Towards Indexing Functions: Answering Scalar Product Queries Arijit Khan, Pouya Yanki, Bojana Dimcheva, Donald Kossmann

Transcription

1 Towards Indexing Functions: Answering Scalar Product Queries Arijit Khan, Pouya anki, Bojana Dimcheva, Donald Kossmann Systems Group ETH Zurich

2 Moving Objects Intersection Finding Position at a future time instance t [x = r cos(ωt) y = r sin(wt)] [x = p + ut y = q + vt] Moving Object Database r, ω p, q, u, v 1/ 15

3 Moving Objects Intersection Finding Find all object pairs that will be within distance S at time instance t A 1 + B 2 + C 3 + D 4 + E 5 + F 6 + G 7 S² Moving Object Database r, ω p, q, u, v 1 = r 2 + p 2 + q 2 + 2rp + 2rq A = 1 2 = 2[u(r-p) + v(r-q)] B = t 3 = -2rp C = 1+ sin(ωt) 4 = -2rq D = 1+ cos(ωt) 5 = -2ru E = t[1+ sin(ωt)] 6 = -2rv F = t[1+ cos(ωt)] 7 = u 2 +v 2 G = t 2 2/ 15

4 Moving Objects Intersection Finding Find all object pairs that will be within distance S at time instance t A 1 + B 2 + C 3 + D 4 + E 5 + F 6 + G 7 S² Moving Object Database r, ω p, q, u, v 1 = r 2 + p 2 + q 2 + 2rp + 2rq A = 1 2 = 2[u(r-p) + v(r-q)] B = t 3 = -2rp C = 1+ sin(ωt) 4 = -2rq D = 1+ cos(ωt) 5 = -2ru E = t[1+ sin(ωt)] 6 = -2rv F = t[1+ cos(ωt)] 7 = u 2 +v 2 G = t 2 Function (known) Query Parameters (unknown) 2/ 15

5 Moving Objects Intersection Finding Find all object pairs that will be within distance S at time instance t A 1 + B 2 + C 3 + D 4 + E 5 + F 6 + G 7 S² Moving Object Database r, ω p, q, u, v Scalar Product Query: (A B C D E F G). ( ) S² Query Parameters (unknown) Function (known) Inequality Parameter (unknown) 2/ 15

6 Moving Objects Intersection Finding Moving Object Database r, ω p, q, u, v Scalar Product Query: (A B C D E F G). ( ) S² Query Parameters (unknown) Function (known) Inequality Parameter (unknown) 2/ 15

7 More Applications: Complex SQL Function Patient ID S B P P P P Patient Dataset for Heart-Rate Prediction ARIMA Time Series Prediction Model: Heart-Rate at time t = S t + B 3/ 15

8 More Applications: Complex SQL Function Patient ID S B P P P P Patient Dataset for Heart-Rate Prediction ARIMA Time Series Prediction Model: Heart-Rate at time t = S t + B Find all patients for whom the predicted heart rate at time t is more than an input threshold H. CREATE FUNCTION Critical_Patient ( INPUT double Threshold, double Time RETURN PatientID FROM Patient WHERE S Time + B H ) Unknown 3/ 15

9 More Applications: Complex SQL Function Patient ID S B P P P P Patient Dataset for Heart-Rate Prediction S Time + B H Unknown Scalar Product Query (Time, 1). (S, B) H Query Parameters (unknown) Function (known) Inequality Parameter (unknown) 4/ 15

10 More Applications: Complex SQL Function Patient ID S B P P P P Patient Dataset for Heart-Rate Prediction S Time + B H Unknown 4/ 15

11 Problem Statement Inequality Query Find all data points x that satisfy: (a, F(x)) b Applications: moving-object-intersection finding half-space range search complex SQL functions Top-k Nearest Neighbor Query Find the top-k data points x satisfying (a, F(x)) b, that also minimize: (a, F(x)) b / a Applications: top-k nearest points to hyper plane active learning 5/ 15

12 Related Work THEOR MACHINE LEARNING Half-space Range Searching - Agarwal et. al. [PODS 98], Matousek et. al. [Computational Geometry 92] Linear Constraint Queries - Goldstein et. al. [PODS 97] Nearest Neighbor Queries - Liu et. al. [ICML 12], Jain et. al. [NIPS 10] DATABASES Top-k Queries with Ranking Function - Chang et. al. [SIGMOD 00], in et. al. [SIGMOD 07], Li et. al. [SIGMOD 05], Ilyas et. al. [ACM Comp. Survey 08], Hristidis et. al. [SIGMOD 01], Ram et. al. [KDD 12] Index for Moving Objects - Nascimento et. al. [R-tree, SAC 98], Sistla et. al. [ICDE 97], Kollios et. al. [PODS 99], Saltenis et. al. [TPR-Tree, SIGMOD 00], Jensen et. al. [B x -Tree, VLDB 04], Tao et. al. [SIGMOD 02], Zhang et. al. [MBR Tree, VLDB J 12] 6/ 15

13 Related Work THEOR MACHINE LEARNING Half-space Range Searching - Agarwal et. al. [PODS 98], Matousek et. al. [Computational Geometry 92] Linear Constraint Queries - Goldstein et. al. [PODS 97] Nearest Neighbor Queries - Liu et. al. [ICML 12], Jain et. al. [NIPS 10] DATABASES Top-k Queries with Ranking Function - Chang et. al. [SIGMOD 00], in et. al. [SIGMOD 07], Li et. al. [SIGMOD 05], Ilyas et. al. [ACM Comp. Survey 08], Hristidis et. al. [SIGMOD 01], Ram et. al. [KDD 12] Index for Moving Objects - Nascimento et. al. [R-tree, SAC 98], Sistla et. al. [ICDE 97], Kollios et. al. [PODS 99], Saltenis et. al. [TPR-Tree, SIGMOD 00], Jensen et. al. [B x -Tree, VLDB 04], Tao et. al. [SIGMOD 02], Zhang et. al. [MBR Tree, VLDB J 12] 6/ 15

14 Planar Index: Geometrical Indexing x 1 x 4 x 3 x 2 x 6 x 5 Query Processing using Planar Index 7/ 15

15 Moving Objects Intersection Finding Find all object pairs that will be within distance S at time instance t A 1 + B 2 + C 3 + D 4 + E 5 + F 6 + G 7 S² Moving Object Database r, ω p, q, u, v Scalar Product Query: (A B C D E F G). ( ) S² Query Parameters (unknown) Function (known) Inequality Parameter (unknown) 7/ 15

16 Planar Index: Geometrical Indexing x 1 x 4 x 3 x 2 x 6 x 5 Query Processing using Planar Index 7/ 15

17 Planar Index: Geometrical Indexing r 1 One Planar Index = Collection of Parallel Hyper-planes Passing through the Data Points r 2 r 3 x 1 r 4 r 5 x 4 x 3 x 2 r 6 x 6 x 5 p 6 p 5 p 4 p 3 p 2 p 1 Query Processing using Planar Index 7/ 15

18 Planar Index: Geometrical Indexing Query: (a, x) b q 2 x 1 x 4 x 3 x 2 x 6 x 5 q 1 Query Processing using Planar Index 7/ 15

19 Planar Index: Geometrical Indexing r 1 Query: (a, x) b q 2 x 1 Accept x 2 q 1 p 1 Query Processing using Planar Index 7/ 15

20 Planar Index: Geometrical Indexing Query: (a, x) b q 2 Reject r 5 x 6 x 5 p 5 q 1 Query Processing using Planar Index 7/ 15

21 Planar Index: Geometrical Indexing Query: (a, x) b q 2 Verify r 3 x 4 x 3 q 1 p 3 Query Processing using Planar Index 7/ 15

22 Planar Index: Geometrical Indexing Accept Query: (a, x) b Verify x 1 x 4 x 2 Reject x 6 x 5 x 3 Query Processing using Planar Index 7/ 15

23 Planar Index: Time and Space Complexity Accept Verify Query: (a, x) b Reject Query Processing using Planar Index Index Time: O(n log n) Index Space: O(n) Query Processing Time: O(d log n + t) ~ O(d n) 8/ 15

24 Multiple Planar Indices Query: (a, x) b Multiple Planar Indices 9/ 15

25 Best Index Selection at Query Time Query: (a, x) b Accept Reject Accept Reject Verify Planar Index at Right is Better for the Given Query 9/ 15

26 Top-k Nearest Neighbor Query Query: (a, x) b; Top-k Closest Points to Query Hyper plane x 3 x 4 x 2 x 1 x 6 x 8 x 7 x 5 Top-k Nearest Neighbor Query 10/ 15

27 Top-k Nearest Neighbor Query Query: (a, x) b; Top-2 Closest Points to Query Hyper plane x 3 x 4 x 2 x 1 x 6 Reject x 8 x 7 x 5 Top-k Nearest Neighbor Query 10/ 15

28 Top-k Nearest Neighbor Query Query: (a, x) b; Top-2 Closest Points to Query Hyper plane Verify x 3 x 4 x 6 x 2 x 1 Process 6, 5 5 Top-2 Buffer Reject x 8 x 7 x 5 Top-k Nearest Neighbor Query 10/ 15

29 Top-k Nearest Neighbor Query Lower Bound Distance x 3 x 2 x 1 Query: (a, x) b; Top-2 Closest Points to Query Hyper plane Process 4 x x 6 Top-2 Buffer x 7 x 5 x 8 Top-k Nearest Neighbor Query 10/ 15

30 Top-k Nearest Neighbor Query Lower Bound Distance x 3 x 2 x 1 Query: (a, x) b; Top-2 Closest Points to Query Hyper plane Process 3 x x 6 Top-2 Buffer x 7 x 5 x 8 Top-k Nearest Neighbor Query 10/ 15

31 Top-k Nearest Neighbor Query Lower Bound Distance x 3 x 4 x 2 x 1 Query: (a, x) b; Top-2 Closest Points to Query Hyper plane Process x 6 Top-2 Buffer x 7 x 5 x 8 Top-k Nearest Neighbor Query 10/ 15

32 Top-k Nearest Neighbor Query Lower Bound Distance x 3 x 4 x 2 x 1 Query: (a, x) b; Top-2 Closest Points to Query Hyper plane Prune Process x 6 Top-2 Buffer Reject x 7 x 5 x 8 Top-k Nearest Neighbor Query 10/ 15

33 List of Experiments Datasets: - Real-World: CMoment, Ctexture, Electricity Consumption - Synthetic: Independent, Correlated, Anti-Correlated List of Experiments: - Efficiency vs. No of Index - Efficiency vs. No of Dimension - Efficiency vs. Randomness of Query - Efficiency vs. Query Selectivity - Pruning Capacity vs. No of Index - Pruning Capacity vs. No of Dimension - Pruning Capacity vs. Randomness of Query - Pruning Capacity vs. Query Selectivity - Scalability of Index Building, Query Processing - Dynamic Index Updating - Memory Usage of Planar Index Experimentally Evaluated Planar Index in: Moving-Object Intersection Top-k Nearest Neighbor Query 11/ 15

34 Dataset and Query Datasets: # Data Points # Dimension # Attribute Range CMoment (Real-World) 68,040 9 ( 4.15, 4.59) Independent (Synthetic) 1,000, (1, 100) Query: Q Q Q d d 75 (Q 1 + Q Q d ) Query Selectivity Randomness of Query (QR): Q i (1,n) 12/ 15

35 Efficiency (Real-World Dataset) # Dimension = 9 # Index = ~ 4.5 times better than Baseline 13/ 15

36 Efficiency (Synthetic Dataset) # Index = 100 Dimension = 2 Dimension = 6 12 ~ 170 times better than Baseline 1.6 ~ 400 times better than Baseline 14/ 15

37 Application: Moving Object Intersection Intersection Finding among 5K 5K Moving Objects Objects moving with uniform velocity 12 ~ 50 times better than Baseline Objects moving with acceleration 27 ~ 55 times better than Baseline 15/ 15

38 Conclusion Scalar product query widely applicable Planar index one generalized index for many problems Application in moving object intersection finding Future Work: Dynamic updates in planar indices based on past query workload Software and Dataset: (Publicly Available)