towards multidimensional skyline analysis jian pei simon fraser university, canada jpei...

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


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


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.


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)


Sky Cube


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


Redundancy in Sky Cube

Does it just happen that skylines in multiple subspaces are identical?


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?


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


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)


C-Group Lattices

C-group lattices Maximal c-group latticesquotient

Where are the skylines?Are they also in good structure?


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


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


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


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


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


Relationship Among Lattices

C-group lattices Maximal c-group lattices

Skyline projection lattices Skyline group lattices

quotient

quotient

sub-lattice sub-lattice


OLAP Analysis on Skylines

• Subspace skylines

• Relationships between skylines in subspaces

• Closure information


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


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


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


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


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


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


Results on Great NBA Players’

• 17,266 records• 4 attributes are selected• 67 skyline records in the full space, 146 decisive

subspaces


# 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


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


Scalability w.r.t Database Size

Independent

Correlated

Anti-correlated


Scalability w.r.t. Dimensionality


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)


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.


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

towards multidimensional skyline analysis jian pei simon fraser university, canada jpei...

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