1. Geometric Travel Planning 1 Stefan Edelkamp (University of Dortmund, Germany) Shahid Jabbar (University of Freiburg, Germany) Thomas Willhalm (Universtiy of Karlsruhe, Germany)

2. Geometric Travel Planning 2 The big question.. What are we doing here ? The Problemo  Digital maps available in the market are very expensive.  Most of those maps do not allow updates.  Not possible to have timed queries. The travel time can change drastically during different kinds of days like, workdays and holidays … Can even change during different times of a day like, from 8 to 9 AM as compared to 10 to 11 PM.

3. Geometric Travel Planning 3  Why not people make their own maps that they can query and update ?  But how ? How to collect the data ? How to process that data ? Global Positioning System (GPS) Receiver + Computational Geometry The big question.. What are we doing here ? The Solution

4. Geometric Travel Planning 4 Data Collection What about the cost of collecting the data ? We say …. You only need some Bananas.

5. Geometric Travel Planning 5 Data Collection

6. Geometric Travel Planning 6 Data format,,, 48.0070783, 7.8189867, 20030409, 100156 48.0071067, 7.8190150, 20030409, 100158 48.0071850, 7.8191400, 20030409, 100200 48.0071650, 7.8191817, 20030409, 100202 48.0071433, 7.8191867, 20030409, 100204 48.0071383, 7.8191883, 20030409, 100206 48.0071333, 7.8191917, 20030409, 100208 48.0071317, 7.8191917, 20030409, 100212

7. Geometric Travel Planning 7 Not everything that glitters is Gold. Filtering + Rounding  Kalman Filter GPS Information + Speed-o-meter reading as the inertial information => removes the outliers  Douglas-Peuker Line Simplification Algorithm Simplifies a polyline by removing the waving affect. Remove all the points that are more than Θ distance away from the straight line between the extreme points

8. Geometric Travel Planning 8 Geometric Rounding Douglas-Peucker’s algorithm results using Hersberger and Snoeyink variant #pointsΘ=10 -7 10 -6 10 -5 10 -4 10 -3 1,2777665582437722 1,7061,5401,16243311725 2,3652,0831,394376287 50,00048,43242,21817,8534,3851,185

9. Geometric Travel Planning 9 Lets sweeeeep … Graph Construction Bentley - Ottmann Line Segment Intersection Algorithm. We have multiple traces. Some of them might be intersecting  Road crossings!!! We need to convert them into a graph to be able to apply different graph algorithms e.g. shortest path searching Seems very simple, just convert: Point  Vertex Segment  Edge

10. Geometric Travel Planning 10 Where am I ?  I am at hotel building and I want to go to the Cinema.  Pity!!! I have no existing trace that pass through the hotel building.  What to do ? Hmmmm …interesting problem  How about going to the nearest place that is in my existing traces ?

11. Geometric Travel Planning 11 Where am I ?  Sounds good.. But how to find that nearest place ?  Voronoi Diagram to the rescue!!!

12. Geometric Travel Planning 12 Node localization Results # points# queries Construc- tion Time Searching Time (sec) Naive Searching Time (sec) 1,277 0.100.3012.60 1,706 0.240.5424.29 2,365 0.331.1443.3 50,000 13.7314.26>10,000

13. Geometric Travel Planning 13 My floppy is too small … how can I carry this file ? Graph Compression

14. Geometric Travel Planning 14 My floppy is too small … how can I carry this file ? Graph Compression (contd…)

15. Geometric Travel Planning 15 My floppy is too small … how can I carry this file ? Graph Compression  Problem: The original layout is destroyed => Restricted to perform the search only on intersection points + start/end points of traces Shortest path only in terms of intersection points + start/end points of traces.  Solution: A compression algorithm that can retain the original layout of the graph also.

16. Geometric Travel Planning 16 My floppy is too small … how can I carry this file ? Graph Compression (contd…) Algorithm Hide the original edges Delete the new edge

17. Geometric Travel Planning 17 My floppy is too small … how can I carry this file ? Graph Compression (contd…) Results of Graph Compression # Nodes # Compressed Nodes Time (sec) 1,4731990.01 1,777740.02 2,4811300.03 54,2674,3910.59

18. Geometric Travel Planning 18 I have to reach the Cinema ASAP.. What to do ? Search  Dijkstra – Single-Source shortest path.  A* - Goal directed Dijkstra

19. Geometric Travel Planning 19 #Points Sweep Time (sec) Search Time (sec) Expansions Dijkstra50,00011.130.2744,009 A*50,00011.130.2018,775 I have to reach the Cinema ASAP.. What to do ? Search (results)

20. Geometric Travel Planning 20 I have to reach Cinema ASAP.. What to do ? Search (results)  Number of queries is much more than the updates.  How about pre-computing some information ?  How about running All-pairs shortest path algorithm and saving all the paths:  Nope … O(n²) space

21. Geometric Travel Planning 21 Accelerating Search Bounding-Box pruning  With every edge, save a bounding box that contains at least the nodes that are reachable from the source node, on a shortest path using that edge. s 2 1 11 1 1 11 1 11 1 1 1 11 11 _/2 _/8 9,24 t 2

22. Geometric Travel Planning 22 Accelerating Search Bounding-Box pruning  In Dijkstra u  DeleteMin(PQ) forall v \in adjacent_edges(u) If t \in BB(u,v)..... Endif endfor

23. Geometric Travel Planning 23 Accelerating Search Bounding-Box pruning Results of 200 queries #Nodes Time + Expansions + Time – Expansions – 1990.346,5960.6019,595 4,3918.1165,72612.88217,430

24. Geometric Travel Planning 24 GPS-R OUTE Architecture

25. Geometric Travel Planning 25 Summary  Presented Map generation from GPS traces.  Geometric-, Graph- and AI-Algorithms  Running GPS Route Planning System  Available Visualisation with Vega  Future: Dynamic Updates Handling of large data sets