Efficient Evaluation of Semi-Skylines

Transcription

1 Efficient Evaluation of Semi-Skylines Markus Endres and Werner Kießling Fifth International Workshop on Ranking in Databases (2011)

2 Outline 1. Skyline Queries and Semi-Skylines 2. The Staircube Algorithm 3. Performance Benchmarks 4. Summary and Outlook 2

3 1. Skyline Queries and Semi-Skylines 3

4 Skyline Queries Find hotels which are cheap and close to the beach! 2.0 Greenwood Wyndham Distance to the beach [km] Orange Hill Atlantis Red Carpet Westwind SunSpree Hilton Sheraton Literature: Price [Euro] The Skyline Operator (Börzsönyi et al., 2001) Preference Formulas in Relational Queries (Chomicki, 2002) Foundations of Preferences in Database Systems (Kießling, 2002) 4

5 Skyline Queries Find hotels which are cheap and close to the beach! 2.0 Greenwood Wyndham Distance to the beach [km] Orange Hill Atlantis Red Carpet Westwind SunSpree Hilton Sheraton Literature: Price [Euro] The Skyline Operator (Börzsönyi et al., 2001) Preference Formulas in Relational Queries (Chomicki, 2002) Foundations of Preferences in Database Systems (Kießling, 2002) 4

6 Skyline Queries Find hotels which are cheap and close to the beach! 2.0 Greenwood Wyndham Distance to the beach [km] Red Carpet Orange Hill Westwind Atlantis better than SunSpree Hilton Sheraton Literature: Price [Euro] The Skyline Operator (Börzsönyi et al., 2001) Preference Formulas in Relational Queries (Chomicki, 2002) Foundations of Preferences in Database Systems (Kießling, 2002) 4

7 Skyline Queries Find hotels which are cheap and close to the beach! 2.0 Greenwood Wyndham Distance to the beach [km] Orange Hill Atlantis Skyline Red Carpet Westwind SunSpree Hilton Sheraton Literature: Price [Euro] The Skyline Operator (Börzsönyi et al., 2001) Preference Formulas in Relational Queries (Chomicki, 2002) Foundations of Preferences in Database Systems (Kießling, 2002) 4

8 Skyline Queries Preference Background (Kießling) Preference: strict partial order x< P y < P on dom(a) means: I like y more than x Preference selection of a preference P σ[p ](R) := {t R t R : t< P t } Skyline / BMO-set / Winnow Weak Order Preference (WOP) Dominance test by a score / level function level P : dom(a) N 0 x< P y level P (x) > level P (y) 5

9 Skyline Queries Preference Background (Kießling) Base preference constructors, e.g. LOWEST (MIN), HIGHEST (MAX), AROUND, POS, NEG,... P := P OS/NEG(Soup, { Chicken, Noodle }, { T omato }) 6

10 Skyline Queries Preference Background (Kießling) Base preference constructors, e.g. LOWEST (MIN), HIGHEST (MAX), AROUND, POS, NEG,... P := P OS/NEG(Soup, { Chicken, Noodle }, { T omato }) Visualization by the BTG / Hasse diagram 6

11 Skyline Queries Preference Background (Kießling) Complex preference constructors, e.g. Pareto / traditional Skyline P 1 P 2 =(A 1 A 2,< P ) (x 1,x 2 ) < P (y 1,y 2 ) (x 1 < P1 y 1 (x 2 < P2 y 2 x 2 = y 2 )) (x 2 < P2 y 2 (x 1 < P1 y 1 x 1 = y 1 ) A tuple is said to dominate another tuple if it is better in at least one dimension and not worse in all other dimensions. 7

12 Skyline Queries Preference Background (Kießling) Complex preference constructors, e.g. Pareto / traditional Skyline P 1 P 2 =(A 1 A 2,< P ) (x 1,x 2 ) < P (y 1,y 2 ) (x 1 < P1 y 1 (x 2 < P2 y 2 x 2 = y 2 )) (x 2 < P2 y 2 (x 1 < P1 y 1 x 1 = y 1 ) A tuple is said to dominate another tuple if it is better in at least one dimension and not worse in all other dimensions. 7

13 Skyline Queries Preference Background (Kießling) Complex preference constructors, e.g. Pareto / traditional Skyline P 1 P 2 =(A 1 A 2,< P ) (x 1,x 2 ) < P (y 1,y 2 ) (x 1 < P1 y 1 (x 2 < P2 y 2 x 2 = y 2 )) (x 2 < P2 y 2 (x 1 < P1 y 1 x 1 = y 1 ) A tuple is said to dominate another tuple if it is better in at least one dimension and not worse in all other dimensions. 7

14 Skyline Queries Preference Background (Kießling) Complex preference constructors, e.g. Pareto / traditional Skyline P 1 P 2 =(A 1 A 2,< P ) (x 1,x 2 ) < P (y 1,y 2 ) (x P 1 < P 1 < P1 y 1 (x 2 < P2 y 2 x 2 = y 2 )) 2 (x 2 < P2 y 2 (x 1 < P1 y 1 x 1 = y 1 ) Left-Semi-Pareto / Left-Semi-Skyline P 1 >P 2 Right-Semi-Pareto / Right-Semi-Skyline No intuitive interpretation of Semi-Skylines but applications 8

15 Constrained Skyline Queries Optimization of Constrained Skyline Queries S σ[p 1 P 2 P 3 ] σ H B M Unoptimized operator tree. SELECT * FROM Soup S, Meat M, Beverage B WHERE S.Cal + M.Cal + B.Cal <= 1100 PREFERRING S.Name IN (Chicken, Noodle) AND M.Name IN (Beef) AND B.Vc HIGHEST 2 σ[p 1 P 2 P 3 ] σ H σ[p 3 < HB.Cal] σ[p 1 < HS.Cal] σ[p 2 < HM.Cal] B - Semi-Skyline Optimization of Constrained Skyline Queries (Endres / Kießling in ADC 2011) - Semi-Skylines and Skyline Snippets (Endres, Books on Demand, 2011) S M Optimized operator tree. 9

16 Semi-Skylines How to compute Semi-Skylines? How to compute Semi-Skylines? 10

17 3. The Staircube Algorithm 11

18 The Staircube Algorithm Better-Than Graph and Pruning Pruning of the BTG (Better-Than Graph) 12

19 The Staircube Algorithm Better-Than Graph and Pruning Pruning of the BTG (Better-Than Graph) 12

20 The Staircube Algorithm Better-Than Graph and Pruning Pruning of the BTG (Better-Than Graph) pl P (a) = m j=1 max(p j) min({max(p i ) level Pi (a i ) 1 i m level Pi (a i ) > 0}) 12

21 The Staircube Algorithm Better-Than Graph and Pruning Pruning of the BTG (Better-Than Graph) pl P (a) = m j=1 max(p j) min({max(p i ) level Pi (a i ) 1 i m level Pi (a i ) > 0}) 12

22 The Staircube Algorithm Pruning for Semi-Pareto Partial Level for right and left part of Semi-Pareto P := P 1... P m < P m+1... P n parl P (t) : dom(a) N 0 N 0 parl P (t) := (level P1 (t), level P2 (t)) Pruning Partial Level for Left-Semi-Pareto: { (plp1 (t), level ppl P (t) = P2 (t)) if P 2 is a WOP (pl P1 (t), pl P2 (t)) otherwise All nodes with a partial level equal or higher than are dominated by t. ppl P (t) 13

23 The Staircube Algorithm Pruning of the BTG of Semi-Pareto P := (P 1 P 2 ) < (P 3 P 4 ) 0,0<0,0 0,0<0,1 0,0<1,0 0,0<1,1 0,0<2,0 0,0<2,1 0,1<0,0 0,1<0,1 0,1<1,0 0,1<1,1 0,1<2,0 0,1<2,1 1,0<0,0 1,0<0,1 1,0<1,0 1,0<1,1 1,0<2,0 1,0<2,1 0,2<0,0 0,2<0,1 0,2<1,0 0,2<1,1 0,2<2,0 0,2<2,1 1,1<0,0 1,1<0,1 1,1<1,0 1,1<1,1 1,1<2,0 1,1<2,1 2,0<0,0 2,0<0,1 2,0<1,0 2,0<1,1 2,0<2,0 2,0<2,1 0,3<0,0 0,3<0,1 0,3<1,0 0,3<1,1 0,3<2,0 0,3<2,1 1,2<0,0 1,2<0,1 1,2<1,0 1,2<1,1 1,2<2,0 1,2<2,1 2,1<0,0 2,1<0,1 2,1<1,0 2,1<1,1 2,1<2,0 2,1<2,1 1,3<0,0 1,3<0,1 1,3<1,0 1,3<1,1 1,3<2,0 1,3<2,1 2,2<0,0 2,2<0,1 2,2<1,0 2,2<1,1 2,2<2,0 2,2<2,1 2,3<0,0 2,3<0,1 2,3<1,0 2,3<1,1 2,3<2,0 2,3<2,1 14

24 The Staircube Algorithm The Algorithm For each input tuple t in R do Compute the BTG node n of t Check if n can be pruned by any pruning level Check for dominance left of the partial level Insert node into the Skiplist Remove worse nodes right of the partial level NIL Data Structure Based on a Skiplist (Pugh 1990) Stores partial level lists Ordered by the scalar product mapping search / insert / remove in O(log n) (1,2) (2,1) (4,0) parllist parllist parllist (1, 0 < 2, 0) (1, 1 < 0, 1) (2, 2 < 0, 0) t 1 t 2 t 3, t 4 15

25 4. Performance Benchmarks 16

26 Performance Benchmarks BNL (Börzsönyi et al., 2001) vs. Staircube Implementation in Preference SQL The Preference SQL System - An Overview (Kießling / Endres, IEEE DEB, Vol. 34, 2011) Synthetic data sets (Börzsönyi 2001) ANTI, COR, IND distributions Vary data cardinality and number of distinct values 17

27 Performance Benchmarks Benchmark 1: Computation time BNL vs. Staircube Semi-Pareto: (P 1 P 2 ) < (P 3 P 4 ) only LOWEST preferences (MIN) n=50k to n=500 tuples BNL Staircube Runtime in sec K 100K 150K 200K 250K 300K K 450K 500K Data cardinality 18

28 Performance Benchmarks Benchmark 2: Influence of different left and right Pareto preference Semi-Pareto: P := (P 1... P i ) < (P i+1... P m ) Fixed n=500k, c=100k 600 BNL Staircube Runtime in sec Dimension m 19

29 5. Summary and Outlook 20

30 Summary and Outlook Summary Semi-Skylines for Constrained Skyline Optimization The Staircube Algorithm Based on BTG pruning Data structure: Skiplist Worst-case complexity O(n log n) 21

31 Summary and Outlook Summary Semi-Skylines for Constrained Skyline Optimization The Staircube Algorithm Based on BTG pruning Data structure: Skiplist Worst-case complexity O(n log n) Outlook Further optimization rules Parallel computation of Skylines using Semi-Skyline intersection 21

32 Questions? A Demo of Preference SQL is available at endres@informatik.uni-aufgsburg.de 22