Cheng, Xie, Yiu, Chen, Sun UV-diagram: a Voronoi Diagram for uncertain data 26th IEEE International...

1Cheng, Xie, Yiu, Chen, Sun

UV-diagram: a Voronoi Diagram

for uncertain data

26th IEEE International Conference on Data Engineering

Reynold Cheng (University of Hong Kong)Xike Xie (University of Hong Kong)Man Lung Yiu (Hong Kong Polytechnic University)Jinchuan Chen (Renmin University of China)Liwen Sun (University of Hong Kong)


Voronoi Diagram

http://www.crowddynamics.co.uk/images/Personal%20Space.jpghttp://www.ics.uci.edu/~eppstein/vorpic.html

http://www.ics.uci.edu/~eppstein/junkyard/nn.html

http://www.ics.uci.edu/~eppstein/


Voronoi Diagram

• Aggregate Query in Sensor Network [Shahabi06a]

• Spatial Skyline Query [Shahabi06b]

• Reverse Nearest Neighbor Query [Yiu07]

• Common Influence Join [Yiu08]

• Uncertain Data Clustering [Kao08]

4

Location Uncertainty

x

y

probabilitydensity function

U

[TDRP98,ISSD99,VLDB04]


UV-diagram(Uncertain Voronoi Diagram)

O1

O3

O7

O2O4

O5

O6

(a)

q

(b)

O1

O3

O7

O2

O5

O6

q2

O4

q1

(a)Voronoi Diagram. (b) UV-Diagram.

6

Probabilistic Nearest Neighbor Query [cheng04]

INPUT1. A query point called q2. A set of n objects

O1,O2,…, On with uncertainty regions and pdfs

OUTPUT A set of (Oi,pi) tuples

– pi is the non-zero probability (qualification probability) that Oi is the nearest neighbor of q

O2

qq

ff

O1

O3

O4

O5

O6


Agenda

• Introduction– Basic Concepts

• Voronoi Diagram in Spatial Database Management• Data Uncertainty

– Applications of UV-diagram

• UV-diagram– Basic concepts of UV-diagram

UV-edge, UV-cell, possible region, outer region…

– ConstructionInitial region construction, I- and C- level pruning, UV-index construction

– Results

• Conclusion• Future work


UV-Diagram: an example

Exponential number of UV-partitions can be generated! UV-cell

Partition AnswerObjects

R1 O1

R2 O2

R3 O3

R4

O1, O2, O3

O2

O1

O3

R2 R1

R3

R7

R4

R6

R5R5

R6

R7

O1, O2

O1, O3

O2, O3

O3

O2

O1

q


UV-cell

Partition AnswerObjects

R1 O1

R2 O2

R3 O3

R4

O1, O2, O3

O2

O1

O3

R2 R1

R3

R7

R4

R6

R5R5

R6

R7

O1, O2

O1, O3

O2, O3

O3

O2

O1

q

•We can use 3 UV-cells to represent 7 UV-partitions.

•The number of UV-cells equals to the number of objects.

10

)( jX i

Cheng, Xie, Yiu, Chen, Sun

Shape of a UV-cell

Bisector

Bisector

Oi Oj

)( jX i

Outer Region of Oi w.r.t Oj

)( jX iInner Region of Oi w.r.t Oj

)( jX iUV-cell is the intersection of inner regions of Oi w.r.t. all other objects

11

O2

O1

O3

R2 R1

R3

R7

R4

R6

R5

O3

O2

O1

q

Basic Method

Example: constructing U1


12

Basic Method

O1

O2

O3

)2(1X

)3(1X

)3()2( 111 XXU

Example: constructing U1


n-1 inner region has to be constructed!

Pruning techniques

Evaluating Ui requires expensive numerical calculations

Reference objects

Candidate Reference objects


UV-diagram(Uncertain Voronoi Diagram)

(a)Voronoi Diagram. (b) UV-Diagram.

O1

O3

O7

O2

O5

O6

O4


Efficient Construction

ii PU Index Level Pruning

Computational Level Pruning

Refinement

Candidate ReferenceObjects Ci

Reference Objects Fi

UV-index Construction

Initial Possible Region ConstructionPossible Region Pi

ii CF

Index level Pruning

Computational level Pruning

Refinement

Index level Pruning

Computational level Pruning


Step 1: Generating a Possible Region

OiOiOi


Step 1: Generating a Possible Region

Oi

d1

d2


Step 2,3: I- and C- Pruning

d

2d-ri

OiOi

d1

d2

v1

v2

O7

O1

O2O3

O4

O5

O8

O6


Step 4. UV-index Construction

• Splitting Condition• Overlap Checking• PNN Query

disk page

leaf node

root

non-leaf node

O3

O2

O1

O3

O2

O1

U1U2 U1 U2

U3U1 U2U1

U1 U3U3U2U1 U3U2U1

U2U3


Experiment Setup

Uncertain Dataset Synthetic: 10k – 80k (30k def)

Real dataset: 17k, 30k, 36k

Uncertainty pdf Gaussian (represented by 20 histogram bars)


Query Performance (ms)


Query Time’s Break-down (Tq)


Query Performance (I/O)


Construction Time


Pruning Ratio


Real Dataset


Conclusion

• We propose UV-diagram, which is a variant of Voronoi Diagram for uncertain data.

• We introduce the concepts of UV-cell and reference objects to efficiently construct UV-diagram.

• We also propose an adaptive index for the UV-diagram.


Future Work

• Use UV-diagram to support various types of queries- Continuous query, imprecise NN query,

reverse NN query, etc.

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THANKS!

• Q & A

• More discussions are welcome in the poster session!

28

Contact: Xike Xie [email protected] of Computer ScienceThe University of Hong Kong


Reference• [shahabi06a] Mehdi Sharifzadeh, Cyrus Shahabi: The Spatial Skyline Queries. VLDB 2006: 751-762• [Shahabi06b] Sharifzadeh, Mehdi and Shahabi, Cyrus: Utilizing Voronoi Cells of Location Data Streams for

Accurate Computation of Aggregate Functions in Sensor Networks. Geoinformatica. 2006• [Kao08] Clustering Uncertain Data using Voronoi Diagrams: Ben Kao; Sau Dan Lee; David Cheung; Wai-Shing

Ho; K. F. chan. ICDM 2008 • [Yiu07] Yiu, Man Lung and Mamoulis, Nikos. Reverse Nearest Neighbors Search in Ad Hoc Subspaces. TKDE

2007 • [Yiu08] M. L. Yiu, N. Mamoulis, and P. Karras. Common Influence Join: A Natural Join Operation for Spatial

Pointsets. In ICDE 2008. • [Zheng06] B. Zheng, J. Xu, W.-C. Lee, and L. Lee, “Grid-partition index: a hybrid method for nearest-neighbor

queries in wireless location-based services,” VLDB J., vol. 15, no. 1, pp. 21–39, 2006. • [cheng04] R. Cheng, D. V. Kalashnikov, and S. Prabhakar, “Querying imprecisedata in moving object

environments,” TKDE, vol. 16, no. 9, 2004.• [TDRP98] P. A. Sistla, O. Wolfson, S. Chamberlain, and S. Dao,“Querying the uncertain position of moving

objects,” in Temporal Databases: Research and Practice, 1998.• [ICDCS07] S. Ganguly, M. Garofalakis, R. Rastogi, and K. Sabnani, “Streaming algorithms for robust, real-time

detection of ddos attacks,” in ICDCS, 2007.• [VLDB04a] A. Deshpande, C. Guestrin, S. Madden, J. Hellerstein, and W. Hong, “Model-driven data acquisition in

sensor networks,” in Proc. VLDB, 2004• [Jooyandeh09] M. Jooyandeh, A. Mohades, and M. Mirzakhah, “Uncertain voronoi diagram,” Inf. Process. Lett.,

vol. 109, no. 13, pp. 709–712, 2009.• [Sember08] J. Sember and W. Evans, “Guaranteed voronoi diagrams of uncertain sites,” in CCCG, 2008.

Cheng, Xie, Yiu, Chen, Sun UV-diagram: a Voronoi Diagram for uncertain data 26th IEEE International...

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