1. Indexing Network Voronoi Diagrams*
Hello, today I want to present this paper.
Indexing Network Voronoi Diagrams.
It come from International Conference DASFAA 2012.
Ugur Demiryurek and Cyrus Shahabi
2012
Speaker: Yihao Jhang
Adviser: Prof. Yuling Hsueh

2. Outline Abstract Introduction Approach Experimental Conclusion
Voronoi Diagram
Network Voronoi Diagram
R-Tree
The Voronoi R-Tree
Approach
Quad-Tree
The Voronoi Quad-Tree
Experimental
Conclusion
This is the outline that I will go through later.
National Chung Cheng University

3. Abstract
First is the 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.
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.
Although the existing approaches successfully utilize the network Voronoi diagram
as a way to partition the space for their specific problems,
there is little emphasis on how to efficiently find and access the network Voronoi cell
containing a particular point or edge of the network.
In this paper,
author study the index structures on network Voronoi diagrams
that enable exact and fast response to contain query in road networks.
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.
So, author propose a method, termed Voronoi-Quad-tree (or VQ-tree for short)
Use Quad-tree to index network Voronoi diagrams to address both of these shortcomings.
Finally, demonstrate the efficiency of VQ-tree via experimental evaluations
with real-world datasets consisting of a variety of large road networks with numerous data objects.
National Chung Cheng University

5. Introduction Next is the introduction.
I will talk about some background techniques.

6. Voronoi Diagram (VD)
𝑃:{ 𝑝 1 , 𝑝 2 ,…, 𝑝 𝑛 } be a set of 𝑛 distinct sites distributed in the Euclidean space.
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑞, 𝑝 𝑖 <𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒(𝑞, 𝑝 𝑗 ) for each 𝑝 𝑗 ∈𝑃 with 𝑗≠𝑖
Each edge of VC( 𝑝 𝑖 ) is a segment of the perpendicular bisector of the line segment connecting 𝑝 to another point of the set P.
First is Voronoi diagram.
Let P be a set of n distinct sites, like generator points.
These generator points can be considered any spatial type of objects,
like gas station or restaurant.
We define the Voronoi diagram of P as the subdivision of the space into n cells,
one for each site in point set P, with the property that a point q lies in the cell corresponding to a site pi
if and only if distance from q to pi less than distance from q to pj.
Each edge of Voronoi cell is a segment of the perpendicular bisector
of the line segment connecting p to another point of the set P.
So we can see this figure.
This figure show s the ordinary Voronoi diagram of eight points where the distance metric in Euclidean.
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7. Voronoi Diagram (cont.)
Definition 1.
The region given by VC( 𝑝 𝑖 ) = 𝑝|𝑑(𝑝, 𝑝 𝑖 )≤(𝑝, 𝑝 𝑗 ) where 𝑑(𝑝, 𝑝 𝑖 ) is the minimum Euclidean distance between 𝑝 and 𝑝 𝑖 .
Definition 2.
Voronoi Diagram (VD) 
𝑉𝐷 𝑃 ={𝑉𝐶 𝑝 𝑖 ,…,𝑉𝐶( 𝑝 𝑛 )}
Here, we have two definition for Voronoi diagram.
Definition 1 is defined for what is Voronoi cell and Definition 2 is defined for what is the Voronoi diagram.
As shown on the figure,
The gray region of p4 represent a cell as defined in definition1.
And the distance for each point in the gray region is nearer to p4 than to other points.
And Voronoi diagram is set of these cells.
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8. Network Voronoi Diagram (NVD)
The Voronoi diagram with a spatial network.
𝑝 1 , 𝑝 2 , and 𝑝 3 are the Voronoi generators (i.e., data objects such as restaurants, hotels).
𝑝 4 to 𝑝 16 are the intersections on a road network that interconnected by a set of edges.
Next, after the Voronoi diagram,
I’m going to talk about the NETWORK Voronoi diagram.
With network Voronoi diagram,
the Voronoi diagram described above is generalized by replacing the Euclidean space with a spatial network,
like a road network.
Let’s see the figure (a).
p1, p2 , and p3 are the Voronoi generators, such as restaurants or hotels.
And p4 to p16 are the intersections on a road network that interconnected by a set of edges.
And figure B shows the NVD of the road Network where each line style corresponds to the shortest path tree
based on the generators p1, p2 and p3.
Each shortest path tree composes a network Voronoi cell and some edges
can be partially contained in different network Voronoi cells.
The border point b1 to b7 are the nodes where the shortest path trees meet as a result of the parallel Dijkstra algorithm.
The border points between any two generator pi and pj are equally distanced from pi and pj.
For example, b1 between p1 and p2 are equally distanced from p1 and p2.
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9. Network Voronoi Diagram (cont.)
This figure shows a real network Voronoi diagram with respect to 50 data objects in Los Angeles road network.
Each network node marked with a different color corresponds to a network Voronoi cell.
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10. Network Voronoi Diagram (cont.)
Definition:
𝑁𝑉𝐷 𝑃 ={ 𝑉 𝑒𝑑𝑔𝑒 𝑝 1 ,…, 𝑉 𝑒𝑑𝑔𝑒 𝑝 𝑛 }
NVD: Network Voronoi Diagram.
𝑉 𝑒𝑑𝑔𝑒 𝑝 𝑖 : Voronoi edge set of 𝑝 𝑖 .
This is a simple definition for Network Voronoi Diagram.
The Vedge is Voronoi edge set of pi
And the Network Voronoi diagram is a set of Vedge.
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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.
Next, I going to introduce the R-tree.
R-trees are tree data structures used for spatial access methods.
And was proposed by Antonin Guttman in 1984.
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.
At the leaf level, each rectangle describes a single object;
at higher levels the aggregation of an increasing number of objects.
This can also be seen as an increasingly coarse approximation of the data set.
Similar to the B-tree, the R-tree is also a balanced search tree,
organizes the data in pages, and is designed for storage on disk.
The figure shows a simple concept of R-tree. MBR and tree structure.
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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.
𝑐𝑜𝑛𝑡𝑎𝑖𝑛(𝑞)
After we introduce the R-tree and the network Voronoi diagram.
Next, I going to talk about The Voronoi R-tree. VR-tree for short.
VR-tree is based on the R-tree that splits the network space with hierarchically nested MBR generated around network Voronoi cells.
Given the location of a query point q,
a contain q function query invoked on VR-tree starts from the root node
and iteratively checks the MBRs with respect to a q to decide
whether or not to further search the child nodes.
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13. The Voronoi R-tree (cont.)
First shortcoming
Inaccurate results for a 𝑐𝑜𝑛𝑡𝑎𝑖𝑛(𝑞) query.
False-negative edges.
But VR-tree has two main shortcomings.
Let’s see the first shortcoming.
VR-tree may yield inaccurate results for a contain query.
This is because VR-tree makes the simplifying assumption
that although the Network Voronoi Diagram is computed based on the network distance metric,
its Network Voronoi Cells are treated as regular polygons and index using the R-tree.
However, such approach may cause misclassification of the network edges in the network Voronoi cell
and hence inaccurate results.
We call these network edges are false-negative edges.
We can see Figure B, although the new edges are included inside the Voronoi cell of p1,
the network distance from any point on the false-negative edges to p3 is shorter than p1.
Thus, with VR-tree, when q is located on false-negative edges.
The function contain q will return incorrect Voronoi generator as the NN.
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14. The Voronoi R-tree (cont.)
This figure depicts the Network Voronoi cell of a particular data object in Los Angeles road network
where border nodes and false-negative edges are marked by light blue and red color respectively.
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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.
And this is Second shortcoming of VR-tree.
VR-tree is inefficient due to non-disjoint partitioning of the space.
And depending on the different topologies of the road network
and the distribution of the objects on the network segments,
The overlapping areas of MBRs of network Voronoi cells may be quite large,
and hence significant computation overhead in traversing R-tree for contain(q) query.
For example:
The parent nodes 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.
The author also implemented VR-tree with R+ tree to reduce the impact of overlapping areas.
However, author observe that the performance of VR+ tree is still less as compared to VQ-tree was proposed by the author.
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16. Approach
Next, I going to present the 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.
Here, I introduce the Quad-tree.
A quadtree is a tree data structure in which each internal node has exactly four children.
Quadtrees are most often used to partition a two dimensional space
by recursively subdividing it into four quadrants or regions. Like this figure.
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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.
There are two types of the quad tree,
first is the point quad tree and second is the region quad tree.
First, I going to introduce the point quadtree.
Point quad tree 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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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).
Second type is the region quadtree.
Region quadtree represents a partition of space in two dimensions
by decomposing the region into four equal quadrants,
subquadrants, and so on with each leaf node containing data corresponding to a specific subregion.
Each node in the tree either has exactly four children, or has no children (a leaf node).
A region quadtree may also be used as a variable resolution representation of a data field.
For example, the temperatures in an area may be stored as a quadtree,
with each leaf node storing the average temperature over the subregion it represents.
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20. The VQ-tree Enables disjoint decomposition of the underlying space.
Color code.
Concept of the region quadtree.
Here, is our main approach proposed by author, The Voronoi Quad-tree.
It enables disjoint decomposition of the underlying space to solve the problem of VR-tree.
The main observation behind VQ-tree is that each color coded area in Left figure is a spatially contiguous region in the network space.
The regions are mutually exclusive as they do not have any overlapping areas
and collectively exhaustive as every location in the network space is associated with at least one generator.
Therefore, an exact approximation of the network Voronoi diagram can be obtained
by using a region quad-tree
where the leaf nodes of the quad-tree correspond to a region in a Voronoi cell in Network Voronoi Diagram.
The VQ-tree algorithm recursively subdivide the quadrants until each quadrant contains only one network Voronoi cell information.
And Right figure illustrates the quad-blocks generated on the road network in Left figure.
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21. The VQ-tree (cont.)
Algorithm 1 presents the outline for VQ-tree. Given a set of N nodes with their color codes and bounding box x1, x2, y1, y2 that contains N as an input.
And the algorithm creates VQ-tree by recursively splitting the quadrants until all the nodes in a quadrant have the same color code.
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22. Experiment
Next is the experiments

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
Author conducted experiments with different spatial networks
and various parameters to evaluate the performance of VQ-tree and VR-tree.
It dataset is used California (CA), Los Angeles (LA) and San Joaquin County (SJ) networks.
And workstation with 2.7 GHz Pentium Core Duo processor and 12 GB RAM.
And experimental parameters value in table 1.
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24. Experiment (cont.) Ratio of False-Negative Edges
First experiment is ratio of false-negative edges.
To identify false-negative edges,
author compare the encoded values of each node based on VR-tree and VQ-tree.
Figure a shows the ratio of false-negative edges of both networks
where the object cardinality ranging from 10 to 1000.
The ratio of incorrectly identified edges is 16% on average in both networks.
And maximum recorded for LA and CA is 24% and 29% respectively.
Figure B illustrates the ratio of false-negative edges with different object distribution for both CA and LA road networks.
We can observe that the number of false-negative edges is less in Gaussian distribution.
This is because as objects are clustered in the spatial network with Gaussian distribution,
the corresponding shortest path trees would be less disperse and hence spatially close border points.
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25. Experiment (cont.) Precomputation Time
Next experiment is compare Precomputation time of VQ-tree with VR-tree.
Figure A shows the Precomputation time with varying network size.
And Figure B illustrates the impact of object cardinality over Precomputation time in LA road network.
We observe that as the number of objects increases, the preprocessing time for both approaches increases.
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26. Experiment (cont.) Ratio of Index Reconstruction Time
Next experiment is compare the index reconstruction time of VQ-tree with VR-tree.
In this experiments,
author update the location of the randomly selected data objects
and measure the index reconstruction overhead in both VR-tree and VQ-tree.
We can observe that VQ-tree outperforms in VR-tree with respect to index reconstruction.
This is because the insert operations in VR-tree are expensive.
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27. Experiment (cont.) Response Time
Final experiment is compare the response time of VQ-tree with VR-tree.
The results indicate that VQ-tree outperforms VR-tree and scales better with large number of data objects.
This is because of the fact that, with VR-tree,
the amount of work often depends on the size of the overlapping areas.
The overlapping areas may belong to more than one NVC
and hence during the search the parent nodes of the overlapping MBRs have to be accessed repeatedly.
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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!
Final we make a conclusion.
VQ-tree enable efficient access to the network Voronoi cells containing a particular point or edge of the network.
Author also intend to pursue this study in two directions.
First is investigate disk organization strategies for Voronoi Quad-tree.
Second , work on incremental index update techniques to avoid node reconstruction overhead due to update in the location of Voronoi generators.
Ok. It’s end for this present.
Thank you for listening.
National Chung Cheng University