Indexing Network Voronoi Diagrams*

1. Indexing Network Voronoi Diagrams* Ugur Demiryurek and Cyrus Shahabi 2012 Speaker: Yihao Jhang Adviser: Prof. Yuling Hsueh

2. Outline • Abstract • Introduction • Voronoi Diagram • Network Voronoi Diagram • R-Tree • The Voronoi R-Tree • Approach • Quad-Tree • The Voronoi Quad-Tree • Experimental • Conclusion National Chung Cheng University

3. Abstract

4. Abstract • The Network Voronoi diagram and its variants have been extensively used in the context of numerous applications in road networks, particularly to efficiently evaluate various spatial proximity queries such as kNN. • Existing index structures, treating a network Voronoi cell as a simple polygon, may yield inaccurate results due to the network topology, and fail to scale to large networks with numerous Voronoi generators. • The author proposed the Voronoi Quad-tree to solve this problem. National Chung Cheng University

5. Introduction

6. Voronoi Diagram (VD) • be a set of distinct sites distributed in the Euclidean space. • for each with • Each edge of VC() is a segment of the perpendicular bisector of the line segment connecting to another point of the set P. National Chung Cheng University

7. Voronoi Diagram (cont.) • Definition 1. The region given by VC() = where is the minimum Euclidean distance between and . • Definition 2. Voronoi Diagram (VD)  National Chung Cheng University

8. Network Voronoi Diagram (NVD) • The Voronoi diagram with a spatial network. • and are the Voronoi generators (i.e., data objects such as restaurants, hotels). • toare the intersections on a road network that interconnected by a set of edges. National Chung Cheng University

9. Network Voronoi Diagram (cont.) National Chung Cheng University

10. Network Voronoi Diagram (cont.) • Definition: NVD: Network Voronoi Diagram. : Voronoi edge set of . National Chung Cheng University

11. R-tree • R-trees are tree data structures used for spatial access methods. • The key idea of the data structure is to group nearby objects and represent them with their minimum bounding rectangle in the next higher level of the tree. • Is a balanced search tree. National Chung Cheng University

12. The Voronoi R-tree • VR-tree for short. • VR-tree is based on the R-tree that splits the network space with hierarchically nested Minimum Bound Rectangles (MBR) generated around network Voronoi cells. National Chung Cheng University

13. The Voronoi R-tree (cont.) • First shortcoming Inaccurate results for a query. • False-negative edges. National Chung Cheng University

14. The Voronoi R-tree (cont.) National Chung Cheng University

15. The Voronoi R-tree (cont.) • Second shortcoming • Inefficient due to non-disjoint partitioning of the space. • Affected by topologies and distribution of the objects. • Example: The parent node(s) of the overlapping MBRs have to be accessed repeatedly in order to search the child nodes that contain q. Thus, with VR-tree the amount of work often depends on the overlapping areas of MBRs. National Chung Cheng University

16. Approach

17. Quad-Tree • A tree data structure in which each internal node has exactly four children. • Used to partition a two dimensional space by recursively subdividing it into four quadrants or regions. National Chung Cheng University

18. The Point Quad-tree • The point quadtree is an adaption of a binary tree used to represent two dimensional point data. It shares the features of all quadtrees but is a true tree as the center of a subdivision is always on a point. National Chung Cheng University

19. The Region Quad-tree • The region quadtree represents a partition of space in two dimensions by decomposing the region into four equal quadrants, sub quadrants, and so on with each leaf node containing data corresponding to a specific sub region. Each node in the tree either has exactly four children, or has no children (a leaf node). National Chung Cheng University

20. The VQ-tree • Enables disjoint decomposition of the underlying space. • Color code. • Concept of the region quadtree. National Chung Cheng University

21. The VQ-tree (cont.) National Chung Cheng University

22. Experiment

23. Experiment • Experimental Setup • VQ-tree vs. VR-tree • Dataset • California (CA), Los Angeles (LA) and San Joaquin County (SJ) • Workstation with 2.7 GHz CPU and 12GB RAM National Chung Cheng University

24. Experiment (cont.) • Ratio of False-Negative Edges National Chung Cheng University

25. Experiment (cont.) • Precomputation Time National Chung Cheng University

26. Experiment (cont.) • Ratio of Index Reconstruction Time National Chung Cheng University

27. Experiment (cont.) • Response Time National Chung Cheng University

28. VQ-tree enable efficient access to the network Voronoi cells containing a particular point or edge of the network. • Intend to pursue this study in two directions. • Investigate disk organization strategies for Voronoi Quad-tree. • Work on incremental index update techniques to avoid node reconstruction overhead due to update in the location of Voronoi generators. Conclusion Q&A Thank you for listening! National Chung Cheng University