Nearest-Neighbor Searching Under Uncertainty

Transcription

1 Nearest-Neighbor Searching Under Uncertainty Wuzhou Zhang Joint work with Pankaj K. Agarwal, Alon Efrat, and Swaminathan Sankararaman. To appear in PODS 2012.

2 Nearest-Neighbor Searching S: a set of n points in R 2 q: any query point in R 2 Find the closest point p S to q S q p Applications Pattern Recognition, Data Compression Statistical Classification, Clustering Databases, Information Retrieval Computer Vision, etc. 2

3 Data Uncertainty Location of data is imprecise: Sensor databases, face recognition, mobile data, etc. What is the nearest neighbor of q now? q 3

4 Our Model and Problem Statement Uncertain point P: represented as a probability density function (pdf) P p 1 p 2 p 3 p 4 w p Expected distance: Ed P, Q = w p w q d p, q. p P q Q w p, w q : probabilities/weights d, : distance function Let P = P 1,, P n, find the expected nearest neighbor (ENN) of Q: P = argmin Ed(P, Q). P P Or an ε-enn P P: Ed P, Q 1 + ε Ed P, Q. 4

5 Previous work and Our contribution Previous work The expected k-nn under L 1 metric: ε-approximation [Ljosa2007] Aggregate nearest neighbor (ANN) under the SUM function [Li2011, Sharifzadeh2010, Lian2008, etc] All based on heuristics Our contribution First nontrivial methods for answering exact or ε- approximate ENN queries with provable performance guarantees 5

6 Summary of results Distance function Squared Euclidean distance Rectilinear metric Euclidean metric (ε-enn) Settings Preprocessing time Space Query time Uncertain data Uncertain query Uncertain data Uncertain query Uncertain data Uncertain query O(n log n + nk) O(n) O(log n) O(n log n) O(n) O(log n + k) O(k 2 n log 3 n) O(k 2 n log 2 n) O(log 3 (kn)) O(n log 2 n) O(n log 2 n) O(k 2 log 3 n) O( n/ε 2 log 2 n log(n/ε) log 1/ε ) O( n/ε 2 log 2 (1/ε)) O(log n/ε ) O(n log n) O(n) O( k/ε 2 log(1/ε) log n) Results in R 2, extends to higher dimensions 6

7 Voronoi Diagram P = p 1,, p n Voronoi cell: Vor p i = x R d d(p i, x) d(p j, x)} Voronoi diagram VD(P): decomposition induced by Vor p i Preprocessing time O(n log n) Space O(n) Query time O(log n) 7

8 Expected Voronoi Diagram P = P 1,, P n Expected Voronoi cell EVor P i = x R d Ed(P i, x) Ed(P j, x)} Expected Voronoi diagram EVD(P): induced by EVor P i An example in L 1 metric 8

9 Minimization diagram F = f 1,, f n The lower envelope of F: L F x = min f i(x) 1 i n M(F): the projection of the graph of L F 9

10 Squared Euclidean distance Uncertain data p = σ 2 = Then, P = R d R d xf P x dx: the centroid of P P 1,, P n x p 2 f P x dx Ed P, q = ǁx pǁ 2 + σ 2. Replace P i P by p i with weight σ i 2 EVD P same as the power diagram WPD P, P = p 1,, p n Preprocessing time Space Query time O(n log n + nk) O(n) O(log n) 10 Remarks: Works for any distribution

11 Rectilinear metric Uncertain data Assume L 1 metric: d p, q = x p x q + y p y q Size of EVD(P): O(k 2 n 2 α(n)) Lower bound Ω(n 2 ) construction 11 α n : the inverse Ackermann function Remarks: Extends to L metric

12 Rectilinear metric Uncertain data (cont.) A near-linear size index exists despite Ω(n 2 ) size of EVD P P i x pij x q y pij y q p ij Linear! x pij x q y pij y q p ij x pij x q + y pij y q x pij x q + y pij y q k k Ed P i, q = j=1 w ij d p ij, q = j=1 w ij x pij x q + y pij y q O(k 2 ) linear pieces! 12

13 Rectilinear metric Uncertain data (cont.) Preprocessing time Space Query time O(k 2 n log 3 n) O(k 2 n log 2 n) O(log 3 (kn)) 13 Remarks: Extends to higher dimensions

14 Euclidean metric (ε-enn) Uncertain data Approximate Ed P, x by g(x) Outside the grid: g x = x p + Ed(P, p ) Inside the gird: g x = Ed P, a + cell size Total # of cells: O(1/ε 2 log(1/ε)) B: the collection of squares C: outermost square 14 Remarks: Extends to any L p metric

15 Euclidean metric (ε-enn) Uncertain data (cont.) Quadtree: 4-way tree in q = P i q C i } and out [q] = P i q C i } B in = B i and B out : generated by Arya s data structure on P = p 1,, p n A linear size approximate EVD(P)! Preprocessing time Space Query time O( n/ε 2 log 2 n log(n/ε) log 1/ε ) O( n/ε 2 log 2 (1/ε)) O(log n/ε ) 15

16 Further work Is there a linear-size index to answer the following queries in sublinear time in the worst case? the nearest neighbor with highest probability the nearest neighbors with probability higher than τ 16

17 Squared Euclidean distance Uncertain query P = p 1,, p n q: the centroid of Q argmin Ed(p, Q) = argmin p P p P ǁq pǁ 2 Preprocessing Compute the Voronoi diagram VD(P) Query Given Q, compute q in O k, then query VD P with q Preprocessing time Space Query time O(n log n) O(n) O(log n + k) 17 Remarks: Extends to higher dimensions and works for any distribution

18 Rectilinear metric Uncertain query Similarly, Ed Q, p : O(k 2 ) linear pieces Preprocessing time O(n log 2 n) Space O(n log 2 n) Query time O(k 2 log 3 n) 18

19 Euclidean metric (ε-enn) Uncertain query P = p 1,, p n Preprocessing time O(n log n) Space O(n) Query time O( k/ε 2 log(1/ε) log n) 19 Remarks: Extends to higher dimensions