On ‘Line graphs’ and Road Networks

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

2. Modelling • Modelling • Real World Virtual World

3. Modelling • Modelling Minimize data storage Resemble real-lifeas much as possible Fast answer

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. Road Network • Spatial problems : Graph theoretical problems • A Shortest path • Travelling Salesman problem (visit all nodes) • Chinese Postman problem (visit all edges) • Etc.

6. Road Network • Mapping of a road network onto a graph • Nodes : intersections and endpoints • Edges : connections between intersections and endpoints 5 7 6 2

7. Road Network • Adding Direction (Different Costs, OneWay) • By means of a Directed Graph : D(N,E,c) 7 5 6 6 2 2 6

8. Road Network • Adding Turn Cost and Prohibitions • Cadwell (1961) • node expansion (Directed or not)

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. Road Network • Adding Turn Cost and Prohibitions • e.g. Jiang et al. • By Using 'Turn Tables’ • For Shortest path same complexity O(nlogn)

11. Road Network • Adding Turn Cost and Prohibitions • E.g. Winter (2002) • Using a line graph

12. Road Network • Adding Turn Cost and Prohibitions • Difference in Navigation Winter Turn Tables

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. Road Network • Adding Turn Cost and Prohibitions • E.g. Winter (2002) • Advantages vs. Normal representation Round tours Cycles

15. Road Network • Adding Turn Cost and Prohibitions • E.g. Winter (2002) • Advantages vs. Normal representation U- turns

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. 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. Road Network • Adding Turn Cost and Prohibitions • Possible solution • Using TurnTables in Combination with the line graph

19. Road Network • Adding Turn Cost and Prohibitions • Possible solution • Turn Table: Defines Line * Line graph

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