ExternalMemoryGeometricDataStructures LarsArge DukeUniversity - - PowerPoint PPT Presentation

ExternalMemoryGeometricDataStructures

LarsArge DukeUniversity

June29,2002

SummerSchoolonMassiveDatasets

LarsArge Externalmemorydatastructures 2

SoFarSoGood

• Yesterdaywediscussed “dimension1.5”problems:

– Intervalstabbing andpointlocation

• Wedevelopedanumberofusefultools/techniques

– Logarithmicmethod – Weight-balancedB-trees – Globalrebuilding

• On Thursdaywealsodiscussedseveraltools/techniques

– B-trees – PersistentB-trees – Constructionusingbuffertechnique

LarsArge Externalmemorydatastructures 3

• MaintainN intervalswithuniqueendpointsdynamicallysuchthat

stabbingquerywithpointxcanbeansweredefficiently

• Solvedusingexternalintervaltree
• Weobtainedthesameboundsasforthe1d case

– Space:O(N/B) – Query: – Updates:I/Os

IntervalManagement

x

) (log N O

B

) (log

B T B N

O +

LarsArge Externalmemorydatastructures 4

IntervalManagement

• Externalintervaltree:

– Fan-outweight-balancedB-tree onendpoints – IntervalsstoredinO(B)secondarystructureineachinternalnode – Queryefficiencyusingfiltering – Bootstrapping usedtoavoidO(B)searchcostineachnode * SizeO(B2)underflowstructureineachnode * ConstructedusingsweepandpersistentB-tree * Dynamicusingglobalrebuilding

$m$blocks

v

) ( B Θ

v ) ( B Θ

LarsArge Externalmemorydatastructures 5

3-SidedRangeSearching

• Intervalmanagementcorrespondstosimpleformof2d rangesearch
• Moregeneralproblem:Dynamic 3-sidederangesearching

– Maintainsetofpointsinplanesuch thatgivenquery(q1, q2,q3),allpoints (x,y)withq1 ≤ x ≤ q2 andy ≥ q3 can befoundefficiently

(x,x) (x1,x2) x x1 x2 q3 q2 q1

LarsArge Externalmemorydatastructures 6

3-SidedRangeSearching:StaticSolution

• Construction:Sweep top-downinsertingx inpersistentB-treeat(x,y)

– O(N/B)space – I/Oconstructionusingbuffertechnique

• Query(q1, q2,q3):Performrangequerywith[q1,q2]inB-treeatq3

– I/Os

• Dynamicusinglogarithmicmethod

– Insert: – Query:

• Improveto?Deletes?

q3 q2 q1

) (log

B T B N

O + ) (log2 N O

B

) log ( N O

B B N

) (log2

B T B N

O + ) (log N O

B

LarsArge Externalmemorydatastructures 7

• Basetreeonx-coordinates withnodesaugmentedwithpoints
• Heapony-coordinates

– Decreasing yvaluesonroot-leafpath – (x,y)onpathfromroottoleafholdingx – Ifv holdspointthenparent(v)holdspoint

InternalPrioritySearchTree

9 16.20 16 19,9 13 13,3 19 20,3 4 5,6 5 9,4 1 1,2 20 19 16 13 9 5 4 4,1 1

LarsArge Externalmemorydatastructures 8

• Linearspace
• Insertof(x,y)(assumingfixedx-coordinateset):

– Compareywithy-coordinateinroot – Smaller:Recursivelyinsert (x,y)insubtree onpathtox – Bigger:Insertinrootandrecursivelyinsertoldpointinsubtree O(log N)update

InternalPrioritySearchTree

9 16.20 16 19,9 13 13,3 19 20,3 4 5,6 5 9,4 1 1,2 20 19 16 13 9 5 4 4,1 1

Insert(10,21)

10,21

LarsArge Externalmemorydatastructures 9

InternalPrioritySearchTree

• Query with(q1, q2,q3)startingatrootv:

– Reportpointinv ifsatisfyingquery – Visitbothchildrenofv ifpointreported – Alwaysvisitchild(s)ofv onpath(s)toq1 and q2 O(log N+T)query

9 16.20 16 19,9 13 13,3 19 20,3 4 5,6 5 9,4 1 1,2 20 19 16 13 9 5 4 4,1 1

4 19 4

LarsArge Externalmemorydatastructures 10

• Naturalidea:Blocktree
• Problem:

– I/Ostofollowpathstoto q1 and q2 – But O(T)I/Osmaybeusedtovisitothernodes(“overshooting”)

• query

ExternalizingPrioritySearchTree

9 16.20 16 19,9 13 13,3 19 20,3 4 5,6 5 9,4 1 1,2 20 19 16 13 9 5 4 4,1 1

) (log N O

B

) (log T N O

B

+

LarsArge Externalmemorydatastructures 11

ExternalizingPrioritySearchTree

• Solutionidea:

– StoreB pointsineachnode * O(B2)pointsstoredineachsupernode * B outputpointscanpayfor“overshooting” – Bootstrapping: * StoreO(B2)pointsineachsupernode instaticstructure

9 16.20 16 19,9 13 13,3 19 20,3 4 5,6 5 9,4 1 1,2 20 19 16 13 9 5 4 4,1 1

LarsArge Externalmemorydatastructures 12

ExternalPrioritySearchTree

• Basetree:Weight-balancedB-treeonx-coordinates(a,k=B)
• Pointsin“heaporder”:

– RootstoresB toppointsforeachofthechildslabs – Remainingpointsstoredrecursively

• Pointsineachnodestoredin“O(B2)-structure”

– PersistentB-treestructureforstaticproblem

• Linearspace

) (B Θ

) (B Θ

LarsArge Externalmemorydatastructures 13

ExternalPrioritySearchTree

• Query with(q1, q2,q3)startingatrootv:

– QueryO(B2)-structureandreportpointssatisfyingquery – Visitchildv if * v onpathtoq1 or q2 * Allpointscorrespondingtov satisfyquery

LarsArge Externalmemorydatastructures 14

ExternalPrioritySearchTree

• Analysis:

– I/Osusedtovisitnodev – nodesonpathtoq1 or q2 – Foreachnodev notonpathtoq1 or q2visited,Bpointsreported inparent(v)

• query

) 1 ( ) (log

2 B T B T B

v v

O B O + = + ) (log N O

B

) (log

B T B N

O +

LarsArge Externalmemorydatastructures 15

ExternalPrioritySearchTree

• Insert(x,y) (assumingfixedx-coordinateset– staticbasetree):

– Findrelevantnodev: * QueryO(B2)-structuretofind Bpointsinrootcorresponding tonodeu onpathtox * Ify smallerthany-coordinates

• fallB pointsthenrecursively

searchinu – Insert(x,y) inO(B2)-structureofv – IfO(B2)-structurecontains>B pointsforchildu,removelowest pointandinsertrecursivelyinu

• Delete:Similarly

u

LarsArge Externalmemorydatastructures 16

• Analysis:

– Queryvisitsnodes – O(B2)-structurequeried/updatedineachnode * Onequery * Oneinsertandonedelete

• O(B2)-structureanalysis:

– Query: – UpdateinO(1)I/Osusingupdate blockandglobalrebuilding

• I/Os

ExternalPrioritySearchTree

u

) (log N O

B

) 1 ( ) / (log

2

O B B B O

B

= + ) (log N O

B

LarsArge Externalmemorydatastructures 17

RemovingFixedx-coordinateSetAssumption

• Deletion:

– Deletepointaspreviously – Deletex-coordinatefrombase treeusingglobalrebuilding

• I/Osamortized
• Insertion:

– Insertx-coordinateinbasetree andrebalance(usingsplits) – Insertpointaspreviously

• Split:Boundaryinv becomesboundaryinparent(v)

) (log N O

B

v v’’ v’

LarsArge Externalmemorydatastructures 18

RemovingFixedx-coordinateSetAssumption

• Split:Whenv splitsB newpointsneededinparent(v)
• Onepointobtainedfromv’ (v’’)using“bubble-up”operation:

– Findtoppointp inv’ – Insertp inO(B2)-structure – Removep fromO(B2)-structureofv’ – Recursivelybubble-uppointtov

• Bubble-up inI/Os

– Followonepathfromv toleaf – UsesO(1)I/Oineachnode

• SplitinI/Os

v’’ v’

)) ( (log v w O

B

)) ( ( )) ( log ( v w O v w B O

B

=

LarsArge Externalmemorydatastructures 19

RemovingFixedx-coordinateSetAssumption

• O(1)amortizedsplitcost:

– Cost:O(w(v)) – Weightbalancedbasetree:insertsbelowvbetweensplits

• ExternalPrioritySearchTree

– Space:O(N/B) – Query: – Updates:I/Osamortized

• Amortizationcanberemovedfromupdateboundinseveralways

– Utilizinglazyrebuilding )) ( ( v w Ω ) (log N O

B

) (log

B T B N

O +

v’’ v’

LarsArge Externalmemorydatastructures 20

Summary:3-sidedRangeSearching

• 3-sidederangesearching

– Maintainsetofpointsinplanesuch thatgivenquery(q1, q2,q3),allpoints (x,y)withq1 ≤ x ≤ q2 andy ≥ q3 can befoundefficiently

• Weobtainedthesameboundsasforthe1dcase

– Space:O(N/B) – Query: – Updates:I/Os

q3 q2 q1

) (log

B T B N

O + ) (log N O

B

LarsArge Externalmemorydatastructures 21

Summary:3-sidedRangeSearching

• Mainproblemindesigningexternalpriority

searchtreewastheincreasedfanout in combinationwith“overshooting”

• Samegeneral solutiontechniquesasinintervaltree:

– Bootstrapping: * UseO(B2)sizestructureineachinternalnode * Constructedusingpersistence * Dynamicusingglobalrebuilding – Weight-balancedB-tree:Split/fuseinamortizedO(1) – Filtering:Chargepartofquerycosttooutput

q3 q2 q1

LarsArge Externalmemorydatastructures 22

Two-DimensionalRangeSearch

• Wehavenowdiscussedstructuresforspecialcases oftwo-

dimensionalrangesearching – Space:O(N/B) – Query: – Updates:

• Cannotbeobtainedforgeneral2d rangesearching:

– queryrequiresspace – spacerequiresquery

q3 q2 q1 q q q3 q2 q1 q4

) (

log log log N N B N

B B B

• )

(log N O

c B

) ( B

N

O ) (

B N

Ω ) (log N O

B

) (log

B T B N

O +

LarsArge Externalmemorydatastructures 23

• Basetree:Fan-outweightbalancedtreeonx-coordinates
• height
• Pointsbeloweachnodestoredin4linearspacesecondarystructures:

– “Right”prioritysearchtree – “Left”prioritysearchtree – B-tree ony-coordinates – Intervaltree

• space

ExternalRangeTree

) (

log log log N N

B B B

O ) (log N

B

Θ ) (

log log log N N B N

B B B

• )

(log N

B

Θ

LarsArge Externalmemorydatastructures 24

• Secondaryintervaltreestructure:

– Connectpointsineachslabiny-order – Projectobtainedsegmentsiny-axis – Intervalsstoredinintervaltree * Intervalaugmentedwithpointertocorrespondingpointsiny- coordinateB-treeincorrespondingchildnode

ExternalRangeTree

) (log N

B

Θ

LarsArge Externalmemorydatastructures 25

• Query with(q1, q2,q3,q4)answeredin topnodewithq1 and q2 in

differentslabsv1 andv2

• Pointsinslabv1

– Foundwith3-sidedqueryinv1 usingrightprioritysearchtree

• Pointsinslabv2

– Foundwith3-sidedqueryinv2 usingleftprioritysearchtree

• Pointsinslabsbetweenv1 andv2

– Answerstabbingquerywithq3 usingintervaltree firstpointaboveq3 ineachoftheslabs – Findpointsusingy-coordinateB-treeinslabs

ExternalRangeTree

) (log N O

B

) (log N O

B

) (log N

B

Θ v1 v2

LarsArge Externalmemorydatastructures 26

ExternalRangeTree

• Queryanalysis:

– I/Ostofindrelevantnode – I/Ostoanswertwo3-sidedqueries – I/Ostoqueryintervaltree – I/OstotraverseB-trees

• I/Os

) (log N O

B

) (log

B T B N

O + ) (log ) (log

log

N O N O

B B N B

B

= + ) (log

B T B N

O + ) (log N O

B

) (log

B T B N

O +

) (log N

B

Θ v1 v2

LarsArge Externalmemorydatastructures 27

ExternalRangeTree

• Insert:

– Insertx-coordinateinweight-balancedB-tree * SplitofvcanbeperformedinI/Os

• I/Os

– Updatesecondarystructuresinallnodeson one root-leafpath * Updateprioritysearchtrees * Updateintervaltree * UpdateB-tree

• I/Os
• Delete:

– Similarandusingglobalrebuilding ) (

log log log N N

B B B

O ) (

log log log2 N N

B B B

O )) ( log ) ( ( v w v w O

B

) (

log log log2 N N

B B B

O

) (log N

B

Θ v1 v2

LarsArge Externalmemorydatastructures 28

Summary:ExternalRangeTree

• 2d rangesearching inspace

– I/Oquery – I/Oupdate

• Optimal amongquerystructures

) (

log log log2 N N

B B B

O ) (log

B T B N

O + ) (

log log log N N B N

B B B

O ) (log

B T B N

O +

q3 q2 q1 q4

LarsArge Externalmemorydatastructures 29

kdB-tree

• kd-tree:

– Recursivesubdivisionofpoint-setintotwohalfusing vertical/horizontalline – Horizontallineonevenlevels,verticalonunevenlevels – Onepointineachleaf

• Linearspaceandlogarithmicheight

LarsArge Externalmemorydatastructures 30

kdB-tree

• Query:

– Recursivelyvisitnodecorrespondingtoregionsintersectedquery – Reportpointintrees/nodescompletelycontainedinquery

• Analysis:

– Numberofregionsintersectinghorizontallinesatisfyrecurrence Q(N)=2+2Q(N/4) Q(N)= – Queryintersectsregions ) ( N O ) ( ) ( 4 T N O T N O + = + ⋅

LarsArge Externalmemorydatastructures 31

kdB-tree

• KdB-tree:

– Blockingofkd-treebutwithB pointineachleaf

• Query asbefore

– AnalysisasbeforeexceptthateachregionnowcontainsB points

• I/Oquery

) (

B T B N

O +

LarsArge Externalmemorydatastructures 32

kdB-tree

• kdB-treecanbeconstructed inI/Os

– somewhatcomplicated

• Dynamicusinglogarithmicmethod:

– I/Oquery – I/Oupdate – O(N/B)space ) (

B T B N

O + ) (log2 N O

B

) log ( N O

B B N

LarsArge Externalmemorydatastructures 33

O-TreeStructure

• O-tree:

– B-treeonvertical slabs – B-treeonhorizontal slabsineachverticalslab – kdB-treeonpointsineachleaf

N

B B N

log

) log ( N

B B N

Θ ) log ( ) (

2 ) log (

2

N B

B N N

B B N

Θ = Θ

N

B B N 2

log

) log ( N

B B N

Θ

N B

B 2

log

LarsArge Externalmemorydatastructures 34

O-TreeQuery

• Performrangesearchwithq1 andq2 invertical B-tree

– QueryallkdB-trees inleavesoftwohorizontal B-treeswithx- intervalintersectedbutnotspannedbyquery – Performrangesearchwithq3 andq4 horizontal B-treeswithx- intervalspannedbyquery * QueryallkdB-trees withrangeintersectedbyquery

N

B B N

log

N

B B N 2

log

N B

B 2

log

LarsArge Externalmemorydatastructures 35

O-TreeQueryAnalysis

• Vertical B-treequery:
• QueryofallkdB-trees inleavesoftwohorizontal B-trees:
• Queryhorizontal B-trees:
• QuerykdB-trees notcompletelyinquery
• QueryinkdB-trees completely

containedinquery:

• I/Os

) ( )) log ( (log

B N B B N B

O N O = ) ( ) log ( ) log (

2 B T B N B T B B B N

O B N B O N O + = + ⋅ ) log ( N O

B B N

) ( )) log ( (log ) log (

B N B B N B B B N

O N O N O = ⋅ ) ( ) log ( ) log ( 2

2 B T B N B T B B B N

O B N B O N O + = + ⋅ ⋅ ) ( B

T

O ) (

B T B N

O + ) log ( 2 N O

B B N

⋅

LarsArge Externalmemorydatastructures 36

O-TreeUpdate

• Insert:

– Searchinvertical B-tree:I/Os – Searchinhorizontal B-tree:I/Os – InsertinkdB-tree:I/Os

• Useglobalrebuilding whenstructuresgrowtoobig/small

– B-treesnotcontainelements – kdB-treesnotcontainelements

• I/Os
• Deletes canbehandled

inI/Ossimilarly ) log ( N

B B N

Θ ) log (

2 N

B

B

Θ ) (log N O

B

) (log N O

B

) (log )) log ( (log

2 2

N O N B O

B B B

= ) (log N O

B

) (log N O

B

LarsArge Externalmemorydatastructures 37

Summary:O-Tree

• 2d rangesearching inlinearspace

– I/Oquery – I/Oupdate

• Optimal amongstructures

usinglinearspace

• Canbeextendedtoworkind-dimensions

withoptimalquerybound

q3 q2 q1 q4

) (log N O

B

) (

B T B N

O + ) ) ((

1

1 B T B N

d

O +

−

LarsArge Externalmemorydatastructures 38

Summary:3and4-sidedRangeSearch

• 3-sided2drangesearching:Externalprioritysearchtree

– query,space, update

• General(4-sided)2d rangesearching:

– Externalrangetree:query,space, update – O-tree:query,space, update

q3 q2 q1 q3 q2 q1 q4

) (

log log log N N B N

B B B

• )

(log

B T B N

O + ) ( B

N

O ) (

B T B N

+ Ω ) (log N O

B

) (log

B T B N

O + ) (log N O

B

) (

log log log2 N N

B B B

O ) ( B

N

O

LarsArge Externalmemorydatastructures 39

Techniques(onefinaltime)

• Tools:

– B-trees – PersistentB-trees – Buffertrees – Logarithmicmethod – Weight-balancedB-trees – Globalrebuilding

• Techniques:

– Bootstrapping – Filtering

q3 q2 q1 q3 q2 q1 q4 (x,x)

LarsArge Externalmemorydatastructures 40

Otherresults

• Manyotherresults fore.g.

– Higherdimensionalrangesearching – Rangecounting – Halfspace (andotherspecialcases)ofrangesearching – Structuresformovingobjects – Proximityqueries

• Manyheuristicstructures indatabasecommunity
• Implementationefforts:

– LEDA-SM(MPI) – TPIE(Duke)

LarsArge Externalmemorydatastructures 41

THEEND