A NEW DATA STRUCTURE TO REPRESENT ROAD NETWORKS Peter Bogaert Vakgroep Geografie Universiteit Gent Krijgslaan 281, S8, 9000 Gent Tel: 09 264 4692 Fax: 09 264 4985 e-mail:
[email protected] Nico Van de Weghe Vakgroep Geografie Universiteit Gent Krijgslaan 281, S8, 9000 Gent Tel: 09 264 4696 Fax: 09 264 4985 e-mail:
[email protected] Ruben Maddens Vakgroep Geografie Universiteit Gent Krijgslaan 281, S8, 9000 Gent Tel: 09 264 4694 Fax: 09 264 4985 e-mail:
[email protected] Leen de Temmerman Vakgroep Geografie Universiteit Gent Krijgslaan 281, S8, 9000 Gent Tel: 09 264 4636 Fax: 09 264 4985 e-mail:
[email protected] Philippe De Maeyer Vakgroep Geografie Universiteit Gent Krijgslaan 281, S8, 9000 Gent Tel: 09 264 4688 Fax: 09 264 4985 e-mail:
[email protected]
Abstract In the domain of Geographic information systems networks are often used to solve spatial problems. The majority of these problems involve path calculation. In order to calculate a realistic path, the real world needs to be represented as accurate as possible in a data structure. In this paper we present a new data structure combining the advantages of two well know network data structures.
1. Introduction When data of the real world is stored on a computer, abstractions of real-life concepts have to be made. The data, necessary to store these concepts, needs to be minimised. On the other hand, the real world needs to be represented as accurate as possible. Ideally, an information system could answer a question in its virtual world that coincides with an exact answer in the real world. Networks – such as road networks – can easily be modelled as a graph (Miller, 2002). Spatial problems on these networks can then be translated to graph-theoretical problems, such as shortest path problems, or optimal (closed) walks visiting all nodes (Hamilton cycle or travelling salesman problem) or all edges (Euler circuit or Chinese postman problem) (Ahuja, 1993). The way in which the graph is modelled, is of the uttermost importance, since this describes how close the model reflects real life. In most cases, intersections and endpoints are represented by a node, and the roads connecting these intersections or endpoints are represented by edges (Miller, 2002). 2. Adding attributes important for navigation In order to implement the concept of one-way streets, a directed graph can be used. Different approaches have been used to model turn costs or restrictions. Kirby and Pots (1969) used an expanded representation of the network. Each intersection is split into dummy nodes, being connected by dummy edges. The turn costs are assigned to the dummy edges (Fig 1). Fig 1. Representation of the expanded network (Kirby and Pots, 1969)
The problem with this solution is the large increase of nodes and edges in the graph. This increases the data storage requirements as well as the calculation time of most spatial
problems, since the complexity of such problems is typically a function of the nodes in a graph. In order to deal with this problem, Jiang et al. (2002) suggested a link-based data structure, where a link-node table is used to represent the connectivity of the road network and a link-link table is used to represent turning restrictions (Fig 2). Fig 2. Representation of the link-based network
Cadwell (1961), Anez et al. (1996), and Winter (2002), suggest an alternative approach. They introduced the concept of the line graph, in which connections between intersections or endpoints are modelled as nodes, and the links between these connections are modelled as edges (Fig 3). Fig 3. Representation of the line graph
As a result the number of data to be stored is reduced compared to the expanded approach. Additionally, the approach allows common algorithms, like the shortest path algorithm, to run on the graph without adaptations. The line graph solution also allows cycles, U-turns and round tours in the results of algorithms that normally exclude these properties if the extended approach is used. However, even in the line graph, nodes sometimes have to be split in order to model a real-life situation. Fig 4. illustrates this problem.
Fig 4. Necessity of splitting nodes using a line graph
A car driving from c to b can pass e without any additional cost. If another driver plans a route from a to b, he can also pass node e, but in node e there is an additional cost for making the U-turn. Splitting these nodes increases data storage and computational time of graph algorithms. 3. Towards a new data structure To deal with this disadvantage, we suggest a new data structure combining both the advantages of the line graph and the link-based approach. Instead of a link-based approach, we suggest a ‘connection link’-based approach in combination with a line graph. This ‘connection link’ can be represented by the line graph of a line graph of a road network in which nodes represent intersections or endpoints, and connections between these nodes are represented by edges (Fig 5). Hence, we denote this structure as a line*line graph. Fig 5. the line*line graph
The overall idea is thus that the information of topological relations reaches a higher level each time a graph is transformed in to its line graph. Table 1 illustrates this idea. Each higher level is the line graph of a lower level. Table 1. Levels of information after the construction of a line graph Level Nodes Edges 0 (a graph) Intersections Road segments 1 (line graph) Road segments Turns 2 (line*line graph) Turns Connections between turns
4. Conclusions and future work We strongly believe that this new data structure will give good results both in calculation time and in data storage for realistic road networks that involve turn costs and prohibitions. In the near future we still have to implement the structure and verify the already promising prospectives. 5. References Ahuja, R.K., Magnanti, T.L. and Orlin, J.B. (1993), Network flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs, NJ. Anez, J., T. de la Barra, and B. Perez (1996), “Dual Graph Representation of Transport Networks”. Transportation Research 30(3), p. 209–216. Caldwell, T. (1961), “On finding minimum routes in a network with turn penalties”. Communications of the ACM 4(2), p. 107–108. Dijkstra, E. W. (1959), “A note on two problems in connection with graphs”. Numerische Mathematik 1, p. 269–271. Jiang J., Han H. and Chen J. (2002), “Modeling turning restrictions in traffic network for vehicle navigation system”. IAPRS, Vol. XXXIV, part 4. Kirby R.F and Potts R.B. (1969), ‘The minimum route problem for networks with turn penalties and prohibitions’. Transport Research 3, p. 397-408. Miller, H.J. and Shaw S.-L. (2001), Geographic Information Systems for Transportation: Principles and Applications. Oxford University Press, Oxford. Winter, S. (2002), “Modeling Costs in Turns of Route Planning”. GeoInformatica 6(4), p. 345-360.
