Computing Compressed Multidimensional Skyline Cube Efficiently

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Computing Compressed Multidimensional Skyline
Cube Efficiently

• ICDE07

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Outline

• Introduction
• Problem Definition
• Stellar Algorithm
• Experimental Result
• Conclusion

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Introduction

• Suppose we want look for a vacation package
• Cheaper package
• Higher hotel-class
• Example

Package-ID Price Hotel-class
a 1600 4
b 2100 1
c 3000 5
Skyline We want to find a set of packages which
are NOT dominated by any other pacakges
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Problem Definition

• In many applications, user may be interested in
  not only the skyline in the full space , but also
  the skyline in various subspaces.
• Can we compute skyline groups and their decisive
  subspaces without searching all subspaces?

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Cont.

• C-group and skyline group
• In subspace B , a set of objects G forms a
  coincident group (or c-group for short) (G , B)
  if all objects in G have the same projection in
  B, i.e.
• We denote the common projection by GB
• Decisive Subspace
• For skyline group (G , B) a subspace
  is called decisive if
• (1) Gc is in the subspace skyline of
  C
• (2) For any object
  ,
• (3) There exists no proper subspace
• such that conditions (1) and (2)
  also hold

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Cont.

• Object e has value 1 in dimension Y which enables
  e as a skyline object in spaces Y and XY.
• No other object shares the same value on Y with e
  .
• Thus (e , XY) is a skyline group with decisive
  subspace Y.

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Cont.

• Theorem (Full space skyline objects)
• For any skyline group (G , B), there exists at
  least one object such that u is in
  the skyline of full space D.
• Seed skyline group
• For a data set S in space D, an object in the
  full space skyline is called a seed object.
• The set of seed object is denoted F(S)

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Example
Oid A B C D
P1 5 6 10 7
P2 2 6 8 3
P3 5 4 9 3
P4 6 4 8 5
P5 2 4 9 3

• F(S) P2, P4 , P5 is the set of seed objects.
• P3 is in the skylines of subspace B,D and BD

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Seed Lattice Lattice of all skyline groups
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Stellar Algorithm

• Dominance matrix Mdom
• In the dominance matrix Mdom , the cell at row Pi
  and column Pj denoted by ,records
  the dimensions on which Pi has smaller value than
  Pj.

Oid A B C D
P1 5 6 10 7
P2 2 6 8 3
P3 5 4 9 3
P4 6 4 8 5
P5 2 4 9 3
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Cont.

• Coincidence matrix Mco
• In the coincidence matrix Mco , the cell at row
  Pi and column Pj denoted by ,recoeds
  the dimension on which Pi and Pj share the same
  values.

Oid A B C D
P1 5 6 10 7
P2 2 6 8 3
P3 5 4 9 3
P4 6 4 8 5
P5 2 4 9 3
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Experimental Result
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Conclusion

• Stellar exploits the skyline groups and avoids
  searching all subspace.
• If most of the subspace skyline objects form a
  unique skyline group in their subspace, then
  Skyey can be faster.

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