Pankaj K. Agarwal , Alon Efrat , Swaminathan Sankararaman , and Wuzhou Zhang

1. PODS 2012 Nearest-Neighbor Searching Under Uncertainty Motivation 1. Squared Euclidean Distance 3. Euclidean Metric (Approximate) Approximate by Outside the grid: Inside the grid: : center of cell Total # of cells: : the centroid of Lemma: same as the weighted Voronoi diagram WVD • Data Location is imprecise… • Sensor databases • Face recognition • Mobile data What is the“nearest neighbor” of 𝑞 now? Model and Problem Statement Pankaj K. Agarwal, AlonEfrat, SwaminathanSankararaman, and Wuzhou Zhang Uncertain point : represented as a probability density function Expected distance: . Find the expected nearest neighbor (ENN) of : Or an -ENN : Cell size: 𝜀 Expected Voronoi cell Expected Voronoi diagram : induced by 2. Metric Size of : A linear size approximate ! Lower bound construction: Each uncertain point has two possible locations, each with probability 0.5. Assume is even: align uncertain points horizontally and vertically. Contribution Conclusion Firstnontrivial methods with theoretical guarantees • First nontrivial methods for ENN queries with theoretical guarantees • ENN is not a good indicator when the variance is large A near-linear size index exists despite size of EVD(𝒫) Open Problem • Linear-size index for most likely NN queries in sublinear time • Index for returning the probability distribution of NNs Linear! Acknowledgements P.A. and W.Z. are supported by NSF under grants CNS-05-40347, CCF-06 -35000, IIS-07-13498, and CCF-09-40671, by ARO grants W911NF-07-1-0376 and W911NF-08-1-0452, by an NIH grant 1P50-GM-08183-01, and by a grant from the U.S.-Israel Binational Science Foundation. A.E. and S.S. are supported by NSF CAREER Grant 0348000. is a piecewise linear function with pieces. Results in , extends to higher dimensions *“uncertain data, exact query”, the following results are under this setting.