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Towards Multidimensional Skyline Analysis Jian Pei Simon Fraser University, Canada http://www.cs.sfu.ca/~jpei jpei@cs.sfu.ca Joint work with Y. Tao, M. Ester and W. Jin

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J. Pei: Towards Multidimensional Skyline Analysis2 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

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J. Pei: Towards Multidimensional Skyline Analysis3 Domination and Skyline A set of objects S in an n-dimensional space D=(D 1, …, D n ) –Numeric dimensions for illustration in this talk For u, v S, 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

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J. Pei: Towards Multidimensional Skyline Analysis4 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.

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J. Pei: Towards Multidimensional Skyline Analysis5 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., VLDB05) –Any subspace skyline queries can be answered (efficiently)

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J. Pei: Towards Multidimensional Skyline Analysis6 Sky Cube

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J. Pei: Towards Multidimensional Skyline Analysis7 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

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J. Pei: Towards Multidimensional Skyline Analysis8 Redundancy in Sky Cube Does it just happen that skylines in multiple subspaces are identical?

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J. Pei: Towards Multidimensional Skyline Analysis9 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?

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J. Pei: Towards Multidimensional Skyline Analysis10 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

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J. Pei: Towards Multidimensional Skyline Analysis11 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 –G B 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)

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J. Pei: Towards Multidimensional Skyline Analysis12 C-Group Lattices C-group latticesMaximal c-group lattices quotient Where are the skylines? Are they also in good structure?

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J. Pei: Towards Multidimensional Skyline Analysis13 Skyline Groups A maximal c-group (G, B) is a skyline group if G B is in the subspace skyline of B How to characterize the subspaces where G B is in the skyline? –(x, ABCD) is a skyline group –If the set of subspaces are convex, we can use bounds

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J. Pei: Towards Multidimensional Skyline Analysis14 Decisive Subspaces A space C B is decisive if –G C 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

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J. Pei: Towards Multidimensional Skyline Analysis15 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 C C B Signature of skyline group Sig(G, B)=(G B, C 1, …, C k ) where C 1, …, C k are all decisive subspaces

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J. Pei: Towards Multidimensional Skyline Analysis16 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 u G and C C B

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J. Pei: Towards Multidimensional Skyline Analysis17 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

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J. Pei: Towards Multidimensional Skyline Analysis18 Relationship Among Lattices C-group latticesMaximal c-group lattices Skyline projection latticesSkyline group lattices quotient sub-lattice

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J. Pei: Towards Multidimensional Skyline Analysis19 OLAP Analysis on Skylines Subspace skylines Relationships between skylines in subspaces Closure information

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J. Pei: Towards Multidimensional Skyline Analysis20 Full Space vs. Subspace Skylines For any skyline group (G, B), there exists at least one object u G 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

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J. Pei: Towards Multidimensional Skyline Analysis21 Computing Skylines in All Subspaces NP-hard –Intuition: the curse of dimensionality – there are an exponential number of subspaces Reduction from frequent itemset mining TidItems 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 Oidabcde O100011 O201000 O310000 O000000 Sup(cde)=# skyline objects in cde - 1

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J. Pei: Towards Multidimensional Skyline Analysis22 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

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J. Pei: Towards Multidimensional Skyline Analysis23 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

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J. Pei: Towards Multidimensional Skyline Analysis24 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

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J. Pei: Towards Multidimensional Skyline Analysis25 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 objects id –A non-skyline object may be absorbed by multiple skyline objects –Recursively reduce the number of objects and shorten the sorted lists

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J. Pei: Towards Multidimensional Skyline Analysis26 Results on Great NBA Players 17,266 records 4 attributes are selected 67 skyline records in the full space, 146 decisive subspaces

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J. Pei: Towards Multidimensional Skyline Analysis27 # 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

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J. Pei: Towards Multidimensional Skyline Analysis28 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

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J. Pei: Towards Multidimensional Skyline Analysis29 Scalability w.r.t Database Size Independent Correlated Anti-correlated

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J. Pei: Towards Multidimensional Skyline Analysis30 Scalability w.r.t. Dimensionality

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J. Pei: Towards Multidimensional Skyline Analysis31 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., ICDE06)

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J. Pei: Towards Multidimensional Skyline Analysis32 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.

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J. Pei: Towards Multidimensional Skyline Analysis33 Thank You! Vancouver, BC, Canada http://members.virtualtourist.com/m/822f5/dc80f/ Trondheim, Norway By Gerold Jung Hong Kong http://lambcutlet.org/gallery/Day_6/Hong_Kong_Island_ skyline_on_a_cloudy_night_around_Central

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J. Pei: Towards Multidimensional Skyline Analysis34 Subspace Skyline Queries Given a set of objects in multidimensional space D, and a subspace D D, find the skyline objects in space D Materializing subspace skylines in all subspaces can be very costly if dimensionality is high

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J. Pei: Towards Multidimensional Skyline Analysis35 Pruning Using Skyline Points Suppose every dimension is normalized to range [0, 1] –A C =(1, 1, …, 1) is called the maximal corner L distance –f(p) = max n i=1 {(1-p[i])} If p sky is a skyline object, then p cannot be a skyline object if –f(p) < min n i=1 {(1-p sky [i])}

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J. Pei: Towards Multidimensional Skyline Analysis36 Searching a Subspace If p sky is a subspace skyline object in D, then any object satisfying the following condition cannot be in the subspace skyline –f(p) < min Di D {(1-p sky [i])} A search algorithm –Compute f(p) for every object p, sort in f(p) descending order –Maintain the current set S sky of skyline objects in D and U=max psky Ssky {min Di D {(1-p sky [i])}} –Scan points in f(p) ascending order, until U > f(p)

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J. Pei: Towards Multidimensional Skyline Analysis37 Example Sorted list: p3, p4, p5, p1, p6, p2, p8, p7 First skyline point p3, U=0.5 Second skyline point p4, U=0.5 Third skyline point p5, U=0.5 Fourth skyline point p1, remove p3, U=0.8 Done!

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J. Pei: Towards Multidimensional Skyline Analysis38 How Effective Is This Simple Idea? A 15-d uniformed data set of 100,000 points, retrieve a 2-d subspace skyline The probability that no point exists in region [0, λ]x[0, λ] is (1-λ) 100,000 < 10% for λ=0.001 All points p with f(p)<0.999 can be pruned! –0.999 15 x100% = 98.5% points in expectation We can build a B+-tree/B-tree to sort all points according to f(p)

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J. Pei: Towards Multidimensional Skyline Analysis39 Using Multiple Anchors Critical Idea: cluster objects and find good anchors for clusters! Details in our ICDE06 paper.

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J. Pei: Towards Multidimensional Skyline Analysis40 A Few Anchors Work Well!

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J. Pei: Towards Multidimensional Skyline Analysis41 I/O Efficiency for Queries

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J. Pei: Towards Multidimensional Skyline Analysis42 Scalability w.r.t. Dimensionality

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