Nearest-Neighbor Searching Under UncertaintyWuzhou Zhang

RIP Final and Masters, March 22, 2012

Joint work with Pankaj K. Agarwal, Alon Efrat, and Swaminathan Sankararaman. To appear in PODS 2012.

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Nearest-Neighbor Searching

ApplicationsPattern Recognition, Data CompressionStatistical Classification, ClusteringDatabases, Information RetrievalComputer Vision, etc.

http://en.wikipedia.org/wiki/Nearest_neighbor_search

𝑆𝑞

𝑝∗

a set of points in

any query point in

Find the closest point to

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Data Uncertainty Location of data is imprecise: Sensor databases, face recognition, mobile data, etc.

𝑞

What is the “nearest neighbor” of now?

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Our Model and Problem Statement Uncertain point : represented as a probability density function (pdf)

Expected distance:

: probabilities/weights: distance function

Let , find the expected nearest neighbor (ENN) of :

Or an -ENN :

0.1 0.2 0.3 0.4

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Previous work and Our contribution Previous work • The expected -NN under 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 -approximate ENN queries with provable performance guarantees

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Summary of resultsDistanc

e functio

n

Settings Preprocessing time Space Query time

Squared Euclidean distance

Uncertain data

Uncertain query

Rectilinear metric

Uncertain data

Uncertain query

Euclidean metric(-ENN)

Uncertain data

Uncertain query

Results in , extends to higher dimensions

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Voronoi Diagram

Voronoi cell: Voronoi diagram : decomposition induced by

Preprocessing time

Space

Query time

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Expected Voronoi Diagram

Expected Voronoi cell

Expected Voronoi diagram : induced by

An example in metric

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Minimization diagram

The lower envelope of :

: the projection of the graph of

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Squared Euclidean distanceUncertain data : the centroid of

Then,

• Replace by with weight • same as the power diagram WPD

Preprocessing time

Space Query time

Remarks: Works for any distribution

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Rectilinear metricUncertain data Assume metric: Size of : Lower bound construction

the inverse Ackermann function Remarks: Extends to metric

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Rectilinear metricUncertain data (cont.) A near-linear size index exists despite size of

linear pieces!

𝑝𝑖𝑗

− (𝑥𝑝 𝑖𝑗−𝑥𝑞)+(𝑦𝑝 𝑖𝑗

− 𝑦𝑞)

− (𝑥𝑝 𝑖𝑗−𝑥𝑞)− ( 𝑦𝑝𝑖𝑗

− 𝑦𝑞)

(𝑥𝑝𝑖𝑗−𝑥𝑞)+ (𝑦 𝑝𝑖𝑗

− 𝑦𝑞)

(𝑥𝑝𝑖𝑗−𝑥𝑞)− ( 𝑦𝑝 𝑖𝑗

−𝑦𝑞 )𝑝𝑖𝑗

Linear!

𝑃 𝑖

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Rectilinear metricUncertain data (cont.)

Preprocessing time

Space Query time

Remarks: Extends to higher dimensions

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Euclidean metric (-ENN)Uncertain data Approximate by

Outside the grid:

Inside the gird:

Total # of cells:: outermost square: the collection of squares

Remarks: Extends to any metric

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Euclidean metric (-ENN)Uncertain data (cont.)

and and : generated by Arya’s data structure on A linear size approximate !

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Quadtree: 4-way tree

Preprocessing time Space Query time

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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

THANKS

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Squared Euclidean distanceUncertain query

: the centroid of

Preprocessing• Compute the Voronoi diagram VD Query• Given , compute in , then query VD with

Preprocessing time

Space Query time

Remarks: Extends to higher dimensions and works for any distribution

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Rectilinear metricUncertain query Similarly, linear pieces

Preprocessing time

Space

Query time

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Euclidean metric (-ENN)Uncertain query

Preprocessing time

Space

Query time

Remarks: Extends to higher dimensions