Generation of constrained network Voronoi diagram using linear tessellation and expansion method
Introduction
The Voronoi diagram (abbreviated as VD), also known as the Voronoi tessellation, is one of the fundamental geometric constructions for partitioning a space populated with interesting elements through a competitive approach. It plays an important role in GIS domain, such as the detection of the service area of point of interest (POI), the neighborhood analysis of point clusters, and the determination of region of influence of urban facilities (Gahegan and Lee, 2000, Gold, 1991, Gold, 1992, Gold, 1994, Hanjoul et al., 1989, Lee and Lee, 2009). It has been suggested that the neighborhood relations defined by Voronoi diagrams are more closely related to human perception than other related data models (Boots et al., 2003, Chen, 1999, Gold, 1991, Okabe et al., 1992, Okabe et al., 2008). The VD construction depends on the space type and the definition of spatial distance. Conventionally, in Euclidean space, a VD is the partitioning of the plane into N polygonal regions, each of which is associated with a given point. The region associated with a point is the locus of points closer to that point by some criterion than to any other given point (Lee & Drysdale, 1981). This type of VD can be called the planar Voronoi diagram (or P-VD, for short) based on the 2D Euclidean distance. For the applications of route navigation and street infrastructure planning in network space, based on the network route distance (such as the shortest path), we can obtain the network VD (N-VD, for short) (Okabe et al., 2008). The N-VD considers the factor that movement in urban areas is constrained by the street network under certain transformation conditions.
The traditional P-VD assumes that the real world is abstracted as an infinitely homogeneous and isotropic space, and the distance between two events or facilities is measured by the Euclidean distance. In the field of location based service (LBS), which is capable of delivering geographic information and geo-processing power to mobile users according to their current location (Beatty, 2002, Huang and Wu, 2008, Jiang and Yao, 2006, Li, 2006, Schiller and Voisard, 2004), the associated urban space with the distributed POI facilities is a finite 1-D space that is usually not homogeneous, and the spatial connection between mobile clients and facilities is actually along networks. Yamada and Thill, 2004, Yamada and Thill, 2007 illustrates the greater efficiency of network distance versus Euclidean distance measures with a typical example of network-constrained point processes, i.e., vehicle crash distribution. Miller (1994) concludes that the planar method tends to overestimate the clustering tendency of network phenomena. However, LBS technology urgently requires a geometric model, such as a VD model, to conduct range queries and nearest-neighbor queries. The LBS providers most frequently receive inquiries such as this: “What is the nearest gas station to my car?”, which is actually among the spatio-temporal queries supported by VD.
Compared with P-VD using the Euclidean distance metric, the N-VD is more appropriate for spatial phenomena or activities constrained by transportation networks, especially in the field of microscopic analysis (Ai and Yu, 2013, Okabe and Okunuki, 2001, Okabe and Suzuki, 1997, Ratcliffe, 2002, Xie and Yan, 2008). Research on N-VD concepts, generation methods and applications has been an active issue for decades. Okabe et al. (2008) formulates six types of generalized N-VD, including directed N-VD, weighted N-VD, kth nearest point N-VD, farthest point N-VD, line N-VD, polygon N-VD and point-set N-VD, which consider the constraints of street directionality, the importance of the feature, and facility data storage type. Related works propose a computational method for constructing the corresponding N-VDs through the extended shortest-path tree (ESPT). This method is “vector-based” using Dijkstra’s algorithm under graph theory (Dijkstra, 1959). Alternatively, Tan, Zhao, and Wang (2012) and Xie and Yan (2008) present methods to construct N-VD by dividing the network into equal-length linear units and, using this model, to estimate the kernel density of traffic accidents occurring in a street network. However, the method by Tan et al. (2012) still calculates the network distance using the shortest-path tree technique, which lacks efficiency for large-scale data sets. However, the method is developed in a static environment, considering neither the difference of travel speed over time nor traffic constraints, such as one-way restriction. By applying these models to the nearest-neighbor query in LBS, the returned closest facility most likely remains the same for each time interval, which is not consistent with the reality due to the mobility of mobile clients and changes in traffic conditions. As noted by Okabe et al. (2008) and Weber and Kwan (2002), the practical constraints of time-constrained traffic capacity must be considered for real dynamic conditions.
Because traffic conditions change over time, it is necessary to dynamically build a time-constrained N-VD. In this study, we attempt to develop a novel approach for dynamically constructing constrained N-VD on time-dependent networks. The proposed method tries to extend the idea of raster extension in a planar space to a network space using the idea of stream flowing. The basic idea is that the stream flows along certain linear channels until it meets others or arrives at the end of a route. In our method, the network space is represented by basic linear units of equal length, which are termed lixels by Xie and Yan (2008). A new distance, “lixel distance”, is defined to measure the travel cost and is set up on the traffic conditions of the corresponding street.
The remainder of the paper is organized as follows. Section 2 presents a method to partition the network into a discrete representation. Section 3 discusses the algorithm to construct N-VD based on the stream flowing idea. A case study and experiment on real street network and facilities data is conducted in Section 4. Section 5 concludes by discussing future works.
Section snippets
Network tessellation
For a planar space, we can use the raster expansion operation to construct P-VD, as illustrated in Fig. 1. The raster data structure that tessellates the planar space into a discrete representation using field theory models the correlation of homogenous space in a global way. Through neighboring operations (such as dilation operator in mathematical morphology) among grid units, one can easily obtain the P-VD (Chen, 1999). The advantage of a discrete data structure can also be extended to the
N-VD construction by stream flow ideas
We think the ordinary Voronoi diagram represents the impact range of each event entity through equal competition. Each entity within the VD cell expands outward to gain growth area and finally encounters the same expansion from neighboring entities. This process is similar to that of the stream flow in network space when considering the expansion route around the network edge. The event entity (facility in the network) acts as the stream source and the generator in N-VD generation. The flows
Data
The algorithm is tested in an experiment system of network analysis, which is developed using the platform of Microsoft Visual C++ 6.0. The test dataset is from a real transportation network system with a set of POI (points of interest) facilities in part of the city of Guangzhou, China, as shown in Fig. 13. The data are provided by the Guangzhou Urban Planning and Design Survey Research Institute. The number of total POI facilities is 74,755, grouped into 20 main categories and 446
Conclusions
Whether in the traditional network analysis or in current IT-driven technology, e.g., LBS, the N-VD plays an important role in applications of road navigation, route finding and infrastructure facility planning. Because the movement in a network is constrained in graph space rather than in 2D planar space, the network analysis needs to replace traditional P-VD with N-VD. Furthermore, in real network applications, there are many conditional constraints impacting the N-VD generation. In this
Acknowledgements
This research is supported by the Open Fund of Key Laboratory of Urban Land Resources Monitoring and Simulation, Ministry of Land and Resources, and the National High-Tech Research and Development Plan of China under Grants No. 2012AA12A404.
References (31)
- et al.
Modeling retail trade areas using higher-order, multiplicatively weighted Voronoi diagrams
Journal of Retailing
(1997) Generating and updating multiplicatively weighted Voronoi diagrams for point, line and polygon features in GIS
Computers & Geosciences
(2008)- et al.
Data structures and algorithms to support interactive spatial analysis using dynamic Voronoi diagrams
Computers, Environment and Urban Systems
(2000) - et al.
The algebraic basis of mathematical morphology I. Dilations and erosions
Computer Vision, Graphics, and Image Processing
(1990) - et al.
Location-based services and GIS in perspective
Computers, Environments and Urban Systems
(2006) - et al.
A generic triangle-based data structure of the complete higher order Voronoi diagrams for distributed disaster and emergency management
Computers, Environments and Urban Systems
(2009) User preferences, information transactions and location-based services: A study of urban pedestrian wayfinding
Computers, Environment and Urban Systems
(2006)- et al.
Locational optimization problems solved through Voronoi diagrams
European Journal of Operational Research
(1997) - et al.
Kernel density estimation of traffic accidents in a network space
Computers, Environment and Urban Systems
(2008) - et al.
Comparison of planar and network K-functions in traffic accident analysis
Journal of Transport Geography
(2004)
Algorithm for constructing network Voronoi diagram based on flow extension ideas
Acta Geodaetica et Cartographica Sinica
Location-based services: Navigation for the masses, at last!
Journal of Navigation
Modelling geographical systems: Statistical and computational applications
A raster-based method for computing Voronoi diagrams of spatial objects using dynamic distance transformation
International Journal of Geographical Information Science
A note on two problems in connexion with graphs
Numerische Mathematic
Cited by (30)
Resilience analysis in road traffic systems to rainfall events: Road environment perspective
2024, Transportation Research Part D: Transport and EnvironmentTransit Voronoi diagrams in multi-mode public transport networks
2022, Computers, Environment and Urban SystemsCitation Excerpt :They also introduced several N-VD variants, such as weighted N-VDs, k-order N-VDs, line N-VDs, polygon N-VDs and point-set N-VDs, However, the dynamic traffic conditions in the road network were not considered. Ai et al. (2015) proposed a raster-based algorithm for constructing N-VDs in time-dependent road networks. This proposed algorithm discretizes road links into a set of linear units of equal length, and employs a raster-based stream flowing approach to construct N-VDs.
An iterative tessellation-based analytical approach to the design and planning of waste management regions
2021, Computers, Environment and Urban SystemsCitation Excerpt :Thiessen polygons are used in different fields and have been recently used in waste management studies (Karimi et al., 2020; Khan et al., 2018; Richter, Karimi, & Ng, 2019). They are a vital structure in GIS (Ai et al., 2015; Mu, 2009), and are drawn using Delaunay Triangulation (Esri, 2020b). Thiessen polygons define a proximal area, where any point within the area is closer to its seed that any other point in the set (Esri, 2020b).
The role of compactness distribution on the development of regionalized waste management systems
2021, Journal of Cleaner ProductionCitation Excerpt :In this study, Centroidal Voronoi Tessellation (CVT) is used to recursively build Thiessen polygons to optimize regions for waste management. CVT is a topic of considerable interest in many different fields (Freeman et al., 2014), and it has been successfully applied in transportation engineering (Ai et al., 2015), land use (Fleischmann et al., 2020), geology (Huang et al., 2020), and hydrology (Di et al., 2020). The percent feature count (equation (1)) considers the count of features within an area, then divides it by the total number of features in the entire tessellation.
Spatiotemporal heterogeneous allocation to support service area response
2020, Computers, Environment and Urban SystemsA mathematical morphology based method for hierarchical clustering analysis of spatial points on street networks
2019, Applied Soft Computing JournalCitation Excerpt :Nevertheless, for detecting multi-scale spatial clusters, sometimes we should turn to another prominent and useful class of method, i.e. hierarchical (agglomerative) clustering, which first defines all objects as individual clusters and then gradually agglomerates/merges the closest pair of clusters until all data points belong to one cluster (or until a desired number of clusters remain). Spatial clustering depends on the type of object space and the definition of geometric distance [12,13]. Previously, most of the hierarchical clustering methods, along with the partitioning clustering and the density-based clustering methods, consider the geographic space as a homogeneous and isotropic space under Euclidean space representation.