A Spatial Index Structure for High Dimensional Point Data

1
A Spatial Index Structure for High Dimensional
Point Data
PK tree

• Wei Wang, Jiong Yang, and Richard Muntz
• Data Mining Lab
• Department of Computer Science
• University of California, Los Angeles

2
Outline

• Introduction
• Structure of PK-tree
• Operations on PK-tree
• Performance
• Conclusions

3
Introduction

• Dynamic spatial index method has been an active
  research area.
• index structure based on spatial decomposition
• PR-Quad tree, K-D tree, K-D-B tree, ...
• No overlapping among sibling nodes
• How to achieve high disk page utilization for
  large dimensionality with skewed data
  distributions remains a challenge.
• R-tree family of index structure
• R-tree, SR-tree, X-tree, ...
• Increasing of overlapping among sibling nodes
  along with increasing dimensionality degrades
  performance severely.

4
Introduction

• PK-tree
• Spatial decomposition
• no overlapping among sibling nodes
• Bound on height
• Bounds on number of children
• Uniqueness for any data set
• independent of order of insertion and deletion
• Solid theoretical foundation
• Fast retrieval and updates

5
Structure of PK-tree

• Recursively rectilinear dividing space

Set notation (e.g., ?, ?, ?, ?, ?, ?, ?) is used
to express relationships among cells.
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Structure of PK-tree

• Space is recursively divided until a level LD
  such that each cell contains at most one point.

7
Structure of PK-tree

• Point cell a non-empty cell at level LD
• A cell C is K-instantiable iff
• C is a point cell, or
• there does not exist (K-1) or less K-instantiable
  sub-cells C1, , CK-1 ? C, such that ?d ? D (d ?
  C ? d ? ?i0K-1 Ci).

8
Structure of PK-tree
Example of a PK-tree of rank 3
9
Structure of PK-tree

• Given a finite set of points D over index space
  C0 and dividing ration R, a PK-tree of rank K
  (Kgt1) is defined as follows.
• The cell at level 0 (C0) is always instantiated
  and serves as the root of the PK-tree.
• Every node else (except the root) in the PK-tree
  is mapped one-to-one to a K-instantiable cell.
• For any two nodes C1 and C2 in the PK-tree, C1 is
  a child of C2 (or C2 is the parent of C1) iff
• C1 is a proper sub-cell of C2, i.e., C1 ? C2, and
• there does not exist C3 in the PK-tree such that
  C1 ? C3 and C3 ? C2.
• Properties existence and uniqueness, bounds on
  node outdegree, bounded storage space, bounds on
  expected height, no overlapping among sibling
  nodes, and so on.

10
Properties of PK-tree
Expected Height of a PK-tree
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Properties of PK-tree

• M-Level Clustering Spatial Distribution
• 0-level uniform distribution over C0 P(d ?
  Ci1 d ? Ci) 1/r
• 1-level Let A ? C0 be some subset of C0 and Ac
  C0 - A. Distributions for points in A and Ac are
  0-level clustering spatial distribution.

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Operations on PK-tree

• Pagination of the PK-tree
• Pick the parameter K and the number of dimensions
  to split at each level such that the maximum size
  node is close to a page size.
• Allocate one node to a page.
• Space utilization can be guaranteed to be at
  least 50 and is much more than 50 in
  experiments.
• Insertion
• First follow the path from the root to locate all
  (potential) ancestors of the inserted leaf cell.
• Then from the leaf level back to the root along
  the same path to make all necessary changes
  (e.g., instantiate or de-instantiate cells).
• Search
• K Nearest Neighbor Query
• Range Query

13
Performance

• Setup Sparc 10 workstation (SunOS 5.5) with 208
  MB main memory and a local disk with 9GB capacity
• Synthetic Data Sets (each contains 100,000
  points)
• u uniform distribution
• c1, c2 20 of data are uniformly distributed and
  80 of data are distributed in disjoint clusters
• Height of generated trees

14
Performance

• Size of index in MB with 100,000 points

15
Performance

• Range query on uniform data distribution

16
Performance

• Range query on clustered data distribution

17
Performance

• KNN query on uniform data distribution

18
Performance

• KNN query on clustered data distribution

19
Performance

• Real data set NASA Sky Telescope Data
• 200,000 two-dimensional points (they are the
  coordinates of crater locations on the surface of
  Mars)

20
Conclusions

• PK-tree employing spatial decomposition to
  ensure no overlapping among sibling nodes but
  avoiding large number of nodes usually resulting
  from a skewed spatial distribution of objects.
• The total number of nodes in a PK-tree is O(N)
  and the expected height of a PK-tree is O(logN)
  under some general conditions.
• Other properties uniqueness, bounds on number of
  children.
• Empirical studies shown that the PK-tree
  outperforms SR-tree and X-tree by a wide margin.