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Towards Multidimensional Skyline Analysis
Jian PeiSimon Fraser University, Canada
http://www.cs.sfu.ca/~jpei [email protected]
Joint work with Y. Tao, M. Ester and W. Jin
J. Pei: Towards Multidimensional Skyline Analysis 2
Searching Flights to Sydney
• Price, travel-time and # stops all matter!• A (long) list of all feasible flights? boring to review• Presenting only some selected flights – how?
– Vancouver Honolulu Sydney ($2100, 19 hours, 1 stop) Good!– Vancouver Honolulu Auckland Sydney ($1980, 24 hours, 2
stop) Also good, cheaper, though longer travel time and more stops– Vancouver Los Angles Honolulu Sydney ($2060, 28 hours,
3 stops) Not good, more expensive, longer travel time, and more stops!
• Skyline routes – all possible trade-offs among price, travel-time and # stops superior to the others
J. Pei: Towards Multidimensional Skyline Analysis 3
Domination and Skyline
• A set of objects S in an n-dimensional space D=(D1, …, Dn)
– Numeric dimensions for illustration in this talk
• For u, vS, u dominates v if – u is better than v in one dimension, and – u is not worse than v in any other dimensions– For illustration in this talk, the smaller the better
• u S is a skyline object if u is not dominated by any other objects in S
J. Pei: Towards Multidimensional Skyline Analysis 4
Finding the Skyline in Full Space
• Many existing methods• Divide-and-conquer and block nested loops by
Borzsonyi et al.• Sort-first-skyline (SFS) by Chomicki et al.• Using bitmaps and the relationships between the
skyline and the minimum coordinates of individual points, by Tan et al.
• Using nearest-neighbor search by Kossmann et al.• The progressive branch-and-bound method by
Papadias et al.
J. Pei: Towards Multidimensional Skyline Analysis 5
Full Space Skyline Is Not Enough!
• Skylines in subspaces– Mr. Richer does not care about the price, how
can we derive the superior trade-offs between travel-time and number of stops from the full space skyline?
• Sky cube – computing skylines in all non-empty subspaces (Yuan et al., VLDB’05)– Any subspace skyline queries can be answered
(efficiently)
J. Pei: Towards Multidimensional Skyline Analysis 6
Sky Cube
J. Pei: Towards Multidimensional Skyline Analysis 7
Understanding Skylines
• Understanding skyline objects– Both Wilt Chamberlain and Michael Jordan are in the full
space skyline of the Great NBA Players, which merits, respectively, really make them outstanding?
– How are they different?
• Finding the decisive subspaces – the minimal combinations of factors that determine the (subspace) skyline membership of an object?– Total rebounds for Chamberlain, (total points, total
rebounds, total assists) and (games played, total points, total assists) for Jordan
J. Pei: Towards Multidimensional Skyline Analysis 8
Redundancy in Sky Cube
Does it just happen that skylines in multiple subspaces are identical?
J. Pei: Towards Multidimensional Skyline Analysis 9
Observations
• a, b and c are in the skyline of (X, Y)– Both a and c are in some subspace
skylines– b is not in any subspace skyline
• d and e are not in the skyline of (X, Y)– d is in the skyline of subspace X– e is not in any subspace skyline
• Why and in which subspaces is an object in the skyline?
J. Pei: Towards Multidimensional Skyline Analysis 10
Subspace Skylines Monotonic?
• Is subspace skyline membership monotonic?– x is in the skylines in spaces ABCD and A, but it is not in
the skyline in ABD – it is dominated by y in ABD
• x and y collapse in AD, x and y are in the skylines of the same subspaces of AD
J. Pei: Towards Multidimensional Skyline Analysis 11
Coincident Groups
• How to capture groups of objects that share values in subspaces?
• (G, B) is a coincident group (c-group) if all objects in G share the same values on all dimensions in B– GB is the projection
• A c-group (G, B) is maximal if no any further objects or dimensions can be added into the group– Example: (xy, AD)
J. Pei: Towards Multidimensional Skyline Analysis 12
C-Group Lattices
C-group lattices Maximal c-group latticesquotient
Where are the skylines?Are they also in good structure?
J. Pei: Towards Multidimensional Skyline Analysis 13
Skyline Groups
• A maximal c-group (G, B) is a skyline group if GB is in the subspace skyline of B
• How to characterize the subspaces where GB is in the skyline?– (x, ABCD) is a skyline group– If the set of subspaces are convex, we can use bounds
J. Pei: Towards Multidimensional Skyline Analysis 14
Decisive Subspaces
• A space CB is decisive if– GC is in the subspace skyline of C
– No any other objects share the same values with objects in G on C
– C is minimal – no C’C has the above two properties
• (x, ABCD) is a skyline group, AC, CD are decisive
J. Pei: Towards Multidimensional Skyline Analysis 15
Semantics
• In which subspaces an object or a group of objects are in the skyline?
• The skyline membership of skyline groups are established by their decisive subspaces– For skyline group (G, B), if C is decisive, then G
is in the skyline of any subspace C’ where CC’B
• Signature of skyline group Sig(G, B)=(GB, C1, …, Ck) where C1, …, Ck are all decisive subspaces
J. Pei: Towards Multidimensional Skyline Analysis 16
Example
The skyline membership of an object is determined by the skyline groups in which it participates
An object u is in the skyline of subspace C if and only if there exists a skyline group (G, B) and its decisive subspace C’ such that uG and C’CB
J. Pei: Towards Multidimensional Skyline Analysis 17
Subspace Skyline Analysis
• All skyline projections form a lattice (skyline projection lattice)– A sub-lattice of the c-group lattice
• All skyline groups form a lattice (skyline group lattice)– A quotient lattice of the skyline projection lattice– A sub-lattice of the maximal c-group lattice
J. Pei: Towards Multidimensional Skyline Analysis 18
Relationship Among Lattices
C-group lattices Maximal c-group lattices
Skyline projection lattices Skyline group lattices
quotient
quotient
sub-lattice sub-lattice
J. Pei: Towards Multidimensional Skyline Analysis 19
OLAP Analysis on Skylines
• Subspace skylines
• Relationships between skylines in subspaces
• Closure information
J. Pei: Towards Multidimensional Skyline Analysis 20
Full Space vs. Subspace Skylines
• For any skyline group (G, B), there exists at least one object uG such that u is in the full space skyline– Can use u as the representative of the group
• An object not in the full skyline can be in some subspace skyline only if it collapses to some full space skyline objects– All objects not in the full space skyline and not
collapsing to any full space skyline object can be removed from skyline analysis
– If only the projections are concerned, only the full space skyline objects are sufficient for skyline analysis
J. Pei: Towards Multidimensional Skyline Analysis 21
Computing Skylines in All Subspaces
• NP-hard– Intuition: the curse of dimensionality – there are an
exponential number of subspaces
• Reduction from frequent itemset mining
Tid Items
T1 {a, b, c}
T2 {a, c, d, e}
T3 {b, c, d, e}
If min_sup=2, a, b, c, d, e, ac, bc, cd, cde, de are frequent itemsets
Oid a b c d e
O1 0 0 0 1 1
O2 0 1 0 0 0
O3 1 0 0 0 0
O0 0 0 0 0 0
Sup(cde)=# skyline objects in cde - 1
J. Pei: Towards Multidimensional Skyline Analysis 22
Subspace Skyline Computation
• Compute the set of skyline groups and their signatures– NP-hard: reduction from frequent closed itemset mining
• Top-down enumeration of subspaces– Similar ideas in skyline cube computation
• For each subspace, find skyline groups and decisive subspaces– Find (subspace) skylines by sorting– Share sorting and use merge-sorting as much as
possible
J. Pei: Towards Multidimensional Skyline Analysis 23
Enumerating Subspaces
• Using a top-down enumeration tree– Each child explores a proper subspace with one
dimension less– All objects not in the skyline of the parent subspace and
not collapsing to one skyline object of the parent subspace can be removed
J. Pei: Towards Multidimensional Skyline Analysis 24
Computing Skylines by Sorting
• Sort all objects in lexicographic ascending order– a-d-b-e-c
• Check objects in the sorted list, an object is in the skyline if it is not dominated by any skyline objects before it in the list– {a, b, c} are skyline objects
J. Pei: Towards Multidimensional Skyline Analysis 25
Efficient Local Sorting
• Not necessary to sort for each subspace– A sorted list in subspace (A, B, C, D) can be used in
subspaces (A), (A, B), (A, B, C)– To generate a sorted list in subspace (B, C, D), we can
use merging sort to merge the sublists of different values on A
• If a non-skyline object collapses to a skyline object, the skyline object “absorbs” the non-skyline object by taking the non-skyline object’s id– A non-skyline object may be “absorbed” by multiple
skyline objects– Recursively reduce the number of objects and shorten
the sorted lists
J. Pei: Towards Multidimensional Skyline Analysis 26
Results on Great NBA Players’
• 17,266 records• 4 attributes are selected• 67 skyline records in the full space, 146 decisive
subspaces
J. Pei: Towards Multidimensional Skyline Analysis 27
# Skyline Groups vs. Dimensionality
• Dimensionality: the complexity of subspaces – A 1-d subspace has only one skyline group– A high-dimensional subspace many have many skyline groups– # skyline groups tends to increase when dimensionality increases
• Number of subspaces– An n-d data set has n 1-d subspaces, 1 n-d (sub-)space, and
n!/[(n/2)!(n/2)!] n/2-d subspaces (if n is even)
• The number of skyline groups in subspaces of dimensionality k depends on the joint-effect of the two factors– When k < n/2, the two factors are consistent– When k > n/2, the two factors are contrasting
J. Pei: Towards Multidimensional Skyline Analysis 28
About the Synthetic Data Sets
• Independent: attribute values are uniformly distributed
• Correlated: if a record is good in one dimension, likely it is also good in others
• Anti-correlated: if a record is good in one dimension, it is unlikely to be good in others
J. Pei: Towards Multidimensional Skyline Analysis 29
Scalability w.r.t Database Size
Independent
Correlated
Anti-correlated
J. Pei: Towards Multidimensional Skyline Analysis 30
Scalability w.r.t. Dimensionality
J. Pei: Towards Multidimensional Skyline Analysis 31
Conclusions
• Skyline analysis is important in many applications– Only skyline objects in the full space may not be enough
• Skyline cube is powerful to answer subspace skyline queries– But it is interesting to ask why an object is in the
subspace skylines, and more• Skyline groups and decisive subspaces –
capturing the semantics of subspace skylines• OLAP subspace skyline analysis• An efficient algorithm to compute skyline groups• Latest progress: An efficient algorithm to query
subspace skylines (Tao et al., ICDE’06)
J. Pei: Towards Multidimensional Skyline Analysis 32
References
• J. Pei, W. Jin, M. Ester, and Y. Tao. "Catching the Best Views of Skyline: A Semantic Approach Based on Decisive Subspaces". In Proceedings of the 31st International Conference on Very Large Data Bases (VLDB'05), Trondheim, Norway, August 30-September 2, 2005.
• Y. Tao, X. Xiao, and J. Pei. "SUBSKY: Efficient Computation of Skylines in Subspaces". In Proceedings of the 22nd International Conference on Data Engineering (ICDE'06), Atlanta, GA, USA, April 3-7, 2006.
J. Pei: Towards Multidimensional Skyline Analysis 33
Thank You!
Vancouver, BC, Canadahttp://members.virtualtourist.com/m/822f5/dc80f/
Trondheim, Norway By Gerold Jung
Hong Konghttp://lambcutlet.org/gallery/Day_6/Hong_Kong_Island_
skyline_on_a_cloudy_night_around_Central