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

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.

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

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

Geometric Travel Planning 5 Data Collection

Geometric Travel Planning 6 Data format,,, , , , , , , , , , , , , , , , , , , , , , , , ,

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

Geometric Travel Planning 8 Geometric Rounding Douglas-Peucker’s algorithm results using Hersberger and Snoeyink variant #pointsΘ= , ,7061,5401, ,3652,0831, ,00048,43242,21817,8534,3851,185

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

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 ?

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

Geometric Travel Planning 12 Node localization Results # points# queries Construc- tion Time Searching Time (sec) Naive Searching Time (sec) 1, , , , >10,000

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

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

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.

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

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, , , ,2674,

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

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

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

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 _/8 9,24 t 2

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

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

Geometric Travel Planning 24 GPS-R OUTE Architecture

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