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Quantile-Based KNN over Multi- Valued Objects Wenjie Zhang Xuemin Lin, Muhammad Aamir Cheema, Ying Zhang, Wei Wang The University of New South Wales, Australia.

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Presentation on theme: "Quantile-Based KNN over Multi- Valued Objects Wenjie Zhang Xuemin Lin, Muhammad Aamir Cheema, Ying Zhang, Wei Wang The University of New South Wales, Australia."— Presentation transcript:

1 Quantile-Based KNN over Multi- Valued Objects Wenjie Zhang Xuemin Lin, Muhammad Aamir Cheema, Ying Zhang, Wei Wang The University of New South Wales, Australia

2 KNN Query Find k objects closest to query q 2 q o1o1 o4o4 o2o2 o3o3 2-NN of q: o 2 and o 4

3 KNN Query Find k objects closest to query q 3 What if objects have multiple values ? q Player A Player B

4 Existing Models for Multi-instances Probabilistic KNN Models – Probability is computed by summing up probabilities of all valid possible worlds – Probable top-k NN: [Bekales, Soliman, Ilyas. VLDB 08] Probabilistic verifiers: [Cheng, Chen, Mokebel, Chow. ICDE 08] 4

5 Probabilistic KNN Model 5 q and have the same probability to be NN of q

6 Probabilistic KNN Model 6 q and have the same probability to be NN of q

7 Probabilistic KNN Model 7 q and have the same probability to be NN of q

8 Probabilistic KNN Model 8 ties Not very sensitive to relative distribution of objects instances Different semantics than multi-value objects !

9 Aggregate Distances Existing: mean, minimal, maximal, etc Generalization: quantile distance 9

10 Applications GIS Location based service Radio Frequency Identify Device … 10

11 Contributions Distance between two multi-value objects – Quantile distance – Quantile group-base distance KNN processing – Linear time algorithm for quantile distance – Pruning Techniques on object & instance level 11

12 Preliminaries Φ-quantile: – The first element in a sorted list with the cumulative weight not smaller than Φ, where Φ is a number in (0, 1]. 12 Sorted elements: 3 6 7 8 8 10 13 15 Weights:.10.15.20.15.10.05.10..15 0.5-quantile0.8-quantile Cumulative weight: 0.45

13 Preliminaries Multi-valued Object: – A set of weighted instances in d-dimensional space – The sum of the weights equals to 1 13

14 Quantile Distances Φ-quantile distance 14 Instance pairs: (q 2, u 1 ) (q 3, u 1 ) (q 3, u 2 ) (q 1, u 1 ) (q 2, u 2 ) (q 1, u 2 ) Weights: 1/8 1/8 1/8 1/4 1/8 1/4 0.25-quantile0.5-quantile ½ ¼ ¼ ½ ½

15 Quantile Distances Φ-quantile group-base distance – Candidate group: a subset S’ of elements set Q x U such that the total weights of elements in is not smaller than Φ. – The minimal total weighted distance among all candidate groups, namely with the minimal 15

16 Quantile Distances Φ-quantile group-base distance 16 Instance pairs: (q 2, u 1 ) (q 3, u 1 ) (q 3, u 2 ) (q 1, u 1 ) (q 2, u 2 ) (q 1, u 2 ) Weights: 1/8 1/8 1/8 1/4 1/8 1/4 Φ = 0.5

17 Problem Definition Given a set of multi-valued objects, a multi- valued query object and Φ in (0, 1], retrieve K objects with smallest Φ-quantile distance and Φ-quantile group-base distance to the query object. 17

18 Framework Seeding – Select k objects and compute quantile distance --- k seeds are selected based on mean – Challenge: efficient computation of quantile distances Refinement: – Determine the final solution – Challenge: efficient pruning techniques 18

19 Φ-quantile Distance Multi-value objects U and V – Overall m=|U| * |V| instance pairs – Existing: Disk-based Quantile computation: O(mlogm) with pruning --- Yiu et al EDBT 06 – Our contribution: O(m) algorithm with pruning Basic Idea: index instances by R-tree; prune entry pairs at higher levels using bounding techniques 19

20 Φ-quantile Distance 20 U1U1 U2U2 0.5 V 1 0.4 V2V2 0.6 (U 1, V 1 ) 0.2 (U 1, V 2 ) 0.3 (U 2, V 2 ) 0.3 (U 2, V 1 ) 0.2 Φ=0.6 Φ=0.6 – 0.2 lower bound upper bound

21 Φ-quantile KNN λ k : the largest distance of the K seeded objects Global R-tree: indexes the MBB of the multi-value objects set Local R-tree: indexes the instances of multi-value objects (query, object set) 21

22 Φ-quantile KNN Distance based pruning rule on Global R-tree 22 Q E from global R-tree ≥λ k

23 Φ-quantile KNN Weight based pruning rule on Global R-tree Entry E: Let Ω denote the set of entries with minimal distance ≤ λ k, if the total weight of entries in Ω is ≤ Φ, then every object in E could be pruned. 23 Q E from global R-tree Q1Q1 Q2Q2 minimal distance: mindist(Q 2, E) ≤ λ k weight(Q2) ≤Φ

24 Φ-quantile KNN Pruning on the Local R-tree of Q and U – Trim the local R-tree of both Q and U by λ k level by level. Discard entries in Q with minimal distance to U larger than λ k, similarly for U. – Denote the total weight of remaining entries as P Q and P U, respectively, if P Q * P U ≤ Φ, then U can be pruned. 24

25 Φ-quantile KNN Pruning on the Local R-tree of Q and U 25 Q U Q1Q1 Q2Q2 U1U1 U2U2 minimal distance: mindist(Q 1, U) ≥ λ k minimal distance: mindist(Q, U 2 ) ≥ λ k weight(Q 2 ) * weight(U 1 ) ≤Φ

26 Φ-quantile KNN Overall processing – 1. Pruning on Global R-tree based on distance – 2. Pruning on Global R-tree based on weights – 3. Pruning on Local R-tree – Φ-quantile distance computation 26

27 Φ-quantile Group-base Distance From |Q| * |U| instance pairs, select a set of pairs such that – the sum of weights is ≥ Φ – the weighted distance is minimized Reduced to Knapsack problem. NP-hard 27

28 Φ-quantile Group-base Distance Approximate algorithm with ratio 2 and time complexity O(m log m), where m is the total number of instance pairs. 28 possible solution: 1.72 (1, 0.28), (3, 0.48) sorted distance: 1 weight: 0.28 2 0.12 3 0.48 4 0.12 possible solution: 1 (1, 0.28), (2, 0.12), (4, 0.12)

29 Φ-quantile Group-base KNN Prune a set of objects on Global R-tree based on distance 29 Q Φ * L ≥ λ k E L

30 Φ-quantile Group-base KNN Prune a set of objects on Global R-tree based on weights – Traverse the local R-tree of Q – Sort entry pairs according to lower bound distance at each level of Q 30

31 Φ = 0.5 31 (1, 0.28)(2, 0.12)(3, 0.48)(4, 0.12) 0.5-quantile d1= 1 * 0.28 + 2 * 0.12 d2= 3 * ( Φ – 0.28 – 0.12) If d1 + d2 ≥ λ k, then E could be removed. Q1Q1 Q2Q2 Q3Q3 Q4Q4 EQGlobal R-tree

32 Φ-quantile Group-base KNN Overall Processing – 1. Distance based pruning on Global R-tree – 2. Weight based pruning on Global R-tree – Φ-quantile Group-base distance calculation 32

33 Experiment Real Dataset – NBA player statistics – 339, 721 records of 1,313 players – 3 dimension Synthetic Dataset – Size: 10, 000 to 50, 000 – Instances: 400 to 2, 000 – Dimension: 2 to 5 – MBB length, spatial distribution, instance distribution, weight distribution … 33

34 Φ-quantile Distance Computation 34

35 Overall Performance Naive algorithms refer to techniques without pruning rules 35

36 Pruning Powers 36

37 Accuracy 37

38 Varying Parameters 38

39 Conclusions Distance between two multi-value objects based on quantile A linear algorithm for quantile distance computation with pruning Efficient KNN processing techniques 39

40 Thanks 40

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