1. On the interaction between multidimensional skylines and functional dependencies
Sofian Maabout
University of Bordeaux. CNRS
Joint work with Nicolas Hanusse, Patrick Kamnang Wanko, Carlos Ordonez

2. Skyline query Id
Dist from INSA
price a
100 50 b 90
200 c
280 d 40 e
240 55 f
245
285 h 95
300
O is in the skyline iff there is no other O’ better than O
Skyline={a, b, c, d} not dominated by any hotel
Intuitively, skyline points represent the best tradeoff

3. Multidimensional skylines
Users are allowed to ask queries using any combination of dimensions
CEO: Best hotels = offering a swimming pool and air conditionning
Student: Best hotels = cheapest and free wifi
Skycube = set of all possible skylines
How to optimize all these multidimensional skylines?
Precompute ALL of them  Full Skycube
Precompute a SUBSET of them  Partial Skycube

4. This talk
How functional dependencies can help full and partial materialization of skycubes

5. Skyline Queries and Data Quality
Discard records with low quality is one dimension of data cleaning
Compare tuples wrt their respective quality parameters
Best tuples = those with best tradeoff wrt quality parameters

6. Skyline Queries and Data QualityId
Name
Phone
Zip
City
Salary t1
Dupont
0123
69000
Bordeaux
[1500,2000] t2
Paul
33000
[1500, 1600]
Dupond
4567
Lyon
[1500, 2000] t3
William
6789
[2000,2000]
Zip  City
Phone Name

7. Skyline Queries and Data QualityId
Name
Phone
Zip
City
Salary t1
Dupont
0123
69000
Bordeaux
[1500,2000] t2
Paul
33000
[1500, 1600] t3
Dupond
4567
Lyon
[1500, 2000] t4
William
6789
[2000,2000] t5
Mary
1357
75000
Paris
[2500,2800]
t1, t3 and t4 involved in Zip  City violation
t1 and t2 involved in Phone  Name violation
t1’s salary is less precise than t2’s

8. Skyline Queries and Data QualityId
Name
Phone
Zip
City
Salary t1
Dupont
0123
69000
Bordeaux
[1500,2000] t2
Paul
33000
[1500, 1600] t3
Dupond
4567
Lyon
[1500, 2000] t4
William
6789
[2000,2000] t5
Mary
1357
75000
Paris
[2500,2800] Id
#FD
Salary
Uncertinty t1 2
500 t2 1
100 t3 t4 t5
300
Sky(#FDs,SU)= {t4, t5}

9. Skylines are not monotone

10. Functional dependencies & multidimensional skylines
A  B
BC A
B A
Theorem: If X Y then Sky(X)  Sky(XY)

11. Closed subspaces X is closed iff XA for every A not in X
The minimal FD’s satisfied by T are
C is closed
AB is not closed

12. Example
sqs
Red : closed subspace

13. Skycube computation
If partial materialization, just stop here

14. Skycube computation
Need of an efficient procedure

15. Mining Closed Subspaces
Intuitive idea:
For every A, find the maximal X st X  A
Every x  X’s is potentially closed
The intersection of these sets of x’s are the closed subspaces
We adapt N. Hanusse, SM: A parallel algorithm for computing borders. CIKM’11

16. Mining Closed Subspaces
Maximal subspaces not determining B

17. Subspace Closure
Let X be a subspace. Let Closed={Y | Y is closed} Then, X+ = smallest Y Closed s.t X  Y

18. Closed Subspaces
ABCD
BCD
ABC
ABD
ACD AB AC AD BC BD CD A B C D

19. Experiments We versus other proposals for fully computing the skycube.
QGS & QGL : Lee et al. VLDBJ’14 and
BUS & TDS: Pei et al. TODS’06
Orion: Raïssi et al. VLDB’10
We versus closed skycubes: a losseless compression technique. Raïssi et al. VLDB’10
Assess query evaluation time

20. Experiments: (1) compute all skylines Synthetic data sets
Independent
Correlated
Anti-correlated

21. Experiments: (1) Full Skycube Synthetic data sets
Speedup = execution time of algorithm X / execution time of our algorithm FMC

22. Experiments: (1) Full Skycube Real Data

23. Experiments: (2) query optimization 1000 random skyline queries
0.31% out of the 2^20 queries are materialized.
49 ms to answer 1K skyline queries from the materialized ones instead of
99.92 seconds from the underlying data.
Speed up > 2000 23 23

24. Experiments: (3) comparison with closed skycubes
Identify equivalent skylines and store just one copy  compression of the whole skylines set
E.g, Sky(C), Sky(D) and Sky(CD) are equivalent

25. Experiments: (3) comparison with closed skycubes
n20K, d=17
n 75K, d=10
n 100K, d=18
Number of materialized skylines (time to find and materialize them)
Synthetic correlated data: n=100K, d=20: MICS=20sec, Closed didn’t finish after 36 hours
More details in N. Hanusse, SM, P. Kamnang Wanko, C. Ordonez: Skycube Materialization Using the Topmost skyline of Functional Dependencies. TODS’16

26. Incomparability Dependencies
Definition: X ↬ Y iff t[X]=t’[X]  t[Y] and t’[Y] incomparable Theorem: Sky(X) satisfies X ↬ Y  Sky(X)  Sky(XY) Property: XY  X ↬ Y

27. Incomparability Dependencies
FDs do not detect Sky(B)  Sky(AB) while Sky(B) satisifes B ↬ A
IncoDs detect that Sky(B)  Sky(BC) because Sky(B) doesn’t satisfy
B ↬ C

28. Prioritized Skyline Expression = Sky(AB & CD) First computes Sky(AB)
If t[AB] = t’[AB] and t Sky(AB), then t and t’ are compared wrt C and D
Kießling. Foundations of preferences in database systems. In VLDB’02,
Chomicki et al. Preference elicitation in prioritized skyline queriesVLDBJ’12
Ciaccia et al. Output-sensitive Evaluation of Prioritized Skyline Queries. Sigmod’15

29. Prioritized Skyline Sky(AB)= {t1, t2, t3, t4}Id A B C D t1 1 3 t2 2 t3 4 t4 5
Sky(AB)= {t1, t2, t3, t4}
t1[AB]= t2[AB] and t1 dominates t2 wrt CD
Sky(AB & CD) = {t1, t3, t4}

30. Prioritized Skyline Let  = X1 & … & Xi & … & Xm
If X1…Xi-1  X and X  Xi then
 ’ = X1& … & Xi\X & … & Xm
AB  C  Sky(AB & CD)  Sky(AB & D)

31. Conclusion
Functional dependencies are helpful for both full and partial skycube materialization
Incomparability dependencies characterize skyline inclusions
Semantic optimization of prioritized skylines with FDs

32. Some Open questions
Is it possible to come up with a Chase like procedure for priotirized skylines semantic optimization?
What about Order dependencies ?
Incremental maintenance
Approximate skylines and approximate FDs
t[A] is preferred to s[A] iff s[A] – t[A] > 
X  Y iff t, s : t[X] ~ s[X]  t[Y] ~ s[Y]

33. Thanks Questions