Pivoting%20M-tree:%20A%20Metric%20Access%20Method%20for%20Efficient%20Similarity%20Search

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Pivoting M-tree A Metric Access Method for
Efficient Similarity Search

• Tomáš Skopaltomas.skopal_at_vsb.czDepartment of
  Computer Science, VŠB-Technical University of
  Ostrava

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Presentation Outline

• Similarity search in Metric Spaces
• M-tree
• PM-tree
• structure
• range queries
• hyper-ring storage
• Experimental Results

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Similarity search in Metric Spaces

• Similarity search methods for content-based
  retrieval in multimedia databases (in Information
  Retrieval resp.)
• Similarity modelled by metric d
• Restriction to metric yields a paradigmatic
  discrepancy with several similarity theories
  nevertheless, the triangular inequality is the
  basic tool for metric region construction leading
  to an efficient similarity search
• Metric queries
• range query (specified by pivot object Q and
  covering radius rQ)
• k-NN query (specified by pivot object Q and
  number of nearest neighbours k)

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Metric Access Methods

• Designed to search in metric datasets in order to
  keep the search costs minimal (number of distance
  computation).
• When searching large multimedia databases also
  the I/O search costs have to be minimized.
• Many MAMs developed so far M-tree, GH-tree,
  GNAT, LAESA, D-index, VP-tree, MVP-tree, SAT, ...
• Majority of the MAMs is not suitable for
  similarity search in large datasets (either a
  static method or high I/O search costs)
• only M-tree and (recently) D-index are suitable
  candidates

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M-tree

• dynamic, balanced, and paged metric tree (like
  e.g. B-tree, R-tree)
• the leaves are clusters of objects
• routing entries in the inner nodes represent
  metric regions, recursively bounding the object
  clusters in leaves
• during query evaluation, the triangular
  inequality allows discarding of irrelevant
  M-tree branches (metric regions resp.)

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PM-tree, motivation

• metric regions in M-tree are unnecessarily large
• ? indexing of large portions of empty space (the
  dead space)
• ? higher probability of intersection with query
  region
• ? less efficient search
• reduction of metric region volume should lead
  to more effective discarding of irrelevant
  subtrees
• the way is to specify a metric region bounding
  all the objects more tightly

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PM-tree, structure

• Pivoting M-tree (PM-tree) a combination of
  M-tree with the pivot-based methods (LAESA-like)
• given a fixed set of p pivots Pi (selected from
  the dataset), a PM-tree region is additionaly
  defined by p hyper-ring regions (Pi , HRi)
• each routing entry contains an array HR of p
  intervals ltHRi.min, HRi.maxgt
• each interval HRi bounds the distances of
  objects to the respective pivot Pi
• intersection of the hyper-sphere and the
  hyper-rings forms a smaller region bounding all
  the objects
• the more pivots, the more thightly bounded
  region

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PM-tree, query processing

• prior to processing of a query (Q,rQ), distances
  d(Q, Pi) for all i p must be computed
• metric region is relevant to a range query just
  in case that all the hyper-rings and the
  hyper-sphere intersect the range query region ?
  the more hyper-rings, the lower probability of
  intersection with query
• ? no additional distance computations are
  needed for the intersection test

M-tree region
PM-tree region
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PM-tree, hyper-ring storage

• The routing entries of PM-tree nodes are enlarged
  by the additional pivot-based information stored
  in HR arrays
• To keep the space overhead minimal, a compact
  storage of HRi intervals is necessary
• A distance histogram for each pivot Pi is
  created, and interval ltdimin, dimaxgt is chosen
  such that e.g. 90 of distances in the distance
  histogram fall into that interval
• Each value HRi.min, HRi.max, is scaled to the
  ltdimin, dimaxgt interval using a single byte,
  i.e. each hyper-ring HRi takes 2 bytes

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Experimental results (synthetic)

• synthetic dataset of 100,000 30-dimensional
  tuples distributed within 1000 clusters, L2
  distance, query selectivity 50 objs.

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Experimental results (images)

• collection of 10,000 images represented by
  256-dimensional vectors (gray histograms), L2
  distance, query selectivity 50 objs.

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Recent results (not included in proceedings)

• Cost models for range queries in PM-tree (?
  ADBIS04)
• Experiments on image dataset (? ADBIS04)
• Optimal k-NN query algorithm for PM-tree cost
  models (to be published...)

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Reference

• 1 Skopal T., Pokorný J., Snášel V. PM-tree
  Pivoting Metric Tree for Similarity Search in
  Multimedia Databases, submitted to ADBIS 2004,
  Budapest, Hungary
• 2 Skopal T. Pivoting M-tree A Metric Access
  Method for Efficient Similarity Search, DATESO
  2004, Desná
• 3 Skopal T., Pokorný J., Krátký M., Snášel V.
  Revisiting M-tree Building Principles. ADBIS
  2003, LNCS 2798, Springer-Verlag, Dresden,
  Germany