On line graphs and road networks

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 much 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). 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). 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). 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). As a result the number of data to be stored is reduced compared to the extended 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 …