A Probabilistic Model for Road S election in Mobile Maps

1. A Probabilistic Modelfor Road Selectionin Mobile Maps Thomas C. van DijkJan-HenrikHaunert W2GIS, 5 April 2013

2. Overview: a little bit of … modelling, interpretation, … algorithms, … evaluation, … outlook, …

3. Road selection focus-and-context maproute map • Depends on user task

4. Road selection destination map Kopf et al.focus maps Yamamoto et al.

5. Motivating example • Pedestrian with a smartphone • Current location available • Wayfinding to a destination • Interest in surroundings • Freedom of movement • Contrast: in-car sat nav

6. From scoring to selection • Maximise score • using given number of road segments • Minimise number of road segments • selecting at least a given amount of score • Additional constraints?

7. Betweenness centrality Consider all shortest paths between all pairs of nodes; count how often each edge is used Folklore interpretation: • People are likely to travel shortest paths • Heavily traveled edges are important

8. Betweenness centrality Consider all shortest paths between all pairs of nodes; count how often each edge is used Probabilistic interpretation: • Start at a uniformly random node,uniformly random destination,take a uniformly random shortest path • Probability that you visit a certain edge

9. PageRank primer • Origin in web page ranking • (Page, Brinn, Motwani, Winograd… Google) • Previous application in predicting human movement (Jiang) • Usually stated as eigenvector calculation • Also: interpretation in terms of random walks

10. Traveler Random Surfer model on road segment • Hypothetical user is at a web page • Transitions along a uniformly random link • All transitions take equal time • Sometimes jumps to a uniformly random page turn road segment

11. Non-uniform transition probability • Walking along a road segment… • Coming up to a crossing… • Where to go…? • Shortest path? Straight on? Anywhere? • What’s the network? • Web pages, hyperlinks?

12. Bidirected line graph

13. Bidirected line graph

14. Bidirected line graph

15. Bidirected line graph

16. Bidirected line graph

17. Non-uniform transition probability • At a node of the line graph… • Follow an arc… • Where to go…? • Straight on? • Constant given road network • Shortest path? • Constant given road network and destination

18. Non-uniform damping • PageRank jumps to uniformly random node • Instead, jump to the current user location! Probabilistic interpretation: • Consider random walk starting at user location • Exponential distribution of walk length • Score of node = probability of going to node

19. Data set • OpenStreetMap • City of Würzburg and surroundings • Lower Franconia, Germany • Approx 105 nodes • Examples here: crop to the city itself • 3786 nodes, 4987 road segments

26. Demo

27. Shortest-path bonus

28. Non-turning bonus

29. Runtime of simple implementation • Calculate PageRank • Loop over nodes • Distribute score to neighbours • Repeat • Select nodes with high score • Runtime: ~100 ms

30. ‘Direct’ selection • Select all nodes with high rank • Select no nodes with tiny rank • O( log n ) time • Correctness holds almost surely • Almost immediately from Borgset al’salgorithm for Significant PageRanks.

31. Really big maps? • Just calculate, then select • Approximation algorithm: • Runtime O( log(n) log(ε-1) ε-1ρ-2)

32. Gaps • Selection is not necessarily connected • Simplified example:

33. Gaps

34. Strokes

35. Overview: a little bit of … modelling, interpretation, … algorithms, … evaluation, … outlook, …

36. Q?

37. A!