Stefan Edelkamp (University of Dortmund, Germany)

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Geometric Travel Planning

• Stefan Edelkamp (University of Dortmund, Germany)
• Shahid Jabbar (University of Freiburg, Germany)
• Thomas Willhalm (Universtiy of Karlsruhe, Germany)

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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.

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The big question .. What are we doing here ?The
Solution

• 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

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Data Collection
What about the cost of collecting the data ?
We say .
You only need some Bananas.
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Data Collection
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Data format

• ltlongitudegt, ltlatitudegt, ltdategt, lttimegt
• 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

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Not everything that glitters is Gold.Filtering
Rounding

• Kalman Filter
• GPS Information Speed-o-meter reading as the
  inertial information gt removes the outliers
• Douglas-Peuker Line Simplification Algorithm
• Simplifies a polyline by removing the waving
  affect.
• Remove all the points that are more than T
  distance away from the straight line between the
  extreme points

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Geometric Rounding Douglas-Peuckers algorithm
resultsusing Hersberger and Snoeyink variant
points T10-7 10-6 10-5 10-4 10-3
1,277 766 558 243 77 22
1,706 1,540 1,162 433 117 25
2,365 2,083 1,394 376 28 7
50,000 48,432 42,218 17,853 4,385 1,185
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Lets sweeeeep Graph Construction

• We have multiple traces.

• 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

• Bentley - Ottmann Line Segment Intersection
  Algorithm.

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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 ?

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Where am I ?

• Sounds good .. But how to find that nearest place
  ?

• Voronoi Diagram to the rescue!!!

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Node localizationResults
points queries Construc-tion Time Searching Time (sec) Naive Searching Time (sec)
1,277 1,277 0.10 0.30 12.60
1,706 1,706 0.24 0.54 24.29
2,365 2,365 0.33 1.14 43.3
50,000 50,000 13.73 14.26 gt10,000
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My floppy is too small how can I carry this
file ?Graph Compression
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My floppy is too small how can I carry this
file ?Graph Compression (contd)
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My floppy is too small how can I carry this
file ?Graph Compression

• Problem
• The original layout is destroyed
• gt
• 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.

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My floppy is too small how can I carry this
file ?Graph Compression (contd) Algorithm
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My floppy is too small how can I carry this
file ?Graph Compression (contd)Results of
Graph Compression
Nodes Compressed Nodes Time (sec)
1,473 199 0.01
1,777 74 0.02
2,481 130 0.03
54,267 4,391 0.59
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I have to reach the Cinema ASAP .. What to do
?Search

• Dijkstra Single-Source shortest path.
• A - Goal directed Dijkstra

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I have to reach the Cinema ASAP .. What to do
?Search (results)
Points Sweep Time (sec) SearchTime (sec) Expansions
Dijkstra 50,000 11.13 0.27 44,009
A 50,000 11.13 0.20 18,775
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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

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Accelerating SearchBounding-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.

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Accelerating SearchBounding-Box pruning

• In Dijkstra
• u ? DeleteMin(PQ)
• forall v \in adjacent_edges(u)
• If t \in BB(u,v)
• .....
• .....
• Endif
• endfor

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Accelerating SearchBounding-Box pruningResults
of 200 queries
Nodes Time Expansions Time Expansions
199 0.34 6,596 0.60 19,595
4,391 8.11 65,726 12.88 217,430
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GPS-ROUTE Architecture
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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