Indexing Network Voronoi Diagrams*Ugur Demiryurek and Cyrus Shahabi2012Speaker: Yihao JhangAdviser: Prof. Yuling Hsueh

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Outline

▪ Abstract

▪ Introduction

▪ Voronoi Diagram

▪ Network Voronoi Diagram

▪ R-Tree

▪ The Voronoi R-Tree

▪ Approach

▪ Quad-Tree

▪ The Voronoi Quad-Tree

▪ Experimental

▪ Conclusion

Abstract

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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.

Introduction

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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.

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Voronoi Diagram (cont.)

▪ Definition 1.

The region given by VC() = where is the minimum Euclidean distance between and .

▪ Definition 2.

Voronoi Diagram (VD)

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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.

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Network Voronoi Diagram (cont.)

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Network Voronoi Diagram (cont.)

▪ Definition:

NVD: Network Voronoi Diagram.

: Voronoi edge set of .

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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.

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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.

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The Voronoi R-tree (cont.)

▪ First shortcoming

Inaccurate results for a query.

False-negative edges.

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The Voronoi R-tree (cont.)

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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.

Approach

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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.

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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.

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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).

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The VQ-tree

▪ Enables disjoint decomposition of the underlying space.

▪ Color code.

▪ Concept of the region quadtree.

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The VQ-tree (cont.)

Experiment

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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

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Experiment (cont.)

▪ Ratio of False-Negative Edges

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Experiment (cont.)

▪ Precomputation Time

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Experiment (cont.)

▪ Ratio of Index Reconstruction Time

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Experiment (cont.)

▪ Response Time

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Conclusion

▪ 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.

Q&A

Thank you for listening!