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

Modelling

Modelling

Real World Virtual World

3

GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 3

Modelling

Modelling

Minimize data storage

Fast answer

Resemble real-lifeas much as possible

4

GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 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

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

Road Network

Mapping of a road network onto a graph

Nodes : intersections and endpoints

Edges : connections between intersections and endpoints

5

26

7

7

GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 7

Road Network

Adding Direction (Different Costs, OneWay)

By means of a Directed Graph : D(N,E,c)

5

26

7

2

6

6

8

GISRUK - Topology and Spatial Databases Workshop – Glasgow 2005 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

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

WinterTurn 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 tablesO (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 tours Cycles

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

ConclusionPossible solution

Combining the advantages of Line Graph and Turn Tables

Levels in Topologic relations with line graph

Future WorkImplementing the different structures and comparing the different ‘real life’ calculation times

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