthenodeisfound soon as the memory can hold all the uncertain constraints. This
thatcontainstheobject. Insertionproceedsdownfromthisnode, means,ifmemoryisavailable,thealgorithmdeliberatelyincreases
ratherthanfromtherootofthetree. Thisapproachavoidsthetop- theuncertainconstraintsinordertoreducethepartitionupdaterate
bottomscanofthetreeforalocationupdate.