1. 1 On ‘Line graphs’ and Road Networks Peter Bogaert, Veerle Fack, Nico Van de Weghe, Philippe De Maeyer Ghent University

2. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 2 2 Modelling Real World Virtual World

3. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 3 3 Modelling Minimize data storage Fast answer Resemble real-life as much as possible

4. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 4 4 Road Network Modelling Specific case of a road network for navigation purposes on the network itself A Graph G(N, E, c) N {a,b,c,d,e,f, …} : a set of nodes E {(a,b) ; (a,c) ; (b,d) ; …} : a set of connections between nodes c : a cost that can be mapped onto each edge

5. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 5 5 Road Network Spatial problems : Graph theoretical problems A Shortest path Travelling Salesman problem (visit all nodes) Chinese Postman problem (visit all edges) Etc.

6. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 6 6 Road Network Mapping of a road network onto a graph Nodes : intersections and endpoints Edges : connections between intersections and endpoints 5 2 6 7

7. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 7 7 Road Network Adding Direction (Different Costs, OneWay) By means of a Directed Graph : D(N,E,c) 5 2 6 7 2 6 6

8. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 8 8 Road Network Adding Turn Cost and Prohibitions Cadwell (1961) node expansion (Directed or not)

9. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 9 9 Road Network Adding Turn Cost and Prohibitions Cadwell (1961), Kirby and Potts(1969) Disadvantage: Data storage Calculation time(e.g. Dijkstra with heaps O(n log n))

10. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 10 Road Network Adding Turn Cost and Prohibitions e.g. Jiang et al. By Using 'Turn Tables ’ For Shortest path same complexity O(nlogn)

11. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 11 Road Network Adding Turn Cost and Prohibitions E.g. Winter (2002) Using a line graph

12. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 12 Road Network Adding Turn Cost and Prohibitions Difference in Navigation Winter Turn Tables

13. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 13 Road Network Adding Turn Cost and Prohibitions E.g. Winter (2002) Better data structure then ‘ node expansion ’ Complexity for SP worse then using turn tables O (n log n) vs. O (e log e)

14. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 14 Road Network Adding Turn Cost and Prohibitions E.g. Winter (2002) Advantages vs. Normal representation Round toursCycles

15. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 15 Road Network Adding Turn Cost and Prohibitions E.g. Winter (2002) Advantages vs. Normal representation U- turns

16. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 16 Road Network Adding Turn Cost and Prohibitions E.g. Winter (2002) Problem concerning specific turns (U-turns) Winter : Splits Nodes (one lane = one node) Doubles number of nodes

17. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 17 Road Network Adding Turn Cost and Prohibitions E.g. Winter (2002) Problem concerning specific turns (U-turn) Winter : Splits Nodes (one lane = one node) Doubles number of nodes

18. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 18 Road Network Adding Turn Cost and Prohibitions Possible solution Using TurnTables in Combination with the line graph

19. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 19 Road Network Adding Turn Cost and Prohibitions Possible solution Turn Table: Defines Line * Line graph

20. GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 20 Conclusions and Future Work Conclusion Possible solution Combining the advantages of Line Graph and Turn Tables Levels in Topologic relations with line graph Future Work Implementing the different structures and comparing the different ‘real life’ calculation times

21. 21 On ‘Line graphs’ and Road Networks Peter Bogaert, Veerle Fack, Nico Van de Weghe, Philippe De Maeyer Ghent University Thank you for your attention