1 0 / 2 3 / 2 0 0 9 Outline I ndexing Land Surface for Efficient kNN Query � Motivation � Related Work � Background Cyrus Shahabi Lu An Tang and Songhua Xing Cyrus Shahabi, Lu-An Tang and Songhua Xing � Indexing Land Surface InfoLab � Query Processing University of Southern California � Performance Evaluation Los Angeles, CA 90089-0781 � Conclusion and Future Work http://infolab.usc.edu 2 Motivation Motivation Yosemite National Park Which is the NEAREST campsite??? 3 4 Motivation Motivation � Applications � Problem � Tourist Applications � Scientific Adventures � To find k Nearest Neighbor � Military Operations based on the Surface Distance . � Geo-realistic Games � Space Explorations � Challenges g � Huge size of surface model Which is the Millions of terrain data for a region of 10km × 10km � NEAREST campsite??? Costly surface distance computation � Tens of minutes on a modern PC for a terrain of 10,000 � No efficient surface index structure � R-tree, Voronoi Diagram cannot apply directly. � 5 6 1
1 0 / 2 3 / 2 0 0 9 Outline Related Work Spatial Database � Motivation kNN Query Processing � Related Work � Background Euclidean Space Road Networks Surface � Indexing Land Surface � Conventional kNN � Query Processing � Reverse kNN � Performance Evaluation � Time-aware kNN � Conclusion and Future Work � Visible kNN 7 8 Related Work Related Work Spatial Database Spatial Database kNN Query Processing kNN Query Processing Euclidean Space Road Networks Surface Euclidean Space Road Networks Surface � Conventional kNN � NN Query: Roussopoulos et al., SI MGOD 1 9 9 5 � Conventional kNN � NN Query: Roussopoulos et al., SI MGOD 1 9 9 5 � I nfluences Set: Korn et al., SI MGOD 2 0 0 0 � Reverse kNN � Reverse kNN � FI NCH Algorithm : W u et al,. VLDB 2 0 0 8 � Time-aware kNN � Time-aware kNN � Visible kNN � Visible kNN 9 10 Related Work Related Work Spatial Database Spatial Database kNN Query Processing kNN Query Processing Euclidean Space Road Networks Surface Euclidean Space Road Networks Surface � Conventional kNN � NN Query: Roussopoulos et al., SI MGOD 1 9 9 5 � Conventional kNN � NN Query: Roussopoulos et al., SI MGOD 1 9 9 5 � I nfluences Set: Korn et al., SI MGOD 2 0 0 0 � I nfluences Set: Korn et al., SI MGOD 2 0 0 0 � Reverse kNN � Reverse kNN � FI NCH Algorithm : W u et al,. VLDB 2 0 0 8 � FI NCH Algorithm : W u et al,. VLDB 2 0 0 8 � Time-aware kNN � Tim e- param eterized queries : Tao et al., SI MGOD 2 0 0 2 � Time-aware kNN � Tim e- param eterized queries : Tao et al., SI MGOD 2 0 0 2 � Continuous NN Search: Tao et al,. VLDB 20 02 � Continuous NN Search: Tao et al,. VLDB 2002 � Visible kNN � Visible kNN � VkNN Query: Nutanong et al., DASFAA 2 0 0 7 11 12 2
1 0 / 2 3 / 2 0 0 9 Related Work Related Work Spatial Database Spatial Database kNN Query Processing kNN Query Processing Euclidean Space Road Networks Surface Euclidean Space Road Networks Surface � Conventional kNN � Query Processing in SNDB : Papadias et al., VLDB 2 0 0 3 � Conventional kNN � SkNN Query : Deng et al., I CDE 2 0 0 6 , VLDB J. 2 0 0 8 � V- based kNN in SNDB: Shahabi et al., VLDB 2004 � Reverse kNN � Reverse kNN � RNN in Large Graphs: Yiu et al., TKDE 2 0 0 6 � Time-aware kNN � CNN Monitoring in RN: Mouratidis et al., VLDB 2006 � Time-aware kNN � Visible kNN � Visible kNN 13 14 Related Work Outline Spatial Database � Motivation kNN Query Processing � Related Work � Background Euclidean Space Road Networks Surface � Indexing Land Surface � Conventional kNN � SkNN Query : Deng et al., I CDE 2 0 0 6 , VLDB J. 2 0 0 8 � Query Processing � Not an increm ental approach � Reverse kNN � Not an exact approach � Performance Evaluation � Time-aware kNN � Conclusion and Future Work � Visible kNN 15 16 Background Background � Triangular I rregular Netw ork ( TI N) Model � Distance Metrics � Triangular Mesh � Euclidean Distance D E ( p,q ) � Network Distance D N ( p,q ) � Digital Elevation Model (DEM) � Surface Distance D S ( p,q ) � D E ( p,q ) ≤ D S ( p,q ) ≤ D N ( p,q ) Delaunay Triangulation * p Euclidean Distance Surface Distance Network Distance q * Com putational Geom etry: Algorithms and Applications (BERG, M., KREVELD, M., OVRMARS, M., 17 18 SCHWARZKOPF, O.) 3
1 0 / 2 3 / 2 0 0 9 Background Background � Shortest Surface Path Com putation � Shortest Surface Path Com putation � Chen-Han (CH) Algorithm * : unfold all the faces of a � Chen-Han (CH) Algorithm * : unfold all the faces of a polyhedron to one plane polyhedron to one plane � Time Complexity: , n is the total number of the vertices � Time Complexity: , n is the total number of the vertices 2 2 O ( n ) O ( n ) on the surface on the surface * Shortest paths on a polyhedron : CHEN, J., HAN, Y., Computational Geometry 1990 * Shortest paths on a polyhedron : CHEN, J., HAN, Y., Computational Geometry 1990 19 20 Background Outline � Shortest Surface Path Com putation � Chen-Han (CH) Algorithm * : unfold all the faces of a polyhedron to one plane � Motivation � Time Complexity: , n is the total number of the vertices 2 O ( n ) on the surface � Related Work 2 � Background B 4 Case 1 ng Unfoldin � Indexing Land Surface 3 A 2 1 1 4 A B Case 2 2 4 � Query Processing C B C A 1 3 � Performance Evaluation 3 Case 3 4 3 A B 1 2 � Conclusion and Future Work …… Case 4 * Shortest paths on a polyhedron : CHEN, J., HAN, Y., Computational Geometry 1990 21 22 Indexing Land Surface Indexing Land Surface � I ntuition – Surface Voronoi Diagram � Tight Surface I ndex TC(p i ) ={ q : q � T and D N Tight Cell ( p i , q ) < D E ( p j , q ) ( ∀ p j ∈ P , p j ≠ p i )} p 3 For any query point For any query point p 2 p 4 q q � TC(p i ), the nearest p 1 q neighbor of q in surface distance is p i . p 7 D S ( p i , q ) ≤ D N ( p i , q ) p 6 < D E ( p j , q ) ≤ D S ( p j , q ) p 5 ( ∀ p j ∈ P , p j ≠ p i )} Too Complex to Build Voronoi Diagram 23 24 4
1 0 / 2 3 / 2 0 0 9 Indexing Land Surface Indexing Land Surface � Storage Scheme � Loose Surface I ndex � R-Tree? LC(p i ) ={ q : q � T and D E Loose Cell � Unlike the Voronoi ( p i , q ) < D N ( p j , q ) ( ∀ p j diagram, tight/loose cell are concave ∈ P , p j ≠ p i )} p 3 polygons in most cases and much more irregular Site p i is guaranteed not Site p i is guaranteed not � All cells are adjacent � All cells are adjacent p 2 p 4 to be the nearest to each other, p 1 causing too much neighbor of q if q is overlapping in R- outside LC(p i ). Tree q � Index both on TC/LC p 7 ∃ p j ∈ P ( p j ≠ p i ) such that p 6 � Solution: SIR-tree D S ( p i , q ) ≥ D E ( p i , q ) > p 5 D N ( p j , q ) ≥ D S ( p j , q ) � An R-tree that is generated on site set P � Leaf node stores: sites inside the corresponding MBR, the pointer to the vertices list of the tight/ loose cell and its neighbor list * For the purpose of clarity, textures on terrain are removed. 25 26 Indexing Land Surface Indexing Land Surface � SIR-Tree � SIR-Tree Insertion � An R-tree that is generated on site set P � Algorithm � Leaf node stores: sites inside the corresponding MBR, 1. locate p in I , find out the loose cell the pointer to the vertices list of the tight/ loose cell LC(r) containing p ; and its neighbor list p .neighbor � LC(r) ’s neighbor; 2 compute TC(p) and LC (p) ; 3 4 4 for each site p i in p .neighbor for each site p i in p .neighbor update LC(p j ) ’s edges according 5 to TC(p) ; update TC(p j ) ’s edges according 6 to LC(p) ; 7 insert p into I ; 8 return I ; 27 28 Indexing Land Surface Outline � More about TSI and LSI � Definitions: � Motivation � TSI , LSI and Neighbor � Please refer to Section 4.1, 4.2 in the paper. � Related Work � Observation: � Background � Given that TSI and LSI are generated for the same site set P the tight and loose cells have common edges; more P , the tight and loose cells have common edges; more � Indexing Land Surface specifically, all the tight cell’s edges are also the edges of loose cells. � Please refer to Section 4.2 Property 3 in the paper. � Query Processing � TSI and LSI Construction � Performance Evaluation � Naïve Index Construction � Conclusion and Future Work � Fast Index Construction � Please refer to Section 4.3 in the paper. 29 30 5
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