Preference Queries over Sets

Transcription

1 Preference Queries over Sets Xi Zhang Jan Chomicki SUNY at Buffalo April 15, 2011 Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

2 Tuple Preferences vs Set Preferences Tuple Preferences Well known preferences: top-k, skyline etc. Set Preferences Preferences between sets of tuples. Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

3 Motivating Example Alice is buying 3 books as gifts. Title Genre Rating Price Vendor a 1 sci-fi 5.0 $15.00 Amazon a 2 biography 4.8 $20.00 B&N a 3 sci-fi 4.5 $25.00 Amazon a 4 romance 4.4 $10.00 B&N a 5 sci-fi 4.3 $15.00 Amazon a 6 romance 4.2 $12.00 B&N a 7 biography 4.0 $18.00 Amazon a 8 sci-fi 3.5 $18.00 Amazon Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

4 Motivating Example Alice is buying 3 books as gifts. Title Genre Rating Price Vendor a 1 sci-fi 5.0 $15.00 Amazon a 2 biography 4.8 $20.00 B&N a 3 sci-fi 4.5 $25.00 Amazon a 4 romance 4.4 $10.00 B&N a 5 sci-fi 4.3 $15.00 Amazon a 6 romance 4.2 $12.00 B&N a 7 biography 4.0 $18.00 Amazon a 8 sci-fi 3.5 $18.00 Amazon She has the following wishes... (C1) Spend as little money as possible. (C2) Get one sci-fi book. (C3) Prioritize (C2) over (C1) Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

5 Motivating Example Alice is buying 3 books as gifts. Title Genre Rating Price Vendor a 1 sci-fi 5.0 $15.00 Amazon a 2 biography 4.8 $20.00 B&N a 3 sci-fi 4.5 $25.00 Amazon a 4 romance 4.4 $10.00 B&N a 5 sci-fi 4.3 $15.00 Amazon a 6 romance 4.2 $12.00 B&N a 7 biography 4.0 $18.00 Amazon a 8 sci-fi 3.5 $18.00 Amazon She has the following wishes... (C1) Spend as little money as possible. (C2) Get one sci-fi book. (C3) Prioritize (C2) over (C1) the cheapest 3 books Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

6 Tuple Preference - Winnow (Chomicki [Cho03]) Tuple Preference: t 1 is preferred to t 2 t 1 C t 2 ô t 1.rating sci-fi ^ t 1.price t 2.price Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

7 Tuple Preference - Winnow (Chomicki [Cho03]) Tuple Preference: t 1 is preferred to t 2 t 1 C t 2 ô t 1.rating sci-fi ^ t 1.price t 2.price Winnow Operator ω C : return all tuples in a database relation r for which there is no preferred tuple in r ω C prq tt P r Dt 1 P r.t 1 C tu Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

8 Tuple Preference - Winnow (Chomicki [Cho03]) Tuple Preference: t 1 is preferred to t 2 t 1 C t 2 ô t 1.rating sci-fi ^ t 1.price t 2.price Winnow Operator ω C : return all tuples in a database relation r for which there is no preferred tuple in r ω C prq tt P r Dt 1 P r.t 1 C tu Set Preference - tuple set tt 1, t 2,..., t k u is preferred to tuple set tt 1 1, t1 2,..., t1 k u Fixed cardinality (k) assumption Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

9 Profile-based Set Preference k-subsets: subsets of relation r, with fixed cardinality k Set Pref. Quantities of Interest Desired Value or Order (C1) total cost (C2) # of sci-fi books 1 (C3) total cost, # of sci-fi books (C2) (C1) Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

10 Profile-based Set Preference k-subsets: subsets of relation r, with fixed cardinality k Set Pref. Quantities of Interest Desired Value or Order (C1) total cost (C2) # of sci-fi books 1 (C3) total cost, # of sci-fi books (C2) (C1) features F 1, F 2,..., F m Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

11 Profile-based Set Preference k-subsets: subsets of relation r, with fixed cardinality k Set Pref. Quantities of Interest Desired Value or Order (C1) total cost (C2) # of sci-fi books 1 (C3) total cost, # of sci-fi books (C2) (C1) features F 1, F 2,..., F m preferences over profiles profile xf 1, F 2,..., F m y Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

12 Features and Set Preferences Features: mini-sql Queries F 1 SELECT sum(price) FROM $S F 2 SELECT count(title) FROM $S WHERE genre= sci-fi Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

13 Features and Set Preferences Features: mini-sql Queries F 1 SELECT sum(price) FROM $S F 2 SELECT count(title) FROM $S WHERE genre= sci-fi Set Preferences s 1 Ï C1 s 2 ô F 1ps 1q F 1ps 2q. s 1 Ï C2 s 2 ô F 2ps 1q 1 ^ F 2ps 2q 1. s 1 Ï C3 s 2 ô pf 2ps 1q 1 ^ F 2ps 2q 1q _pf 2ps 1q 1 ^ F 2ps 2q 1 ^ F 1ps 1q _pf 2ps 1q 1 ^ F 2ps 2q 1 ^ F 1ps 1q F 1ps 2qq F 1ps 2qq. Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

14 Additive Features Assume additive features in this talk. Efficient optimizations exist for additive features. Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

15 Additive Features Assume additive features in this talk. Efficient optimizations exist for additive features. Additive Feature A feature F is additive iff for every subset s of relation r, and every t P r s Fps Y ttuq Fpsq fptq Fpttuq fptq Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

16 Computing the Best Sets Naive Algorithm Generate all k-subsets of relation r and compute their profiles. Run the winnow operator over all the profiles and get the best profiles. Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

17 Computing the Best Sets Naive Algorithm Generate all k-subsets of relation r and compute their profiles. Run the winnow operator over all the profiles and get the best profiles. Too many candidate k-subsets! Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

18 Computing the Best Sets Naive Algorithm Generate all k-subsets of relation r and compute their profiles. Run the winnow operator over all the profiles and get the best profiles. Too many candidate k-subsets! Example k 3, r 1000 ñ candidate subsets Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

19 Superpreference Optimization Goal: Generate as few candidate k-subsets as possible Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

20 Superpreference Optimization Goal: Generate as few candidate k-subsets as possible Superpreference Find a superpreference ( ) over the relation r, such that t 1 t 2 ô s 1 Y tt 1 u Ï C s 1 Y tt 2 u. for every (k-1)-subset s 1 of r containing neither t 1 nor t 2. Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

21 Superpreference Optimization Goal: Generate as few candidate k-subsets as possible Superpreference Find a superpreference ( ) over the relation r, such that t 1 t 2 ô s 1 Y tt 1 u Ï C s 1 Y tt 2 u. for every (k-1)-subset s 1 of r containing neither t 1 nor t 2. Pruning Condition Let coverptq tt 1 P r t 1 tu, i.e. all tuples preferred to t. Dt P s, coverptq s ñ s is not a best subset. Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

22 Superpreference Optimization Goal: Generate as few candidate k-subsets as possible Superpreference Find a superpreference ( ) over the relation r, such that t 1 t 2 ô s 1 Y tt 1 u Ï C s 1 Y tt 2 u. for every (k-1)-subset s 1 of r containing neither t 1 nor t 2. Pruning Condition Let coverptq tt 1 P r t 1 tu, i.e. all tuples preferred to t. Dt P s, coverptq s ñ s is not a best subset. Systematic Construction Possible if all the features are additive. Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

23 Example - Superpreference Set preference: (C5)X(C6) (C5) Alice wants to spend as little money as possible on sci-fi books. (C6) Alice wants the average rating of books to be as high as possible. Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

24 Example - Superpreference Set preference: (C5)X(C6) (C5) Alice wants to spend as little money as possible on sci-fi books. (C6) Alice wants the average rating of books to be as high as possible. Features: F 5 SELECT sum(price) FROM $S WHERE genre= sci-fi F 6 SELECT avg(rating) FROM $S Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

25 Example - Superpreference Set preference: (C5)X(C6) (C5) Alice wants to spend as little money as possible on sci-fi books. (C6) Alice wants the average rating of books to be as high as possible. Features: F 5 SELECT sum(price) FROM $S WHERE genre= sci-fi F 6 SELECT avg(rating) FROM $S Profile preference: s 1 Ï C s 2 F 5 ps 1 q F 5 ps 2 q ^ F 6 ps 1 q F 6 ps 2 q Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

26 Example - Superpreference Set preference: (C5)X(C6) (C5) Alice wants to spend as little money as possible on sci-fi books. (C6) Alice wants the average rating of books to be as high as possible. Features: F 5 SELECT sum(price) FROM $S WHERE genre= sci-fi F 6 SELECT avg(rating) FROM $S Profile preference: s 1 Ï C s 2 F 5 ps 1 q F 5 ps 2 q ^ F 6 ps 1 q F 6 ps 2 q Superpreference formula (assuming price 0) t 1 t 2 t 1.rating t 2.rating ^ t 2.genre sci-fi ^pt 1.price t 2.price _ t 1.genre sci-fi q. Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

27 M-relation Goal Avoid redundancy in generating profiles Book: Title Genre Rating Price Vendor a 1 sci-fi 5.0 $15.00 Amazon a 2 biography 4.8 $20.00 B&N a 3 sci-fi 4.5 $25.00 Amazon a 4 romance 4.4 $10.00 B&N a 5 sci-fi 4.3 $15.00 Amazon a 6 romance 4.2 $12.00 B&N a 7 biography 4.0 $18.00 Amazon a 8 sci-fi 3.5 $18.00 Amazon a 9 romance 4.0 $20.00 Amazon a 10 history 4.0 $19.00 Amazon Redundancy Example profile Γ pta 1, a 2, a 7 uq = profile Γ pta 1, a 2, a 9 uq = x15.00, 13.8y Exchangeable Tuples a 7, a 9 For any 2-subset s of Bookzta 7, a 9 u Profile Γ tf 5, F 6u F 5 SELECT sum(price) FROM $S WHERE genre= sci-fi F 6 SELECT sum(rating) FROM $S profile Γ ps Y ta 7 uq profile Γ ps Y ta 9 uq Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

28 M-relation Generation Book: Title Genre Rating Price Vendor a 7 biography 4.0 $18.00 Amazon a 8 sci-fi 3.5 $18.00 Amazon a 9 romance 4.0 $20.00 Amazon a 10 history 4.0 $19.00 Amazon M-relation: A 5 A 6 A cnt m 7,9,10 $ m 8 $ Profile Γ tf 5, F 6 u F 5 SELECT sum(price) FROM $S WHERE genre= sci-fi F 6 SELECT sum(rating) FROM $S M-relation Generation SQL SELECT CASE WHEN r.genre= sci-fi THEN r.price ELSE 0 END AS A 5, r.rating AS A 6, count(*) AS A cnt FROM r GROUP BY A 5, A 6 Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

29 Set Preference via M-relation Set preference over the original relations ñ set preference over its M-relation Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

30 Conclusions and Future Work Conclusions A formal logic + SQL framework for specifying restricted set preferences and implementing set preference queries. Effective optimizations yielding improvements of several orders of magnitude. Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

31 Conclusions and Future Work Conclusions A formal logic + SQL framework for specifying restricted set preferences and implementing set preference queries. Effective optimizations yielding improvements of several orders of magnitude. Future Work Query optimization for non-additive features. Set preference elicitation. Embedding best-subset generation in relational query languages. Additional set ranking or browsing techniques for result navigation. Relaxing the fixed cardinality assumption: Superpreference depends on it assumption while M-relation does not. Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

32 Thank you! Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

33 Dataset and Set Preferences Dataset 8000 book quotes from Amazon Schema: xtitle, genre, rating, price, vendory Features F 5 SELECT sum(price) FROM $S WHERE genre= sci-fi F 6 SELECT sum(rating) FROM $S F 7 SELECT sum(rating) FROM $S WHERE genre= sci-fi F 8 SELECT sum(price) FROM $S F 9 SELECT count(title) FROM $S WHERE genre= sci-fi and price < F 10 SELECT sum(rating) FROM $S WHERE rating >= 4.0 Set Preferences Set Pref. Name Profile Schema Γ Profile Pref. Formula C SP1 xf 5, F 6 y F 5 ps 1 q F 5 ps 2 q ^ F 6 ps 1 q F 6 ps 2 q SP2 xf 9, F 10 y F 9 ps 1 q F 9 ps 2 q ^ F 10 ps 1 q F 10 ps 2 q SP3 xf 11, F 12 y F 11 ps 1 q F 11 ps 2 q ^ F 12 ps 1 q F 12 ps 2 q Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

34 Performance of Different Algorithms NAIVE SUPER MREL SM MS 1 Set Pref 1 # of sets in Generation (g) 1e+009 1e+008 1e+007 1e K Input Size (n) Generate Percentage e-005 SUPER MREL SM MS NAIVE SUPER MREL SM MS 1 Set Pref 2 # of sets in Generation (g) 1e+009 1e+008 1e+007 1e K Input Size (n) Generate Percentage e-005 SUPER MREL SM MS Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

35 Performance of Different Algorithms Set Pref 3 Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

36 Related Work Guha et al. [GGK 03] Problem: find an optimal subset of tuples Set property: aggrpaq parameter Set preference: objective function min { max Binshtok et al. [BBS 07] Problem: find a optimal subset of items Set property: predicate Set preference: TCP-net or scoring function Consider subsets of any cardinality, in the fixed-cardinality case, it is subsumed by our framework desjardins and Wagstaff [dw05] Consider fixed-cardinality set preference Consider two set features: diversity and depth Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

37 References I Maxim Binshtok, Ronen I. Brafman, Solomon Eyal Shimony, Ajay Mani, and Craig Boutilier. Computing optimal subsets. In AAAI, pages , Jan Chomicki. Preference formulas in relational queries. ACM Trans. Database Syst., 28(4): , Marie desjardins and Kiri Wagstaff. DD-pref: A language for expressing preferences over sets. In AAAI, pages , Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21

38 References II Sudipto Guha, Dimitrios Gunopulos, Nick Koudas, Divesh Srivastava, and Michail Vlachos. Efficient approximation of optimization queries under parametric aggregation constraints. In VLDB, pages , Xi Zhang, Jan Chomicki (SUNY at Buffalo) Preference Queries over Sets April 15, / 21