Skyline Query Processing for
Incomplete Data
Mohamed E. Khalefa Mohamed F. Mokbel Jus
tin J. Levandoski
Department of Computer Science and Engineering, University of
Minnesota, Minneapolis, MN, USA
ICDE 2008
1
Outline
 Introduction
 Problem Formulation
 Methods and Algorithms
 Experiment Results
 Conclusion
2
Introduction

Existing skyline algorithms assume:
1. Date are complete (all dimensions are available for all data )
2.Transitive relation.
p1
p2
p3
(2,3,6)
(1,2,4)
(1,1,1)
p1 dominates p2, p2 dominates p3 => p3 dominates p1.
3
(Cont.)

If data is incomplete:
1.Some dimensions are no value.
2.No transitive relation.
p1
p2
p3
(2,3,-)
(1,-,8)
(-,4,2)
p1 dominates p2, p2 dominates p3.
But p1 don’t dominates p3.
p3 dominates p1.
Cycle and no transitive relation!!
4
Problem Formulation
 Dominance Relation for Incomplete data:
1.There is at least one dimension ui where both P.ui and Q.ui are
known, and P.ui > Q.ui .
2.For all other dimensions j, j ≠i, either P.uj is unknown, Q.uj is
unknown, or P.uj ≥Q.uj .
 Example:
p1
(2,3,-)
p2
p3
(1,-,8)
(-,4,-)
p1 dominates p2.
p2 don’t domninate p3, and p3 don’t domninate p2.
5
(Cont.)
 Bitmap representation:
0: unknown dimension 1:know dimension
example:
point value
Bitmap
p1
(2,3,-), 110
p2 (1,-,4), 101
p3 (-,-,1). 001
p1.B and p2.B=100<-comparable
p1.B and p3.B=000<-incomparable
6
Methods and Algorithms
 The Replacement Algorithm.
 The Bucket Algorithm.
 The ISkyline Algorithm.
7
The Replacement Algorithm
 Replace unknown dimension by
 .
 Use traditional Skyline algorithm to get Ssky 
p1
p2
P3
(4,-,-,8)
(6,3,-,9)
(-,-,-,10)
Incomplete Data
p2
P3
(6,3,-,9)
(-,-,-,10)
Ssky
8

Replace  
p1
(4, , ,8)
p2
(6,3, ,9)
P3
(  ,  , ,10)
Complete Data

P3
(-,-,-,10)
Ssky
The Bucket Algorithm
 To divide all incoming points into distinct buckets where
all points in each bucket have the same bitmap
representation.
 Skylines of each bucket: local skyline.
 Collect all local skyline in one list, termed candidate
skyline.
 Perform an exhaustive pairwise comparison among all
points to get the query answer.
9
(Cont.)
Global Skyline
Local Skyline
4
1
10
Candidate Skyline
(Cont.)
 In general, performance is better than the replacement
algorithm because candidate list is likely to be smaller than
set Ssky  in the replacement algorithm.
 Candidate skylines may be excessive size
 Missing a chance to use the bucket data to reduce the
comparisons
11
The ISkyline Algorithm
 Virtual Points
 Shadow Points
 The ISkyline Algorithm
12
Virtual Points
P1,P2,Q1,P3,P4依序進入
13
Shadow Points
Q1 dominates P3=> add virtual point Q1v to P’s local_skyline
Q4 is dominated by P3. But we just check “local skyline”.
Q4 don’t be dominated.
14
(Cont.)
 Shadow Points: points that are only dominated by virtual
points.
Q1 is dominated by S4v.
Q3 is dominated by S4v.
15
The ISkyline Algorithm
 Phase I:Insert P,
1.If P is dominated by real point in Local Skyline=>Remoed P.
2.If P is dominated by virtual point in Local skyline
=>Insert to shadow skyline point.
3.If P is local skyline point=>Insert to the Candidate
skyline.( Phase II)
 Phase II:the number of the Candidate skyline>t=>Insert to
the global skyline
16
(Cont.)
Global skyline
Candidate skyline
P1(6,4,-)
Node P = 110 Node Q= 101 Node R= 011
17
P1(6,4,-)
t=2
(Cont.)
Global skyline
Candidate skyline
P1(6,4,-)
P1
P1(6,4,-)
Node P = 110 Node Q= 101 Node R= 011
18
t=2
(Cont.)
Global skyline
Candidate skyline
P1(6,4,-)
P1
P1(6,4,-)
Node P = 110 Node Q= 101 Node R= 011
19
t=2
(Cont.)
Global skyline
Candidate skyline
P1(6,4,-)
P1 Q1
Q1(9,-,1)
Node P = 110 Node Q= 101 Node R= 011
20
P1(6,4,-)
Q1(8,-,1)
t=2
(Cont.)
Global skyline
Candidate skyline
P1(6,4,-)
P1 Q1
Q1(9,-,1)
Node P = 110 Node Q= 101 Node R= 011
21
P1(6,4,-)
Q1(9,-,1)
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P1(6,4,-)
P1 Q1
Q1(9,-,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
22
P1(6,4,-)
Q1(9,-,1)
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P1(6,4,-)
P1(6,4,-)
Q1(9,-,1)
R1(-,3,1)
Q1 R1
Q1(9,-,1)
R1(-,3,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
23
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P2(9,3,-)
P1(6,4,-)
Q1 R1
Q1(9,-,1)
R1(-,3,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
24
P1(6,4,-)
Q1(9,-,1)
R1(-,3,1)
P2(9,3,-)
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P2(9,3,-)
P1(6,4,-)
Q1 R1 P2
Q1(9,-,1)
R1(-,3,1)
t=2
|Candidate skyline|>2
Insert to Global skyline
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
25
P1(6,4,-)
Q1(9,-,1)
R1(-,3,1)
P2(9,3,-)
Compare against Shadow skyline
(Cont.)
Global skyline
Candidate skyline
Q1v(8,-,-)
P2(9,3,-)
P1(6,4,-)
Q1 R1 P2
Q1(9,-,1)
R1(-,3,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
26
P1(6,4,-)
Q1(8,-,1)
R1(-,3,1)
P2(9,3,-)
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(8,-,-)
P2(9,3,-)
P1(6,4,-)
R1 is dominated by P1
Q1 R1 P2
Q1(9,-,1)
R1(-,3,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
27
P1(6,4,-)
Q1(8,-,1)
R1(-,3,1)
P2(9,3,-)
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P2(9,3,-)
P1(6,4,-)
Q1 P2
Q1(9,-,1)
R1(-,3,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
28
P1(6,4,-)
Q1(9,-,1)
R1(-,3,1)
P2(9,3,-)
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P2(9,3,-)
P1(6,4,-)
Q1 P2
Q1(9,-,1)
R1(-,3,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
29
P1(6,4,-)
Q1(9,-,1)
R1(-,3,1)
P2(9,3,-)
Q2(6,-,1)
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P2(9,3,-)
P1(6,4,-)
Q1 P2
Q1(9,-,1)
Q2(6,-,1)
R1(-,3,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
30
P1(6,4,-)
Q1(9,-,1)
R1(-,3,1)
P2(9,3,-)
Q2(6,-,1)
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P2(9,3,-)
P1(6,4,-)
Q1 P2
Q1(9,-,1)
Q2(6,-,1)
R1(-,3,1)
R2(-,6,5)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
31
P1(6,4,-)
Q1(9,-,1)
R1(-,3,1)
P2(9,3,-)
Q2(6,-,1)
R2(-,6,5)
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P2(9,3,-)
P1(6,4,-)
Q1 P2
Q1(9,-,1)
Q2(6,-,1)
R2(-,6,5)
R1(-,3,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
32
P1(6,4,-)
Q1(9,-,1)
R1(-,3,1)
P2(9,3,-)
Q2(6,-,1)
R2(-,6,5)
R2 dominates R1
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P2(9,3,-)
P1(6,4,-)
Check Candidate skyline and Global skyline
Q1 P2
R2
Q1(9,-,1)
Q2(6,-,1)
R2(-,6,5)
R1(-,3,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
33
P1(6,4,-)
Q1(9,-,1)
R1(-,3,1)
P2(9,3,-)
Q2(6,-,1)
R2(-,6,5)
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P2(9,3,-)
P1(6,4,-)
Q1 and P2 are dominated by R2
Q1 P2
R2
Q1(9,-,1)
Q2(6,-,1)
R2(-,6,5)
R1(-,3,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
34
P1(6,4,-)
Q1(9,-,1)
R1(-,3,1)
P2(9,3,-)
Q2(6,-,1)
R2(-,6,5)
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P2(9,3,-)
P1(6,4,-)
Global skyline: Global skyline
R2
Q1(9,-,1)
Q2(6,-,1)
R2(-,6,5)
R1(-,3,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
35
Candidate skyline
P1(6,4,-)
Q1(9,-,1)
R1(-,3,1)
P2(9,3,-)
Q2(6,-,1)
R2(-,6,5)
t=2
(Cont.)
Global skyline
Candidate skyline
Q1v(9,-,-)
P2(9,3,-)
P1(6,4,-)
Result is Global skyline:Q2
R2
Q1(9,-,1)
Q2(6,-,1)
R2(-,6,5)
R1(-,3,1)
Node P = 110 Node Q= 101 Node R= 011
P1(6,4,-)
Shadow skyline
36
P1(6,4,-)
Q1(9,-,1)
R1(-,3,1)
P2(9,3,-)
Q2(6,-,1)
R2(-,6,5)
t=2
Experiment Results
37
(Cont.)
38
(Cont.)
39
Conclusion
 Base on traditional skyline Query: the Replacement
Algorithm and the Bucket Algorithm.
 New method: the ISkyline Algorithm.
 The performance of the ISkyline Algorithm is the best of
three.
40