ClusteringBased Similarity Search in Metric Spaces with Sparse Spatial Centers

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Clustering-Based Similarity Search in Metric
Spaces with Sparse Spatial Centers
SOFSEM 2008 Theory and Practice of Computer
Science
Nový Smokovec, Slovakia, January 2008

• Nieves Brisaboa, Oscar Pedreira, Diego Seco,
• Roberto Solar, Roberto Uribe
• Databases Lab, University of A Coruña, Spain
• Dept. of Computer Engineering, Universidad de
  Magallanes, Chile

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Outline

• Introduction
• Basic concepts
• Cluster center selection
• SSSTree
• Experimental results
• Conclusions and future work

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Introduction

• Traditional databases
• Data has a well defined structure (e.g.,
  VARCHAR(15))
• Exact searching, using equality/inequality
  comparisons
• SELECT name FROM Student WHERE city Poprad
• Non-structured databases
• Exact searching is not possible !
• Similarity search is a common operation in these
  domains
• Some examples

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Introduction

• Similarity search retrieve from the database the
  objects
• similar to one given as a query

r
q
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Outline

• Introduction
• Basic concepts
• Cluster center selection
• SSSTree
• Experimental results
• Conclusions and future work

v
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Basic concepts

• Distance functions (metrics)
• The evaluation of d has a high computational
  cost.
• The comparison of the query with all the objects
  in the database makes similarity search a costly
  operation
• Indexes are built on the database to avoid the
  comparison of the query with all the objects in
  the database.

d (latest, greatest) 3 g r e a t
e s t l a t e s t
d(
) 1.5435
,
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Basic concepts

• Metric space universe of objects Distance
  function
• E.g. Collection of words Edit distance
• The triangle inequality, base of indexing
  algorithms
• ? x, y, z ? U, d(x, y) ? d(x, z) d(z, y)

p
q
xi
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Basic concepts

• Pivot-based algorithms

Query
p1 p2 p3 pk-2 pk-1 pk
x1 x2 x3 xn
p1 p2 p3 pk-2 pk-1 pk
x1 x2 x3 xn
q
q
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Basic concepts

• Clustering-based algorithms

Indexing
Query
Index processing
q
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Basic concepts

• Clustering-based algorithms
• Bisector Tree.
• Generalized-hyperplane Tree.
• Geometric Near-neighbor Access Tree.
• Voronoi Tree.
• M-Tree.
• Spatial Approximation Tree.
• SSS Tree

• Pivot-based algorithms
• Burkhard-Keller Tree.
• Fixed-Queries Tree.
• Fixed-Queries Array.
• Vantage Point Tree.
• Multi-Vantage Point Tree.
• AESA.
• Linear AESA.

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Basic concepts
Example of clustering-based algorithm BST
v
v
v
v
?
?
v
Metric space
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Basic concepts

• GNAT is build and it is used in a similar way
• In this case each node has m childs
• And each node stores the distances among cluster
  centers, which are used during the search

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Motivation

• What are the problems of these algorithms?
• The number of clusters in each node has to be
  stated beforehand
• Cluster centers are selected at random
• The partition of the space is not the most
  adequate to the topology of the dataset

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Motivation

• SOLUTION?
• In each node, create a partition adapted to the
  topology of that subspace, both in the number of
  clusters as in the position of the cluster
  centers
• I.e., do NOT partition the space at random
• Select the appropriate number of cluster centers
• And place those centers in strategic positions

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Outline

• Introduction
• Basic concepts
• Cluster center selection
• SSSTree
• Experimental results
• Conclusions and future work

v
v
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Cluster center selection

• We want each cluster to be created in function of
  the topology and complexity of the space we are
  partitioning
• We need a selection strategy which
• Determines by itself the proper number of
  clusters
• Selects good cluster centers
• We use Sparse Spatial Selection Pedreira and
  Brisaboa, 07
• Initially selected for pivot-based algorithms
• It dynamic and adapts the selected points to the
  spaces characteristics

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Sparse Spatial Selection

• An object is selected as a center, if its
  distance to the other centers is greater than Ma
• Thus, we select only the centers necessary to
  cover the space

• M maximum distance
• 0 lt a lt 1

a 0.5
M
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Outline

• Introduction
• Basic concepts
• Cluster center selection
• SSSTree
• Experimental results
• Conclusions and future work

v
v
v
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SSSTree

• Clustering-based indexing algorithm
• Main differences with previous approaches
• The number of branches of each node is not fixed
• Each cluster associated to a node is partitioned
  in the necessary number of clusters, not in a
  fixed number
• Cluster centers are strategically selected

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SSSTree - Indexing

• SSS is applied in each node to select the cluster
  centers
• The index (a tree) is recursively build until
  reaching the leaves when the clusters are small
  enough

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SSSTree - Indexing

• PROBLEM
• SSS needs the maximum distance of the space (M)
  to select the cluster centers. But this distance
  is difference in each cluster
• Instead of computing it (brute force), we can get
  a good estimation two times the covering radius

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SSSTree - Search

• As with previous algorithms, we traverse the tree
  from root to leaves discarding clusters which do
  not contain any object in the result

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Outline

• Introduction
• Basic concepts
• Cluster center selection
• SSSTree
• Experimental results
• Conclusions and future work

v
v
v
v
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Experimental results

• Test datasets
• Synthetic collections of vectors
• Collections of 1,000,000 of vectors of dimension
  8,10,12, and 14, using the Euclidean distance.
  Gauss distribution.
• Collections of words (edit distance).
• 69.069 words from the English dictionary.
• 51.589 words from the Spanish dictionary.
• Collection of images
• 40,050 images from the NASA video and image
  archive

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Experimental results

• Search efficiency in collections of vectors

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Experimental results

• Search efficiency in collections of words

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Experimental results

• Search efficiency in collections of images

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Outline

• Introduction
• Basic concepts
• Cluster center selection
• SSSTree
• Experimental results
• Conclusions and future work

v
v
v
v
v
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Conclusions

• We present a new clustering-based indexing
  algorithm
• Adaptive
• Previous approaches need to state beforehand a
  fixed number of clusters in each node of the tree
• And sthe cluster centers were selected at random
• SSTree determines itself the optimal number of
  clusters and places them in strategies positions
• Efficient
• Experimental results show that SSSTree is more
  efficient than previous proposals in most of the
  situations

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And

• Thanks for your attention