Indexing Land Surface for Efficient knn Query

Transcription

1 Indexing Land Surface for Efficient knn Query Cyrus Shahabi, Lu-An Tang and Songhua Xing InfoLab University of Southern California Los Angeles, CA

2 Outline q Mo+va+on q Related Work q Indexing Land Surface q Query Processing q Performance Evalua+on q Conclusion and Future Work 2

3 Disaster Response Mo+va+on q Why surface? q The availability of the data (imagery, elevation) of Earth surface at very high resolution q The quick geo-realistic rendering of terrain surface on the computer display (e.g., Google Earth ) q The prevalence of GPS equipped sensors q Various Applications Online Map Outdoor Activities Military Operations Geo-realistic Game 99% Coverage 1.3 Million Images 30-Meter Resolution June 2009: the most complete terrain map of the Earth's surface has been published through the Battlefield Simulator collaboration between NASA (USA) and the ASTER (Japan). Space Exploration 3

4 Mo+va+on q Why is this problem challenging? q Huge size of surface model q Millions of triangles within a region of 100 km 2 Delaunay Triangulation * Triangular Meshes Oracle Spatial 11g 4

5 Mo+va+on q Why is this problem challenging? q Huge size of surface model q Millions of triangles within a region of 10km 10km q Costly surface distance computation q Tens of minutes on a modern PC for a terrain of 10,000 triangular meshes Chen-Han Algorithm: O(n 2 ) * Shortest paths on a polyhedron: CHEN, J., HAN, Y., Computational Geometry

6 Mo+va+on q Why is this problem challenging? q Huge size of surface model q Millions of triangles within a region of 100 km 2 q Costly surface distance computation q Tens of minutes on a modern PC for a terrain of 10,000 triangular meshes Chen-Han Algorithm: O(n 2 ) * Shortest paths on a polyhedron: CHEN, J., HAN, Y., Computational Geometry

7 Mo+va+on q Why is this problem challenging? q Huge size of surface model q Millions of triangles within a region of 100 km 2 q Costly surface distance computation q Tens of minutes on a modern PC for a terrain of 10,000 triangular meshes 2 1 A C B 4 2 Unfolding Case 1 Case 2 1 A A C B 3 2 B Case A B 1 2 Chen-Han Algorithm: O(n 2 ) Case 4 * Shortest paths on a polyhedron: CHEN, J., HAN, Y., Computational Geometry

8 Mo+va+on q Distance Metrics q Euclidean Distance DE (p,q) q Network Distance DN (p,q) q Surface Distance DS (p,q) q DE (p,q) DS (p,q) DN (p,q) p Other Approximations? NOT Restricted on Edges Surface Distance Euclidean Distance Restricted on Edges Network Distance q 8

9 Projec+ve Surface Paths q Projective Surface Paths are not necessarily better than the network paths to approximate the actual surface paths. Orthographic Projection The Orthogonal Plane A Projective Surface Path Network Path B 9

10 Outline q Mo+va+on q Related Work q Indexing Land Surface q Query Processing q Performance Evalua+on q Conclusion and Future Work 10

11 Related Work Spatial Database knn Query Processing Euclidean Space Road Networks Surface Conventional knn Reverse knn Time-aware knn Visible knn 11

12 Related Work Spatial Database knn Query Processing Euclidean Space Road Networks Surface F Conventional knn ü NN Query: Roussopoulos et al., SIMGOD 1995 Reverse knn Time-aware knn Visible knn 12

13 Related Work Spatial Database knn Query Processing Euclidean Space Road Networks Surface Conventional knn F Reverse knn Time-aware knn ü NN Query: Roussopoulos et al., SIMGOD 1995 ü Influences Set: Korn et al., SIMGOD 2000 ü FINCH Algorithm: Wu et al,. VLDB 2008 Visible knn 13

14 Related Work Spatial Database knn Query Processing Euclidean Space Road Networks Surface Conventional knn Reverse knn F Time-aware knn Visible knn ü NN Query: Roussopoulos et al., SIMGOD 1995 ü Influences Set: Korn et al., SIMGOD 2000 ü FINCH Algorithm: Wu et al,. VLDB 2008 ü Time-parameterized queries : Tao et al., SIMGOD 2002 ü Continuous NN Search: Tao et al,. VLDB

15 Related Work Spatial Database knn Query Processing Euclidean Space Road Networks Surface Conventional knn Reverse knn Time-aware knn F Visible knn ü NN Query: Roussopoulos et al., SIMGOD 1995 ü Influences Set: Korn et al., SIMGOD 2000 ü FINCH Algorithm: Wu et al,. VLDB 2008 ü Time-parameterized queries : Tao et al., SIMGOD 2002 ü Continuous NN Search: Tao et al,. VLDB 2002 ü VkNN Query: Nutanong et al., DASFAA

16 Related Work Spatial Database knn Query Processing Euclidean Space Road Networks Surface Conventional knn Reverse knn Time-aware knn ü Query Processing in SNDB : Papadias et al., VLDB 2003 ü V-based knn in SNDB: Shahabi et al., VLDB 2004 ü RNN in Large Graphs: Yiu et al., TKDE 2006 ü CNN Monitoring in RN: Mouratidis et al., VLDB 2006 Visible knn 16

17 Related Work Spatial Database knn Query Processing Euclidean Space Road Networks Surface F Conventional knn ü SkNN Query : Deng et al., ICDE 2006, VLDB J Reverse knn Time-aware knn Visible knn 17

18 Related Work Spatial Database knn Query Processing Euclidean Space Road Networks Surface F Conventional knn Reverse knn Time-aware knn ü SkNN Query : Deng et al., ICDE 2006, VLDB J q Not an incremental approach q Not an exact approach Visible knn 18

19 Outline q Mo+va+on q Related Work q Indexing Land Surface q Query Processing q Performance Evalua+on q Conclusion and Future Work 19

20 Indexing Land Surface q Intuition Surface Voronoi Diagram q Too Complex to Build Voronoi Diagram 20

21 Indexing Land Surface q Tight Surface Index p3 Tight Cell TC(pi)={q: q T and DN (pi, q) < DE(pj, q) ( pj P, pj pi)} p2 q p1 p4 For any query point q TC(pi), the nearest neighbor of q in surface distance is pi. p5 p6 p7 DS (pi, q) DN (pi, q) < DE(pj, q) DS (pj, q) ( pj P, pj pi)} 21

22 Indexing Land Surface q Loose Surface Index p3 Loose Cell LC(pi)={q: q T and DE (pi, q) < DN(pj, q) ( pj P, pj pi)} p2 p5 q p6 p1 p7 p4 Site pi is guaranteed not to be the nearest neighbor of q if q is outside LC(pi). pj P (pj pi) such that DS(pi, q) DE(pi, q) > DN(pj, q) DS(pj, q) 22

23 Indexing Land Surface q Storage Scheme q R-Tree? q Unlike the Voronoi diagram, tight/loose cell are concave polygons in most cases and much more irregular q All cells are adjacent to each other, causing too much overlapping in R- Tree q Index both on TC/LC q Solution: SIR-tree q Same as VOR-Tree ü X Efficient NOT Efficient * For the purpose of clarity, textures on terrain are removed. 23

24 Outline q Mo+va+on q Related Work q Indexing Land Surface q Query Processing q Performance Evalua+on q Conclusion and Future Work 24

25 Query Processing q Nearest Neighbor Query q If the query point falls into one tight cell, its nearest neighbor could be identified immediately without any surface distance computation. p2 q p3 p1 p4 q Our experiment shows about 75% queries fall into one of these tight cells. p5 p6 p7 25

26 Query Processing q Nearest Neighbor Query q If the query point falls out of all tight cells, we need to unfold all loose cells that contain the query point to compute its surface distance to the candidates. p2 p3 p1 p4 q Search (i.e., number of candidates we need compute distance to) is localized in loose cells. p5 p6 q p7 26

27 Query Processing q Nearest Neighbor Query q If pi is the nearest neighbor of q, then the shortest surface path from q to pi is inside the loose cell LC(pi). * q Computation (i.e., unfolding: invocation of CH algorithm) is localized in loose cells. p2 p5 p3 p1 p6 q p7 p4 * Please refer to Section 4.2 Property 4 of the paper for proof. 27

28 Outline q Mo+va+on q Related Work q Indexing Land Surface q Query Processing q Performance Evalua+on q Conclusion and Future Work 28

29 Performance Evalua+on q Dataset * q Eagle Peak (EP) at Wyoming State, USA q 10.7km 14km, 1.4M sampled points. q Bearhead (BH) at Washington State, USA q Similar size as above, 1.3M sampled points. q Uniformly distributed Point of Interest Eagle Peak (EP) Bearhead (BH) * 29

30 Performance Evalua+on q Competing Approaches q Surface Index (SI) q Exact and quick answer q Range Ranking (RR) q Approximate and quick answer q Chen Han Algorithm (CH) q Exact and slow answer 30

31 Performance Evalua+on q Query Efficiency, I/O cost vs. Value of k q The difference in improvement of SI over CH increases for larger k Response Time (Sec) 1.2E E+05 Accessed TIN Triangles E E E (a) Query Efficiency, d=2, EP k 2.0E E (e) I/O Cost, d=2, EP k 31

32 Performance Evalua+on q Accuracy vs. Value of k q The accuracy of RR drops dramatically when the value of k increases. q The accuracy of SI stays at 100%. Accuracy Ratio 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% (a) Query Accuracy on EP k Accuracy Ratio 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% (b) Query Accuracy on BH k 32

33 Outline q Mo+va+on q Related Work q Indexing Land Surface q Query Processing q Performance Evalua+on q Conclusion and Future Work 33

34 Conclusion and Future Work q Conclusion q We extend the traditional knn Query to the space constrained with the third dimension. q We construct two complementary indexing schemes, namely Tight Surface Index (TSI) and Loose Surface Index (LSI) to reduce the invocation of the costly surface distance computation. q SI significantly outperforms its competitors in accuracy and efficiency. q Future Work q Further evaluate its performance with synthetic datasets. q Study variations of sknn such as the continuous sknn query, dynamic sknn query and visible sknn query. 34

35 References Shahabi, C.; Tang, L.- A. & Xing, S. Indexing land surface for efficient knn query. PVLDB, 2008, 1,

36 Con9nuous Monitoring of Nearest Neighbors on Land Surface Instructor: Cyrus Shahabi

37 Preliminaries q Continuous nearest neighbor monitoring in road networks (Mouratidis et al, VLDB 06) q Shortest Path Spanning Tree (Dijkstra Tree) 1. Low fan-out 2. Tall and thin BOUNDARY 37

38 Preliminaries q Surface Expansion Tree (SE Tree) source q Observation 1: SE Tree is fat and short. q Observation 2: These shortest surface paths rarely share common edges. q Observation 3: These shortest surface paths do not cross each other. 38

39 Preliminaries q Continuous Monitoring (k = 3) p 6 p 3 p 4 s p 2 p 1 Old New Result Boundary q More incoming objects q Original Result Boundary contains more than k objects q Identify the kth object and shrink the result boundary p 5

40 Preliminaries Ø Ø Ø q Continuous Monitoring (k = 3) DRAWBACKS Expansion is slow; Search Area may be large; SE Tree is fat and short. p 4 p 6 p 3 p 5 s p 1 Expansion Boundary Result Boundary p 2 Search Region q More outgoing objects q Original Result Boundary contains less than k objects q Compute the Expansion Boundary q Expand SE tree to Expansion boundary and identify the kth object inside the search area

41 Surface Index based Approach q Intuition q Motivated by Partitioning Shortest Paths on road networks q Scalable Network Distance Browsing in Spatial Databases: Samet et al., SIGMOD 2008 (best paper award) Is there a way to partition paths on surface? V 3 V 2 X V 4 V 5 V 1 V 6 The shortest path from X to any vertex of one region must be inside that region. 41

42 Surface Index based Approach q Intuition q Motivated by Partitioning Shortest Paths on road networks q Scalable Network Distance Browsing in Spatial Databases: Samet et al., SIGMOD 2008 (best paper award) source Too many partitions 42

43 Surface Index based Approach q Surface Shortest Path Container q Built based on the Observations that shortest paths do not cross each other and hardly share common edges. edges opposite to the source point 43

44 Surface Index based Approach q Angular Surface Index (ASI) q Hierarchically index these containers ASI Tree ü ü ü ADVANTAGES Size: Much Smaller Regularity: Well Balanced Efficiency: Tall and Thin

45 Surface Index based Approach q Query Processing q The core is the same as using SE Tree q Search is localized in Containers only Search Region p p 2 3 s p 6 p 2 p 3 s p 6 p 4 p 4 p 5 p 1 p 1 p 5

46 Surface Index based Approach q More Details q Definitions/Properties of Containers, Equidistance Line and ASI-tree q Please refer to Section 5.1, 5.2 and 5.3 in the paper. q Constructions of Containers q Please refer to Section in the paper. q Algorithms for Query Processing q Please refer to Section 5.4 in the paper.

47 Conclusion and Future Work q Contribution q First study the Continuous knn query on Surface; q Show the concept of expansion tree for land surface does not work as SE-tree suffers from intrinsic defects: it is fat and short; q Propose a superior approach that partitions SE-Tree into hierarchical chunks of pre-computed surface distances which overcome the above deficiency. q Experimentally verify the applicability of proposed methods. q Future work q Study the Continuous knn query problem for arbitary moving query point. 47

48 References Songhua Xing, Cyrus Shahabi, Bei Pan, Con+nuous Monitoring of Nearest Neighbors on Land Surface, VLDB