Metric based KNN indexing

1
Metric based KNN indexing

• Lecturer Prof Ooi Beng Chin
• Presenters Frankie Chan HT00-3550Y
• Tan Zhenqiang HT01-6163J

2
Outline

• Introduction
• Examples
• SR-Tree
• MVP-Tree
• Future works
• Conclusion
• References

3
Introduction

• Metric based queries consider relative
  distance of object/point from a given query point
• Most commonly used metric is the Euclidean
  metric

4
Metric based queries

• Variations
• used with joins queries find 3 closest
  restaurants for each of 2 different theaters
• used with spatial queries find KNN to east of a
  location

5
SR-Tree

• The SR-tree An Indexing Structure for
  High-Dimensional Nearest Neighbor Queries.
• Norio Katayama Shinichi Satoh
• Multimedia Info. Research Div. Software Research
  Div.
• National Institute of Informatics National
  Institute of Informatics

6
SR-Tree Introduction

• SR-tree stands for Sphere/Rectangle-Tree
• SR-Tree is an extension of the R-tree and the
  SS-tree
• A region of the SR-tree is specified by the
  intersection of a bounding sphere and a bounding
  rectangle

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The R-Tree Structure
8
The SS-Tree Structure
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Definitions (SR-Tree)

• The diameter of a bounded region means
• diameter of a bounding sphere for the SS-Tree
• diagonal of a bounding rectangle for the
• R-Tree

10
Properties

• Bounding rectangles divides points into smaller
  volume regions. But tends to have longer
  diameters than bounding spheres, especially in
  high-dimensional space.
• Bounding spheres divides points into
  short-diameter regions. But tends to have larger
  volumes than bounding rectangles.

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Properties

• SR-Tree combined the use of bounding sphere and
  bounding rectangle, as the properties are
  complementary to each other.

12
The SR-Tree Structure
13
Bounded regions
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Indexing Structure

• The structure of the leaf L
• The structure of the node N

15
Insertion Algorithm

• SR-tree insertion algorithm is based on
• SS-trees centroid-based algorithm
• Descend down the tree and choose a
• subtree with the centroid nearest to
• the new entry
• SR-tree algorithm updates both
• bounding spheres (diff. from SS-tree)
• and bounding rectangles (same as
• R-tree)

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Insertion Algorithm

• Bounding sphere computation
• Center, X (X1 ,X2 ,..,XD)
• Radius, r

17
Deletion Algorithm

• SR-tree deletion algorithm is similar to
• that of the R-tree
• If entry deletion do not cause
• leaf/node under-utilisation, then just
• remove it
• Otherwise, remove under-utilised
• leaf/node and reinsert all orphaned
• entries

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Nearest Neighbour Search

• Algorithm ordered depth-first search
• It finds a number of points nearest to the
• query, to make a candidate set
• Then it revises the candidate set, when it
• visits every leaf whose region overlaps the
• range of the candidate set
• After it visited all leave, the final candidate
  is
• the search result

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Definition (NN search)

• Minimum Distance (MINDIST) euclidean distance
  from query point to the bounded region
• Minimax Distance (MINMAXDIST) minimum value of
  all the maximum distances between the query point
  and points on each n axes respectively

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Definition (NN search)
21
Search pruning

• Region R1 with MINDIST greater than MINMAXDIST of
  another region R2 is discarded, because it cannot
  contain NN (downward pruning)
• Actual distance from query point P to a given
  object O which is greater than MINMAXDIST of a
  region, is discarded (upward pruning)
• Region with MINDIST greater than the actual
  distance from query point P to an object O, is
  discarded (upward pruning)

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Recursive procedure (leaf node)

• If Node.type LEAF then
• For I 1 to Node.count
• dist objectDist(Pt,Node.region)
• if (dist lt Nearest.dist) then
• Nearest.dist dist
• Nearest.region Node.region

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Recursive procedure (non-leaf node)

• Else / non-leaf node gt order, sort, prune
  visit node /
• genBranchList(Pt,Node,branchList)
• sortBranchList(branchList)
• / perform downward pruning /
• last pruneBranch(Node,Pt,NearestBranchList)
• For I 1 to last
• newNode Node.branch
• nearestNeighborSearch(newNode,Pt,Nearest)
• / perform upward pruning /
• last pruneBranchList(Node,Pt,Nearest,
• branchList)

24
Performance Analysis - Insertion

• Insertion cost of R-tree, SS-tree and SR-tree
  (uniform dataset)

25
Performance Analysis - Query

• Performance of VAMSplit R-tree, SS-tree and
  SR-tree
• (Uniform dataset)

26
Performance Analysis - Query

• Performance of VAMSplit R-tree, SS-tree and
  SR-tree
• (Real dataset)

27
SR-tree average volume diameter

• Average volume diameter of the leaf-level
  regions of R-tree, SS-tree and SR-tree (real
  dataset)

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Strengths

• SR-Tree divides points into regions with small
  volumes and short diameter.
• Division of points into smaller regions improves
  disjointness.
• Smaller volume and diameter enhances the nearest
  neighbour queries performance.

29
Weaknesses

• SR-tree suffers from the fanout problem
  (branching factor max node entries)
• The node size grows as dimensionality increases
• The reduction of the fanout may requires more
  nodes to be read on queries
• Possibly affect query performance

30
MVP-Tree

• Indexing Large Metric Spaces For Similarity
  Search Queries
• Tolga Bozkaya
  Meral Ozsoyoglu
• Oracle corporation
  Dept. Comp Eng Sci
• 
  Case Western Reserve Univ.

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Outline

• Main idea.
• Algorithm basis
• How to build mvp-tree based on the given data.
• How to do similarity search in mvp-tree.
• Performance analysis and comparison based on
  experiments.

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Main Idea

• Triangle Inequality. Distance based Indexing
• Adopt more vantage points and levels
• Pre-computed distances are kept in leaf node.
• Use pre-computed distances to prune query
  branches.

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Algorithm Basis

• vp1,vp2,vp3 are vantage points.
• Q is given query point
• p1 belongs to points set
• The more vantage points the more unnecessary
  query branches are pruned.
• The distance between two vantage points is
  normally the larger the better

34
How to construct mvp-tree(m,k,p)

• 1) If S 0 then create an empty tree and
  quit.
• 2) If S k then Create leaf node L, put all
  data to L, Quit.
• 3) Choose first vantage point Svp1, Keep
  distances in arrays.
• 4) Divide S into m groups with same cardinalities
  based on the distances between Svp1 to points in
  S. And keeps distances as well.
• 5) For the first v above group
• 5.1) Choose last point in previous group as
  new vantage point.
• 5.2) Divide the group into m sub-groups with
  same cardinalities based on the distances
  between Svp1 to points in S. And keeps distances
  as well.
• 6) Recursively create mvp-tree on the mv
  sub-groups based on the steps from 1) to 5).

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Example

• Sv1first vantage point(level 1)
• Sv2vantange points(level 2)
• D1..kthe distances between
• data points in leaf node and
• vantage points
• x.Pathpthe distances between
• the data point and the first p vantage points
  along the path from the root to the leaf node
  that keeps it

36
How to do similarity search

• Depth-first process. Q is the given query object.
  r is the distance.
• 1) For i1 to m
• If d(Q, Svi) r then Svi is in the answer
  set.
• (Svi is the ith vantage points in current
  node.)
• 2) If current node is leaf node
• For all data points ( Si) in the node,
• If for all vantage points Sv,
• d(Q, Sv) - r d(Si, Sv) d(Q,
  Sv) r holds, and
• for all i1..p ( PATHi - r
  Si.PATHi PATHi r ) holds,
• then compute d(Q, Si). If d(Q, Si) r,
  then Si is in the answer set.
• 3) If the current node is an internal node
• for all i1..m
• if d(Q, Svi) r Mi then
  recursively search the first branch
• (Mi is the maximum of the distances
  between Svi to those points in its child node)

37
Comparison Experiment

• Performance results for the queries with the data
  set where data points form several physical
  clusters

38
Performance Analysis

• Experiments to compare mvp-trees with vp-trees
  but use only one vantage point at each level, and
  do not make use of the pre-computed distances
  show that mvp-tree outperforms the vp-tree by 20
  to 80 for varying query ranges and different
  distance distributions.
• For small query ranges, experiments with
  Euclidean vectors showed that mvp-trees require
  40 to 80 less distance computation compared to
  vp-trees.
• For higher query ranges, the percentagewise
  difference decrease gradually, yet mvp-trees
  still perform better, making up to 30 less
  distance computations for the largest query
  ranges used.
• Experiments on gray-level images using the data
  set with 1151 images show mvp-trees performed up
  to 20-30 less distance computations.

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Strengths

• Based on the thoughts of triangle inequality,
  more vantage points more unnecessary query
  branches pruned, the longer distances among
  vantage points the better and reusing
  pre-computed distances as much as possible.
• Mvp-tree is flatter than vp-tree.
• It is also balanced because of the way it is
  constructed.
• Experiments show that it is more efficient than
  vp- tree and M-tree.

40
Weaknesses

• 1.Construction cost O(nlogmn) distances
  computations
• 2.Additional storage cost are very high. There
  must be an array of size of p in every data point
  in leaf node.
• 3.Updating and inserting data points maybe
  lead to reconstruction of the mvp-tree.
• 4.If the insertions cause the tree structure
  to be skewed (that is, the additions of new data
  points change the distance distribution of the
  whole data set), global restructuring may have to
  be done, possibly during off hours of operation.
• 5.As the mvp-tree is created from an initial
  set of data objects in a top-down fashion, it is
  a rather static index structure.

41
Conclusion

• metric based indexing can be effective for high
  dimensional and non-uniform datasets (eg.
  Image/video similarity indexing)
• future work
• algorithm to perform in both dynamic static
  database environment
• analyse the use of metric with other attributes
  to enable range queries

42
References

• N. Katayama and S. Satoh. The SR-tree An
  Indexing Structure for High-Dimensional Nearest
  Neighbor Queries. Proc. ACM SIGMOD Int. Conf. on
  Management of Data, pages 369--380, Tucson,
  Arizona, 1997.
• R. Kurniawati, J. S. Jin, and J. A. Shepherd. The
  SS -tree An Improved Index Structure for
  Similarity Searches in a High-Dimensional Feature
  Space. SPIE Storage and Retrieval for Image and
  Video Databases V , pages 110--120, 1997.
• N. Beckmann, H.P. Kriegel, R. Schneider, and B.
  Seeger. The Rtree An Efcient and Robust Access
  Method for Points and Rectangles. In Proc. ACM
  SIGMOD Intl. Symp. on the Management of Data,
  pages 322--331, 1990.
• N. Roussopoulosi, S. Kelley, and F. Vincent.
  Nearest Neighbor Queries. Proc. ACM SIGMOD, San
  Jose, USA, pages 71-79, May 1995.
• Tolga Bozkaya and Meral Ozsoyoglu. Indexing Large
  Metric Spaces For Similarity Search Queries.
  Association for Computing Machinery transactions
  on Database System, pages 1-34, 1999.
• Roberto Figueira Santos Filho, Agma Traina,
  Caetano Traina Jr and Christos Faloutsos.
  Similarity Search Without Tears The OMNI-Family
  Of All-Purpose Access Methods. ICDE 2001.