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High-Dimensional Similarity Search
using Data-Sensitive Space
Partitioning ┼
Sachin Kulkarni1 and Ratko Orlandic2
1
Illinois Institute of Technology, Chicago
2 University of Illinois at Springfield
Database and Expert Systems Applications 2006
┼
Work supported by the NSF under grant no. IIS-0312266.
Outline
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Problem Definition
Existing Solutions
Our Goal
Design Principle
GardenHD Clustering and Γ Partitioning
System Architecture and Processes
Results
Conclusions
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Problem Definition
• Consider a database of addresses of clubs
• Typical queries are:
[d2]
– Find all the clubs within 35 miles of 10 West
31st. Street, Chicago.
– Find 5 nearest clubs
[d1]
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Problem Definition
• K-Nearest Neighbor (k-NN) Search:
– Given a database with N points and a query point q in
some metric space, find k  1 points closest to q. [1]
• Applications:
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Computational geometry
Geographic information systems (GIS)
Multimedia databases
Data mining
Etc.
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Challenge of k-NN Search
• In High-dimensional feature spaces indexing
structures face the problem of dead space (KDBTrees) or overlaps (R-tree).
• Volume and area grows exponentially with respect
to number of dimensions.
• Finding k-NN points is costly.
• Traditional access methods are at par with
sequential scan – “Curse of dimensionality”
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Existing Solutions
• Approximation and dimensionality reduction.
• Exact Nearest Neighbor Solutions
• R-tree
• SS-tree
• SR-tree
• VA-File
• A-tree
• iDistance
• Significant effort in finding the exact nearest
neighbors has yielded limited success.
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Goal
• Our goal:
– Scalability with respect to dimensionality
– Acceptable pre-processing (data-loading) time
– Ability to work on incremental loads of data.
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Our Solution
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•Clustering
•Space partitioning
•Indexing
0
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Design Principle
• “multi-dimensional data must be grouped
on storage in a way that minimizes the
extensions of storage clusters along all
relevant dimensions and achieves high
storage utilization”.
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What does it Imply?
• Storage organization must maximize the
densities of storage clusters
• Reduce their internal empty space
• Improve search performance even before the
retrieval process hits persistent storage
• For best results, employ a genuine clustering
algorithm
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Achieving the Principles
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Data space reduction:
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Detecting dense areas (dense cells) in the
space with minimum amounts of empty
space.
Data clustering:
–
Detecting the largest areas with the above
mentioned property, called data clusters.
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GardenHD Clustering
• Motivated by the stated principle.
• Efficiently and effectively separates disjoint
areas with points.
• Hybrid of cell- and density-based clustering that
operates in two phases.
• Recursive space partition -  partitioning.
• Merging of dense cells.
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 partitioning
1
 region1
 region3
 Region4
0
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• G no. generators
• D no. dimensions,
• No. regions = 1+(G–1)D
 subspace
• Space partition is
compactly represented by
a  filter (in memory).
 subspace
 region0
 region2
1
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Data-Sensitive Gamma Partition
• DSGP :– Data-Sensitive Gamma Partition
3
KDB-Trees
2
1
4
Effective
boundaries
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System Architecture
Data Clustering
“Data-Sensitive”
Space Partitioning
Incremental
Data Loading
Data Loading
Data Retrieval
Region Search
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Similarity Search
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Basic Processes
• Each region in space represented by
separate KDB-tree
– KDB-trees perform implicit slicing
• Initial and incremental loading of data
– Dynamic assignment of multi-dimensional data
to index pages
• Retrieval
– Region and k-nearest neighbor search
– Several stages of refinement
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Similarity Search - GammaNN
• Nearest neighbor search using GammaNN.
Region
Representatives
Query Point
Clipped
portions to
be queried
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Query Hyper-sphere
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Region Search
3
2
1
4
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Experimental Setup
• PC with 3.6 GHz CPU, 3GB RAM, and 280GB
disk.
• Page size was 8K bytes.
• Normalized D-dimensional space [0,1]D.
• The GammaNN implementations with and without
explicit clustering are referred to here as ‘data
aware’ and ‘data blind’ algorithms, respectively.
• Comparison with Sequential Scan and VA-File.
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Datasets
• Data:
– Synthetic data
• Up to 100 dimensions, 100,000 points.
• Distributed across 11 clusters—one in the center and 10 in
random corners of the space
– Real data
• 54-dimensional, 580,900 points, forest cover type
(“covtype”).
• Distributed across 11 different classes.
• UCI Machine learning repository.
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Metrics
• Pre-processing time
– Time of space partitioning, I/O and the time for
data loading (i.e., the construction of indices
plus insertion of data).
– For VA-File, only the time to generate the vector
approximation file.
• Performance
– Average page access for k-NN queries.
– Time to process k-NN queries.
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Experimental Results
Pre-processing time for the three algorithms
covtype data, 54 dim , 580900 points
160
Time in seconds
14 0
120
100
80
60
40
20
0
Data Aware
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Data Blind
VA-File
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Cumulative time for 100 queries
10 NN, synthetic data
Time in seconds
450
400
350
300
250
200
150
100
50
0
Sequential Scan
VA-File
Data Blind
Data Aware
10 20 30 40 50 60 70 80 90 100
Number of dimensions
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Avg page accesses
Performance Synthetic Data
Average page accesses for 100 queries
10 NN, synthetic data
3000
Sequential Scan
Data Blind
VA-File
Data Aware
2500
2000
1500
1000
500
0
10 20 30 40 50 60 70 80 90 100
Number of dimensions
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Cumulative time for 100 queries
10 NN, real data
Time in seconds
1200
1000
800
600
400
200
0
S. Scan
Data Blind
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VA-File
Data Aware
Avg page accesses
Performance Real Data
Average page accesses for 100 queries
10 NN, real data
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
S. Scan
Data Blind
VA-File Data Aware
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Progress with k in k-NN
1400
1200
1000
800
600
400
200
0
Data Blind
Sequential Scan
VA-File
Data Aware
1
10
100
Number of nearest neighbors
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Progress of page accesses with respect to
k for k-NN, real data
Avg page accesses
Time in seconds
Progress in time with respect to value
of k for k-NN, real data
10000
8000
6000
4000
2000
0
Sequential Scan
Data Blind
Data Aware
VA-File
1
10
100
Number of nearest neighbors
25
Incremental Load of Data
12
10
8
6
4
2
0
Data Aware
Incremental load
Data Aware
full load
200k 300k 400k 500k
Number of points
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Average page accesses vs number of points
Avg page accesses
Time In seconds
Cumulative time vs number of points
150
100
Data Aware
Incremental load
50
0
200k 300k 400k 500k
Data Aware
full load
Number of points
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Conclusions
• Comparison of the data-sensitive and data-blind
approach clearly highlights the importance of clustering
data on storage for efficient similarity search.
• Our approach can support exact similarity search while
accessing only a small fraction of data.
• The algorithm is very efficient in high dimensionalities
and performs better than sequential scan and the VAFile technique.
• The performance remains good even after incremental
loads of data without re-clustering.
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Current and Future Work
• Incorporate R-trees or A-trees in place of
KDB-trees.
• Provide facility for handling data with
missing values.
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References
1.
Fagin, R., Kumar, R., Shivakumar, D.: Efficient similarity search and classification
via rank aggregation, Proc. Proc. ACM SIGMOD Conf., (2003) 301-312
2.
Orlandic, R., Lukaszuk, J.: Efficient high-dimensional indexing by superimposing
space-partitioning schemes, Proc. 8th International Database Engineering &
Applications Symposium IDEAS’04, (2004) 257-264
3.
Orlandic, R., Lai, Y., Yee, W.G.: Clustering high-dimensional data using an
efficient and effective data space reduction, Proc. ACM Conference on Information
and Knowledge Management CIKM’05, (2005) 201-208
4.
Jagdish H. V., Ooi B. C., Tan K. L., Yu C., Zhang R., iDistance: An Adaptive B+Tree Based Indexing Method for Nearest Neighbor Search, ACM Transactions on
Database Systems, Vol. 30, No. 2, (2005): 364-395.
5.
Weber, R., Schek, H.J., Blott, S.: A quantitative analysis and performance study for
similarity search methods in high-dimensional spaces, Proc. 24th VLDB Conf.,
(1998) 194-205
6.
Sakurai, Y., Yoshikawa, M., Uemura, S., Kojima, H.: The A-tree: An index structure
for high-dimensional spaces using relative approximation, Proc. 26th VLDB Conf.,
(2000) 516-526
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Questions ?
[email protected]
http://cs.iit.edu/~egalite
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