University of Minnesota 1 Exploiting Page-Level Upper Bound (PLUB) for Multi-Type Nearest Neighbor...

1University of Minnesota

Exploiting Page-Level Upper Bound (PLUB) for Multi-Type Nearest Neighbor

(MTNN) Queries

Xiaobin Ma

Advisor: Shashi Shekhar

Dec, 2005

2 University of Minnesota

Outline

Motivation Problem statement Related work and our contributions Proposed algorithm and cost model Experiment design and results Conclusion and future work


Motivation

GIS applications Find shortest path Through one point from each of different feature types


A Running Example

Three feature types:

red(g), green(g), black(b)

q is query point

Route with solid red line is shortest route

Routes with dashed lines are other possible routes

q


Basic Concepts

<P1,P2,…,Pk> ordered point sequence and P1,P2,…,Pk are from k

different (feature) types of data sets R(q, P1,P2,…,Pk)

a route from q through points P1,P2,…, and Pk d(R(q, P1,P2,…,Pk))

distance of route R(q, P1,P2,…,Pk) Multi-Type Nearest Neighbor (MTNN)

ordered point sequence <P1’,P2

’,…,Pk’> such that

d(R(q,P1’,P2

’,…,Pk’)) is minimum among all possible

routes d(R(q, P1

’,P2’,…,Pk

’)) is MTNN distance MTNN query

A query finding MTNN


Problem Statement for MTNN Query

Given: A query point Distance metric k different (feature) types of spatial objects with data

points numbers N1, N2, N3, … ,Nk respectively R-tree for each data set

Find: Multi-type nearest neighbor (MTNN) Objective: Minimize length of route from query

point covering an instance of each feature Constraint:

Correctness: The tour should be the shortest path for the query point and the given collection of spatial query feature types

Completeness: Only the shortest path is returned as the query result


Related Work

Optimal sequence route (OSR) query [Kolahdozan et. al. Tech 05-840 USC]

Optimal algorithms (RLORD) Focus on optimal algorithms for specified

permutation of feature types Point-based algorithms

Trip plan query (TPQ) [Li et. al. SSTD 05] Heuristic algorithms

Give approximate results


RLORD Example

q is query point

Search order is <r, b, g>

R(q,r2,b2, g2) is greedy route

Radius of circle is d(R(q,r2,b2,g2))

q

b2b15

b12b1

g2

g10

g12g13

g1

g6

g8

g11

g3

g9g14

g1g5

b6 b13

b17

b10

b5

b8b9

b3

b14

b4

b11g16

g7g4

r2

r9

r10

r11

r14r13

r7

r4

r5

r6

r3

r12

r1

r8r15


RLORD Running Iterations

Use backward search strategy O=<g,b,r> First iteration - examine feature type g

<g2>, <g3>, <g4>, <g5>,<g7>,<g9>, <g10>, <g12>, <g13>, <g14>, <g15>, <g16> in a set R

Second iteration - examine next feature type in O For every point bi in black set,

iterate on every partial route <gj>in R: IF d(R(q, bi)) + d(R(bi,gj)) < d(R(q,r2,b2,g2)) THEN put <bi,gj> into a set R1

keep ordered sequence <bi,gj> in R1 such that d(R(bi,gj)) + d(R(gj)) is minimum

<b1,g13>, <b2,g2>, <b3,g3>, <b4,g3>, <b6,g14>, <b7,g14>, <b11,g3>, <b12,g13>, <b13,g14>, <b14,g3>, <b15,g13> in a set R2

R <- R2 Examine next feature type and repeat above procedure

until all types of data are examined


Our Contributions

Formalized a new nearest neighbor search problem – Multi-Type Nearest Neighbor (MTNN) query problem

Proposed a new algorithm, i.e., Page Level Upper Bound (PLUB) based algorithm

Evaluated the proposed algorithm via cost model and experiment


Key Ideas of PLUB

Prune search space at page level Create candidate leaf page sequences Search candidate MTNN in these candidate leaf

page sequences


Page Level Upper Bound (PLUB) Algorithm

Step 1: First upper bound search Use basic R-tree based nearest neighbor search

algorithm to find an initial upper bound as current upper bound, using greedy strategy

Step 2: R-Tree search Prune search space with current upper bound and form

a set of leaf node candidate sequences, using page level pruning approach

Step 3: Subset search Search candidate MTNN in leaf node candidate

sequences Go to step 2 until going thought all permutation of

feature types, using candidate MTNN distance as current upper bound


B1

G1

R2

R1

B2

B4

RLUB – An Example

q

b2b15

b12b1

g2

g10

g12g13

g1

g6

g8

g11

g3

g9g14

g1g5

b6 b13

b17

b10

b5

b8b9

b3

b14

b4

b11g16

g7g4

r2

r9

r10

r11r14

r8

r15

r13

r7

r4

r5

r6

r3

r12

r1

Inputs q: query point Euclidean distance R-tree for each feature

B3

G2

G3

G4

R3

R4

R(q,r2,b2,g2) is greedy route Radius of circle is d(R(q,r2,b2,g2)) = 3.37 Rectangles are leaf pages in R-trees


B1

G1

R2

R1

B2

B4

RLUB – An Example

q

b2b15

b12b1

g2

g10

g12g13

g1

g6

g8

g11

g3

g9g14

g1g5

b6 b13

b17

b10

b5

b8b9

b3

b14

b4

b11g16

g7g4

r2

r9

r10

r11r14

r8

r15

r13

r7

r4

r5

r6

r3

r12

r1

B3

G2

G3

G4

R3

R4


UB

E?

R1 B1 G1 2.04 N

R1 B1 G3 6.2 Y

R1 B1 G4 4.27 Y

R1 B3 G1 7.53 Y

R1 B3 G3 6.54 Y

R1 B3 G4 4.29 Y

R1 B4 G1 4.02 Y

R2 B1 3.7 Y

R2 B3 G4 3.43 Y

R2 B4 5.17 Y

R4 B1 4.08 Y

R4 B3 7.94 Y

R4 B4 7.56 Y

Leaf page upper bound calculation (current search bound 3.37)

Only leaf node sequence <R1,B1,G1> left


B1

G1

R2

R1

B2

B4

RLUB – An Example

q

b2b15

b12b1

g2

g10

g12g13

g1

g6

g8

g11

g3

g9g14

g1g5

b6 b13

b17

b10

b5

b8b9

b3

b14

b4

b11g16

g7g4

r2

r9

r10

r11r14

r8

r15

r13

r7

r4

r5

r6

r3

r12

r1

B3

G2

G3

G4

R3

R4


Search candidate MTNN in <R1,B1,G1>(time unit p-p)

1st iteration <g2><g10><g12>

<g13> Time 4

2nd iteration <b12,g13,><b1,g13>

<b2,g2><b15,g13> Time 4x4+4=20

3rd iteration <r10,b15,g13,><r9,b15,g1

3><r2,b2,g2> <r11,b1,g13>

Time 4x4+4=20 Output

Shortest distance route R(q,r11,b1,g13) and distance value 3.16


Running Results of RLORD

First iteration (time unit p-p) <g2>, <g3>, <g4>, <g5>,<g7>,<g9>, <g10>, <g12>, <g13>,

<g14>, <g15>, <g16> Time 11

Second iteration <b1,g13>, <b2,g2>, <b3,g3>, <b4,g3>, <b6,g14>, <b7,g14>,

<b11,g3>, <b12,g13>, <b13,g14>, <b14,g3>, <b15,g13> Time 11x12+12=144

Third iteration <r1,b11,g3>, <r2,b2,g2>, <r3,b11,g3>, <r8,b1,g13>, <r9,b15,g13>,

<r10,b15,g13>, <r11,b1,g13>, <r12,b11,g3>, <r13,b1,g13>, <r14,b1,g13>, <r15,b1,g13>

Time 12x11+11=143 R(q,r11,b1,g13) is shortest among all routes

Shortest distance value 3.16


Running Time Comparison Table

R-R: rectangle to rectangle distance P-P: point to point distance

R-R P-P

PLUB 17 44

RLORD 0 298

RLORD has no R-R distance calculation, but has much more P-P calculation

Cost of R-R < 2 x cost of P-P


Cost Model for PLUB (For One Permutation)

CR-T + CLF + CPN CR-T : cost of R-tree traversal to find all R-tree leaf

nodes intersected by the circle with radius of current upper bound, centered at query point q

CLF : cost of page level leaf node search for R-tree candidate leaf node sequences

CPN : cost of point level search for candidate MTNN in candidate leaf node sequences


CR-T Model of PLUB

CR-T : R-tree traversal cost CPR :cost of point to rectangle distance calculation N t,i : number of all the tree nodes visited in feature

type i tree traversal

CR-T = CPR x Σ N t,i (i= 1, …, k)


CLF Model of PLUB

CLF: search of R-tree candidate leaf node sequences

NR-R : Number of leaf nodes visited in candidate leaf node sequences search

CR-R : cost of rectangle to rectangle distance calculation

CLF = NR-R x CR-R


CPN Model of PLUB

CPN : search MTNN in candidate leaf node sequences FLS : leaf node candidate sequence filtering ability ratio nl : average point number in leaf node for all feature types

pi : page number of feature type i CP-P :cost of point to point distance calculation

Cls : cost of search MTNN in single leaf node sequence Cls = CP-P x (nl +(nl x nl) + nl + (nl x nl) + … + nl + (nl x nl)

(k-1 items) = (k-1) (nl x (nl +1)) x CPP

CPN = Cls x Π pi x (1- FLS) i = 1,…,k


Cost Model for R-Lord (For One Permutation)

CR-T‘+ CPS CR-T‘: cost of R-tree based coarse pruning, i.e. find all

data points inside initial upper bound CR-T‘ = CR-T + CP-P x nl x (p1+ p2 +p3 +…+ pk-1+ pk ) CPS : cost of candidate MTNN search in remaining

subsets CP-P :cost of point to point distance calculation CPS = CP-P x nl x (p1 + nl x p1xp2 + (p2+ nl x p2xp3 )+ …

+ (pk-1+ nl x pk-1 x pk )


Cost Model Summary of PLUB and RLORD( one permutation)

In random or approximate random datasets, FLS is not big enough, PLUB takes more time.

In clustered datasets, FLS tends to be very big. When 1-FLS <(nl x (p1 + nl x p1xp2 +(p2+ nl x p2xp3 )+…+ (pk-1+ nl x pk-1 x pk ))) /((k-1)

nl x (nl +1) x Π pi )

PLUB runs faster than RLORD For clustered datasets, it becomes true when clusters becomes

more compact Left side: remaining ratio (r-ratio) Right side: comparison ratio (c-ratio)

General Form Approximate Form

PLUB CR-T + CLF + CPN CP-P x (k-1) nl x (nl +1) x Π pi x (1- FLS)

RLORD CR-T‘+ CPS

CP-P x nl x (p1 + nl x p1xp2 + (p2+ nl x p2xp3 ) + … + (pk-1+ nl x pk-1 x pk )


Experiment Design


Synthetic Data Sets Generation

Randomly generate cluster center in rectangle with bottom-left (0,0) and top-right point (10000,10000)

Constraint: the minimum distance between two cluster centers is minCCDist

Around every cluster center, generate cluster member points

Maximum distance from member point to cluster center is ClusterSize

Simplified maximum cluster center distance is determined by:

maxCCDist = 10000.0/(int)(sqrt(CN)+1) Thus minimum cluster center distance when generating

cluster center is as follows: minCCDist = BCF x maxCCDist

Then the cluster size is: ClusterSize = ICF x minCCDist


Experiment Parameters

Feature Types:2-7 Between-cluster Compactness Factor (BCF):

0.1-1.0 In-cluster Compactness Factor (ICF):0.1-0.5 Cluster Number(CN):20,50,100,200


Synthetic Datasets Example

BCF=0.5,ICF=0.5,CN=20,Feature Type=2

BCF=0.5,ICF=0.3,CN=20,Feature Type=2


Experiment Setup & Data Sets

Setup C / Pentium-IV 3.2GHz / Linux / 1GB Memory / Synthetic

data

Synthetic data Scalability test in terms feature types Effect of data sets density Effect of Between-cluster compactness factor Effect of In-cluster compactness factor


Scalability Test

Parameters Fixed:

BCF=0.1, ICF = 0.1, CN=20

Variable: feature types (2-7)

Trend PLUB is much

faster when number of features is high


Effect of Data Sets Density

Parameters Fixed: FT = 7,

BCF=0.1, ICF=0.5

Variable: cluster number (20,50,100,200)

Trend PLUB is always

faster than RLORD for all densities of data sets


Effect of Between-cluster Compactness Factor


ICF=0.3,CN=50, Variable: BCF

(0.1-1.0)



Top: execution time v.s. BCF

Trend PLUB is faster

than RLORD when BCF is less than 0.7

PLUB is slower than RLORD when BCF is bigger than 0.7



Bottom: Remaining ratio (r-ratio) and comparison ratio (c-ratio) v.s. BCF

Trend Ratios increase as

BCF increase Remaining ratio is

less than comparison ratio when BCF is less than 0.8



Contradiction? Remaining ratio

increases, which means the pruning ratio decreases, the execution time decreases

when BCF increases, there are less leaf nodes intersected with current search bound. Thus the total possible candidate leaf node sequences decrease dramatically



Key information when remaining

ratio is less than comparison ratio, PLUB runs faster

when remaining ratio is greater than comparison ratio, PLUB takes more time than RLORD.


Effect of In-cluster Compactness Factor


BCF=0.1,CN=50,

Variable: ICF (0.1-0.5)

Trend PLUB is always

faster than RLORD for ICF from 0.1 to 0.5


Conclusion and Future Work

Formalized MTNN query problem Proposed PLUB based algorithm for MTNN

query Compared PLUB and RLORD

Design heuristic algorithms to tackle MTNN query problem in large number of feature types


References

[1] M. Kolahdouzan, M. Sharifzadeh and C. Shahabi. The Optimal Sequenced Route Query. IN USC, CS Dept, Tech. Report 05-840, 2005

[2] Feifei Li, Dihan Cheng, Marios Hadjieleftherious, George Kollios and Shang-Hua Teng. On Trip Planning Queries in Spatial Databases. SSTD 2005.

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