M-Tree: An Efficient Access Method for Similarity Search in Metric Space

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M-Tree An Efficient Access Method for Similarity
Search in Metric Space

• Presenters
• Amool Gupta
• Amit Sharma

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MOTIVATION

• Basic problem that it addresses?(Why)
• Other techniques to solve same problem and how
  this one is step ahead?
• Basic Fundamentals of this Indexing structure

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Similarity Search Problem
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Similarity Searching

• Effectiveness
• - The way of formulating the similarity measures
  a model of human perception
• Efficiency
• - The way of achieving the required performance
  over huge volumes of data index structure

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Examples of Distance Functions

• Lp metric function( vectors)
• L1 Manhattan distance
• Euclidean Distance
• Linfinity
• Edit Distance (for String)
• Hausdorff distance
• Earth movers distance
• Quadratic form distance

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Metric Spaces-An abstraction of Similarity

• A metric space M (D,d) is a pair, where
• D is a domain (universe) of values, and
• d is a distance function that, ? x,y,z ? U,
  satisfies the metric axioms
• d(x,y) 0, d(x,y) 0 ? x y (positivity)
• d(x,y) d(y,x) (symmetry)
• d(x,y) d(x,z) d(z,y) (triangle inequality)
• All the distance functions seen in the previous
  examples are metrics, and so are the (weighted)
  Lp-norms
• The only distance seen so far that does not fit
  the metric framework is the DTW
• Metric indexes only use the metric axioms to
  organize objects, and exploit the triangle
  inequality to prune the search space

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• Limitations of SAMs
• SAMs are limited to indexing of DB Objects
  represented by means of feature values in
  Multi-dimensional vector space (we need more
  generic indexing strategy)
• Dissimilarity of object measured by Lp distance
  between feature values
• Assumes distance computation Trivial
• Limitations of Metric Tress
• Does not support dynamic database environment
• Reduces distance computations but Pays no
  attention to I/O costs

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What is a relative distance?

• OA AB OB
• AB OA OB
• AB relative position of B w.r.t A

B
A
O
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M-Tree

• Key ideas is to Some how reduce distance
  computation and at same time reduce I/O.
• M-Tree partition objects on the basis of their
  relative distance as measured by specific
  distance function and stores this objects into
  nodes.

r(Or)
P(Or)
Or
root
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M-Tree Structure

• Leaf Nodes stores all indexed db objects by
  their key or feature values.
• Internal Nodes Called routing nodes.
• Routing objects Or is associated with
• Or feature value of DB object
• Ptr(T(Or)) pointer to root of sub tree T(Or)
• r(Or) covering radius or maximum relative
  distance of objects in sub tree T(Or) from
  routing object Or
• d(Or , P(Or)) distance of routing object from
  its parent object P(Or)

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M-Tree Structure

• Leaf Node Entry for database object.
• Oj feature value of DB object.
• oid(Oj) object key
• d(Oj P(Oj)) distance of Oj from its parent P(Oj)

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Processing Queries

• Generally SAM try to prune tree for a given Query
  and main emphasis is on developing efficient
  pruning method which reduces no of disk access
  but once a tree is pruned it is required to
  compute distance of query point Q from each point
  in pruned tree.
• On the contrary emphasis of M- Tree is on pruning
  as well as to reduce computation of distance
  which is achieved by maximizing use of pre
  computed distance stored in nodes of M-Tree

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Range Query

• Given query point Q ,
• Maximum search distance r(Q)
• Range query range(Q, r(Q)) is all objects Oj such
  that d(Oj , Q ) lt r(Q)

r(Q)
Or is of our interest if intersection occurs How
To detect intersection using pre Computed
distances? If relative distance between Q and
Or is Less then sum of covering radii of two
Intersection is found.
Q
r(Or)
P(Or)
Or
root
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Range Query

• Leaf node

Object in leaf node is a solution to range Query
if it lies in its covering radii. We can again
use relative distance to Find weather object
lies in covering radii Or not
r(Q)
Q
P(Or)
Oj
root
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Algorithm for Range Queries
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K nearest neighbors queries

• Given query point Q ,
• An integer k gt 1
• k-NN is NN(Q,k) is k indexed objects which have
  shortest distance to Q

Q
Max Bound
Min Bound
r(Or)
P(Or)
Or
root
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SPLIT MANAGEMENT

• M-Tree grows bottom-up fashion
• Overflow of node N is managed by splitting N into
  two new nodes N and N(newly created)
• PARTITIONING Distributing entries are among N
  and N
• PROMOTE Two entries are promoted as routing
  objects and moved to parent level

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SPLIT MANAGEMENT

• If the split node is a leaf, then the covering
  radius of a promoted object, say Op1, is set to
• r(Op1) maxd(Oj,Op1 )Oj ? N1
• whereas if overflow occurs in an internal node
• r(Op1 ) maxd(Or,Op1) r(Or)Or ? N1

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SPLIT POLICIES

• Specific implementation of Promote and Partition
  method defines a split policy
• Ideal split policy should promote two objects and
  partition other objects so obtained regions have
• - Minimum volume
• - Minimum Overlap
• How it is different from SAM??

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PROMOTE Choosing Routing objects

• M_RAD minimum Radii sum
• mM_RAD minimizes maximum of two Radii
• M_LB_DIST maximum lower bound on distance
• RANDOM
• SAMPLING

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PARTITIONING-Distribution of Entries

• Generalized Hyperplane(Unbalanced split (why?))
  Assign each object Oj ?N to the nearest routing
  object
• if d(Oj,Op1 ) d(Oj,Op2 ) then
• assign Oj to N1, else
• assign Oj to N2.
• Balanced Compute d(Oj,Op1) and d(Oj,Op2 ) for
  all
• Oj ? N. Repeat until N is empty
• Assign to N1 the nearest neighbor of Op1 in N
  and remove it from N
• Assign to N2 the nearest neighbor of Op2 in N
  and remove it from N.

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Experimental Results

• Assumed constant node size
• Tested all split policies
• Results
• Balanced partition method has shown to put
  significant overhead and increased th I/O cost
• Fastest split policy observed to be RANDOM and
  slowest m_RAD
• Average volume covered per page(quality of tree
  construction) M_LB_DIST proved effective

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Experimental Results(2)
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I/O cost
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Avg Volume per page
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I/O cost
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I/O cost for M-Tree R-Tree
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• Thanks