Clustering Geometric Data Streams

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ClusteringGeometric Data Streams

• Jirí Skála

Ivana Kolingerová
ZCU/FAV/KIV 2007
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Talk outline

• motivation
• background
• existing solution
• improvements
• experiments observations
• conclusion

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Motivation

• why data streams?
• geometric models growing larger
• Stanfords Michelangelo project (David 28 mil.
  vertices, St. Matthew 187 mil. vertices)
• 187106 points 3 coordinates 8 bytes 4.5 GB
• must be processed out-of-core
• why clustering?
• use hierarchical clustering to create
  multiresolution model
• various LOD in different parts

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Background data stream

• ordered set of data
• data coming online or stored on HDD
• too large to fit in main memory
• viewed only in order random access extremely
  inefficient or even impossible
• processed in one or very few linear scans

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Background clustering

• grouping similar elements together
• vertices, DB entries, documents
• similarity most often measured as Euclidean
  distance
• k-means, k-median clustering
• facility location
• clients and facilities
• facility cost

k-means
k-median
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Facility location

• no data streams yet
• introduced by Charikar and Guha, 1999
• initial solution iteratively refined by local
  improvements local search algorithm
• initial solution
• points taken in random order
• first point always a facility
• others become facility with probability p d /
  fc
• if d / fc gt 1 then p 1
• otherwise connect to closest existing facility

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Facility location

• pick a point at random (new facility candidate)
• compute function gain
• pay for opening a facility
• inspect all points and compare distance to
  facility
• inspect facilities and determine whether they can
  be closed
• if gain gt 0 then perform reassignments closures
• repeated m log m times?

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Facility location
New facility candidate
After reassignments closures
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Data stream clustering

• proposed by Guha et al., 2000
• data stream processed in blocks
• clustering within each block
• cluster centers given weight and passed to higher
  level
• when higher level full, clustered again
• distances multiplied by point weights

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Data stream clustering

• time for video

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Improvements

• limiting the search space
• inspect only points whose reassignment can
  improve the solution
• i.e., those assigned to facilities within 2 fc
  radius
• does not work for weighted points

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Improvements

• modification from k-median to facility location
• choosing the facility cost
• equal to the diagonal of bounding box
• weight normalization
• we need to keep weights around 1, i.e., average
  weight equal to 1
• divide weights by their average

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Experiments setting the facility cost

• high setting
• aggressive clustering
• low number of large clusters
• low setting
• moderate clustering
• many small clusters
• set facility cost equal to diagonal of bounding
  box
• affects memory and running time

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Experiments setting the facility cost
diagonal
2 diagonal
1/2 diagonal
1/4 diagonal
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Experiments input point distribution

• many authors rely on data being ordered
• usually true
• presented algorithm can handle unordered data
• as well
• there may be a problem
• with few outliers

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Experiments input point distribution
1st block
2nd block
3rd block
higher level
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Experiments input point distribution
1st block
2nd block
3rd block
higher level
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Experiments block size

• affects memory requirements
• required memory
• somewhat affects clustering result
• affects running time
• required iterations

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Experiments number of iterations

• m log m iterations necessary for a
  constant-factor approximation
• for large blocks running time grows unpleasantly
• 0.1 m iterations seem to be enough for data with
  clusters even less

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Experiments number of iterations

• 1640 points

6560 iterations
164 iterations
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Conclusion

• modified data stream approach to facility
  location
• introduced facility weight normalization
• improvement to limit the number of points
  inspected
• experiments
• discussion of parameter settings
• description of algorithm behavior

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References

• M. Charikar, S. Guha, Improved Combinatorial
  Algorithms for the Facility Location and k-Median
  Problems. Proc. 40th Sympos. on Foundations of
  Computer Science, 1999, pp. 378-- 388.
• S. Guha, N. Mishra, R. Motwani, L. OCallaghan,
  Clustering Data Streams. In Proceedings of the
  Annual Symposium on Foundations of Computer
  Science. IEEE, November 2000
• L. O'Callaghan, N. Mishra, A. Meyerson, S. Guha,
  R. Motwani, Streaming-Data Algorithms for
  High-Quality Clustering. In Proceedings of IEEE
  International Conference on Data Engineering,
  March 2002.
• S. Guha, A. Meyerson, N. Mishra, R. Motwani, L.
  O'Callaghan, Clustering data streams Theory and
  practice. IEEE Trans. Knowl. Data Eng 15, 3
  (2003), 515-528.