Nearest Neighbours Search using the PM-tree

1
Nearest Neighbours Search using the PM-tree

• Tomáš Skopal1 Jaroslav Pokorný1 Václav
  Snášel2

1 Charles University in Prague Department of
Software Engineering Czech Republic
2 VSB - Technical University of
OstravaDepartment of Computer Science Czech
Republic
2
Presentation Outline

• Similarity search in Metric Spaces
• M-tree
• the structure
• k-NN search
• PM-tree (an extension of M-tree)
• motivation
• the structure
• k-NN search
• Experimental Results

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Similarity search in Metric Spaces

• Similarity search
• methods for content-based retrieval in multimedia
  databases
• the similarity measure is often modelled by a
  metric d (satisfying triangular inequality,
  symmetry, reflexivity, non-negativity)
• similarity queries (query by example) realized as
  metric queries
• range query (Q , rQ) (specified by a query object
  Q and covering radius rQ)
• k-NN query (Q , k) (specified by a query object Q
  and number of nearest neighbours k)
• Metric Access Methods (MAMs)
• designed to search in metric datasets in order to
  keep the search costs minimal
• search costs number of distance computations
  I/O costs
• only distances between objects are used for
  indexing (the structure of object
  representation is not used for indexing)
• many MAMs are not suitable for similarity search
  in large datasets
• either a static method or high I/O search costs
• M-tree and (recently) D-index are the only
  suitable candidates so far

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M-tree (metric tree)

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

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k-NN search in the M-tree

• branch-and-bound algorithm (similar to that of
  R-tree)
• modification of range query algorithm, but the
  query radius rQ is dynamic
• rQ decreasing from infinity to the distance to
  the k-th neighbour
• utilized two structures priority queue PR and
  sorted array NN
• PR stores requests for nodes not-filtered from
  the search yet
• request of form routing entry to a node N,
  dmin(N), where dmin(N) is the lower bound
  distance from Q to all possible objects in N,
  i.e. dmin(rout. entry to N) max 0 , d(Q ,
  Oi) rOi
• where (Oi , rOi ) is region of the Ns routing
  entry (requests in PR sorted by dmin(N))
• NN stores k candidate objects (or distance upper
  bounds)
• at the end of algorithm run, NN contains the
  result, i.e. the k nearest neighbours
• entry of form candidate object Oi, d(Q,Oi) or
  - , dmax(N), where dmax() is the upper bound
  distance from Q to all possible objects in N, i.e
  
• dmax(rout. entry to N) d(Q , Oi) rOi
• PR stores only requests with dmin() lt dmax(),
  other requests are removed from PR
• i.e. such requests are removed, which do not
  overlap the dynamic query region (Q , rQ)
• Query processing the requests in PR are
  processed in FIFO manner ? a node N is retrieved,
  while PR and NN structures are updates by
  routing/ground entries of N
• PR is initialized to ( root , 8 ), NN is
  initialized by k entries -,8 to ( - ,8 , -
  ,8 , ... )
• optimal in I/O costs (the same I/O costs as range
  query (Q , d(Q , NN5) ) )

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k-NN search in M-tree example (k2)
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k-NN search in M-tree example (k2)
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k-NN search in M-tree example (k2)
5 nodes accessed, the same nodes accessed by
range query (Q , d(Q,O5) )
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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 question is how to specify a compact metric
  region bounding all the objects more tightly ?
  generalization of the M-tree for another metric
  region shape representations

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PM-tree region

• utilization of global pivots (inspired by
  LAESA-like methods)
• given a fixed set of p global pivots Pi
  (selected from (a part of) the dataset)
• p hyper-ring regions (Pi , HRi) are defined for
  each routing entry
• array HR of p intervals ltHRi.min , HRi.maxgt
• each interval HRi bounds the distances of
  objects to the respective pivot Pi
• PM-tree region M-tree region HR array
  (pivots Pi shared by all PM-tree regions)
• intersection of the hyper-sphere and the
  hyper-rings forms a smaller region bounding all
  the objects in leaves
• the more pivots, the more tightly bounded region
• PM-tree is built the same way as M-tree is
  built, i.e. the hyper-rings only cut off the
  M-tree sphere

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PM-tree, query processing

• distances d(Q , Pi) for all i p must be
  computed prior to processing a query
• metric region (Oi , rOi , HR) is relevant to
  (intersected by) a range query (Q , rQ) just in
  case that all the hyper-rings and the
  hyper-sphere overlap 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

Q
Q
M-tree region
PM-tree region
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k-NN search in the PM-tree

• 3 modifications of M-trees k-NN algorithm
• different intersection test between query region
  (Q, rQ) and PM-tree region (Oi , rOi , HR)
• ??t1..p d(Pt , Q) rQ HRt.max ? d(Pt , Q)
  rQ HRt.min
• different dmin construction ( possible distance
  increase to the farthest hyper-ring)
• dmin(rout. entry to N) max 0, d(Q , Oi) rOi
  , HRfarthest
• HRfarthest max??t1..p d(Pt , Q) HRt.max
  , HRt.min d(Pt , Q)
• different dmax construction ( possible distance
  decrease to the farthest object in the nearest
  hyper-ring)dmax(rout. entry to N) max d(Q ,
  Oi) rOi , HRnearest HRnearest min ?t1..p
  d(Q , Oi) HRt.max

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k-NN search in PM-tree example (k2)
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k-NN search in PM-tree example (k2)
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k-NN search in PM-tree example (k2)
5 nodes accessed, the same nodes accessed by
range query (Q , d(Q,O5) )
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Experimental Results (synthetic datasets)

• synthetic vector datasets (4D 60D) 100,000
  tuples 1000 clusters
• disk page sizes 1 KB 4 KB index sizes 4.5 MB
  55 MB

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Experimental Results(image database)

• WBIIS image database appr. 10,000 256D-vectors
  (gray histograms)
• disk page size 32 KB index sizes 16 MB 20 MB

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References

• 1 Skopal T., Pokorný J., Snášel V. PM-tree
  Pivoting Metric Tree for Similarity Search in
  Multimedia Databases, ADBIS 2004, Budapest,
  Hungary
• 2 Skopal T. Pivoting M-tree A Metric Access
  Method for Efficient Similarity Search, DATESO
  2004, Desná, Czech Republic
• 3 Skopal T., Pokorný J., Krátký M., Snášel V.
  Revisiting M-tree Building Principles. ADBIS
  2003, Dresden, Germany, LNCS 2798, Springer
• 4 Skopal T.
• Metric Indexing in Information Retrieval
• PhD thesis, VSB-Technical University of Ostrava
• http//urtax.ms.mff.cuni.cz/skopal/phd/thesis.pd
  f