GPS-based Navigation in Static and
Dynamic Environments
Master’s Thesis Presentation
Shahid Jabbar
Institut für Informatik
Universität Freiburg
Supervisor: PD Dr. Stefan Edelkamp
Co-supervisor: Prof. Dr. Th. Ottmann
GPS-based Navigation
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The big question ..
What are we doing here ?
Das 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 let 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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What is GPS ?
 A collection of 24 geo-stationary satellites.
 Gives the position of an object in terms of its
longitude, latitude, and height.
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Data Collection
What about the cost of collecting the data ?
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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.
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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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Graph Construction
Results of Line sweep
# GPS
Points
# Nodes in
Graph
Time to
sweep
1,277
1,473
0.42
1,706
1,777
0.27
2,365
2,481
0.37
50,000
54,267
11.13
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we are very
much
dependent
on k
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Where am I ?
 I am at building 101 and I want to go to CinemaxX.
 Schade!!! I have no existing trace that pass through
building 101.
 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 ?
 Voronoi Diagram to the rescue!!!
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Node localization
Results
# points
Naive
Construc- Searching
# queries
Searching
tion Time
Time
Time
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
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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 (contd…)
Results of Graph Compression
# Nodes
# Compressed
Nodes
Time
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 CinemaxX ASAP .. What to do ?
Search
 Dijkstra – Single-Source shortest path.
 A* - Goal directed Dijkstra
 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
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… O(n²) space
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Accelerating Search
Bounding-Box pruning (Wagner, Willhalm)
 With every edge, save a bounding box that contains at least
all the nodes that can be reached on a shortest path
originating from that particular edge.
b3
b2
t
e3
b1
b5
e2
e1
e4
e5
b4
s
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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Search Models
 Basic model

Shortest path
 Time model

Shortest + fastest path
 Absolute-time model

Timed queries
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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
–
–
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Schade Meldung
Dynamics
 Disturbances on road: A road accident or a traffic jam
 A road not usable at all  Edge weight = +inf
 Probably one lane of the road is still opened  Edge
weight increases by some delta
 Consequence:



The pre-computed information becomes invalid and useless.
Re-computing bounding boxes is very expensive.
This disturbance is temporary.
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Types of Disturbances
Area affected by
a disturbance
Affected Edge
Disturbances as Geometrical
Objects Model
Individual Edge Model
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Affect of disturbances on
pre-computed information
 Which information has become invalid ?


Everything ?
Nope, only those bounding boxes that have
intersections with affected edges are potentially
affected.
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Affect of disturbances on
pre-computed information
b3
b2
t
e3
b1
b5
e2
e1
e4
e5
b4
s
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Graph update – off-line approach
• Introduce the affect of disturbances on the graph
•
•
Simple for Individual Edge model – just increase the
edge weight of the affected edge.
A bit complex for Disturbances as Geometrical Object
model.
•
•
Problem: We need all the edges that are covered by a rectangle.
Solution: WindowQuery using Segment trees
• Perform search on the updated graph
• Use pruning information only if it is not affected
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Graph update – off-line approach
in Disturbances as Geometrical Objects model
b3
b2
t
e3
b1
b5
e2
e1
e4
e5
b4
s
Rectangle Intersection Problem or more precisely
Red – Blue Rectangle Intersection Problem
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Exploration time checking –
on-line approach
Observations:
1. Since the weights are always increased, if the
shortest path is not affected, it remains to be the
shortest path.
2. It is possible that some of the constraints
have terminated and no longer be there by
the time the mobile object will reach that
area.
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Exploration time checking –
on-line approach
General Strategy:
Before exploring an edge e, check if e is affected or not
 if e is affected then check whether the
constraints would be valid by the time e would
be traversed.


if constraints are valid then declare the search
procedure as invalid and use standard Dijkstra or A*.
else continue.
 else continue.
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Exploration time checking –
on-line approach
v
e
u
Disturbances
s
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Future Issues




Handling of large data sets
The compressed edges should not be considered
straight => Curved Edges
Visualization of route on a topographic map.
Bridges => 3D navigation.
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Thesis download
 http://www.informatik.uni-freiburg.de/~jabbar/thesis.pdf
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