Nearest-Neighbor Searching Under UncertaintyWuzhou Zhang

Department of Computer Science, Duke University

PODS, May 23, 2012

Joint work with Pankaj K. Agarwal, Alon Efrat, and Swaminathan Sankararaman

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Nearest-Neighbor Searching

ApplicationsDatabases, Information RetrievalStatistical Classification, ClusteringPattern Recognition, Data CompressionComputer Vision, etc.

𝑆

𝑝∗

a set of points in

any query point in

Find the closest point to

𝑞

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Voronoi Diagram

Voronoi cell: Voronoi diagram : decomposition induced by

Preprocessing time

Space

Query time

𝑝𝑖

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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:

. Find the expected nearest neighbor (ENN) of :

Or an -ENN : 𝑞 𝑄

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Previous work Uncertain data

ENN• The ENN under metric: ε-approximation [Ljosa2007]• No bounds on the running time

Most likely NN• Heuristics [Cheng2008, Kriegel2007, Cheng2004, etc]

Uncertain queryENN• Discrete uniform distribution: both exact and O(1)

factor approximation [Li2011, Sharifzadeh2010, etc] • No bounds on the running time

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Our contribution

Distance

function

Settings Preprocessing time Space Query time

Squared Euclidean distance

Uncertain data

Uncertain query

metric

Uncertain data

Uncertain query

Euclidean metric(-ENN)

Uncertain data

Uncertain query

Results in , extends to higher dimensions

First nontrivial methods for ENN queries with provable performance guarantees !

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Expected Voronoi cell

Expected Voronoi diagram : induced by

An example in metric

Expected Voronoi Diagram

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: the centroid of

Lemma:

• same as the weighted Voronoi diagram WVD

Squared Euclidean distanceUncertain data

Preprocessing time

Space Query time

Remarks: Works for any distribution

�̂�𝜎 2

𝑃∈𝒫 𝑞

Ed (𝑃 ,𝑞)|∨𝑞−�̂� ||2

�̂�𝜎 2

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metricUncertain data Size of : Lower bound construction

the inverse Ackermann function Remarks: Extends to metric

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

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:

Remarks: Extends to any metric

8 Ed (𝑃 , �̂�)/ 𝜀�̂�

Cell size: 𝜀

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

A linear size approximate !

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Preprocessing time

Space Query time

𝑔𝑃 1

𝑔𝑃 2

𝑞

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Conclusion and future work Conclusion

First nontrivial methods for answering exact or approximate ENN queries with provable performance guarantees

ENN is not a good indicator when the variance is large Future work

Linear-size index for most likely NN queries in sublinear time Index for returning the probability distribution of NNs

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

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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