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
<longitude>, <latitude>, <date>, <time>
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 => 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
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Geometric Rounding
Douglas-Peucker’s algorithm results
using Hersberger and Snoeyink variant
#points Θ=10-7
10-6
10-5
10-4
10-3
1,277
766
558
243
77
22
1,706
2,365
1,540
2,083
1,162
1,394
433
376
117
28
25
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
Some of them might be intersecting
 Road crossings!!!
•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 localization
Results
# points
Naive
Searching
ConstrucSearching
# queries
Time
tion Time
Time
(sec)
(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
>10,000
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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
=>
 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
Hide the
original edges
Delete the
new edge
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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)
Search
Sweep
#Points
Time Expansions
Time (sec)
(sec)
Dijkstra 50,000
A*
50,000
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0.27
44,009
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 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.
1
1
1
1
1
_/8
1
t
1
1
1
1
1
1
1
_/2
1
_/2
2
_/2
_/2
s
2
1
1
1
9,24
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Accelerating Search
Bounding-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 Search
Bounding-Box pruning
Results of 200 queries
#Nodes
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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Time Expansions
–
–
23
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
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