1. Nearest Neighbor Searching Under Uncertainty
Wuzhou Zhang
Supervised by Pankaj K. Agarwal
Department of Computer Science
Duke University

2. Nearest Neighbor Searching (NNS)
Applications
Pattern Recognition, Data Compression
Statistical Classification, Clustering
Databases, Information Retrieval
Computer Vision, etc.

3. Nearest Neighbor Searching Under Uncertainty
Discrete pdf
Continuous pdf

4. Nearest Neighbor In Expectation
_________

5. Bisector In Case Of Gaussian
For Gaussian distribution, bisector is a line!
Hard to get explicit formula!
Figure:

6. Squared Distance Function
bisector is simple and beautiful!
In case of discrete pdf, bisector is also a line! In both cases, compute the Voronoi diagram, solve it optimally! However, not a metric !

7. Sampling Continuous Distributions
Sometimes working on continuous distributions is hard….
Lower bounds on other metrics and distributions are also possible….
Let’s focus on discrete pdf then….

8. Expected Nearest Neighbor In L1 Metric (Manhattan metric)

9. Expected Nearest Neighbor In L1 Metric ( cont. )
Source: Range Searching on Uncertain Data [P.K.Agarwal et al. 2009]

10. Geometric Reduction

11. Building Block: Half-Space Intersection and Convex Hulls
Upper hulls correspond to lower envelopes, an example in 2D
Source: page 252 – 253, Computational Geometry: Algorithms and Applications, 3rd Edition[Mark de Berg et al. ]

12. Segment-tree Based Data Structures for Expected-NN In L1 Metric

13. Segment-tree Based Data Structures for Expected-NN In L1 Metric ( cont

14. Segment-tree Based Data Structures for Expected-NN In L1 Metric ( cont
Size of data structure
Preprocessing time
Query time
Summary of the result

15. Approximate L2 Metric
It’s a metric when P is centrally symmetric!

16. Approximate L2 Metric ( cont. )
More complex!

17. Future Work Approximate the expected NN in L2 metric
Work harder
in the near future!
Approximate the expected NN in L2 metric
Study the complexity of expected Voronoi diagram
Study the probability case

18. Questions? Thanks! Main References:
[1] Pankaj K. Agarwal, Siu-Wing Cheng, Yufei Tao, Ke Yi: Indexing uncertain data. PODS 2009:
[2] Pankaj K. Agarwal, Lars Arge, Jeff Erickson: Indexing Moving Points. J. Comput. Syst. Sci. 66(1): (2003)
Questions?