Euler Trails and Circuits
Euler Trails and Circuits
Tarik Reza TohaMd. Moyeen Uddin
Department of Computer Science and EngineeringBangladesh University of Engineering and Technology,
Dhaka, Bangladesh
December 7, 2015
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Euler Trails and Circuits
The Euler Tour
1 The Euler Tour
2 Euler Trails and Circuits
3 Fleury’s Algorithm
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Euler Trails and Circuits
The Euler Tour
Let’s Try to Clean Our City
Garbage Collector Truck
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Euler Trails and Circuits
The Euler Tour
The Story Begins ...
The bridge over the river Pregolya, Kaliningrad, Russia
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Euler Trails and Circuits
The Euler Tour
Back to The 18th CenturyThe City of Konigsberg, Prussia
The seven bridges around the island Kneiphof over Pregel river
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Euler Trails and Circuits
The Euler Tour
Leonardo Euler (1707-1783)e ix = cos x + i sin x
Euler wrote, “This question isso banal, but seemed to meworthy of attention in that[neither] geometry, nor algebra,nor even the art of counting wassufficient to solve it”
On August 26, 1735, Eulerpresents Konigsberg bridgeproblem with a general solutionin “Solutio problematis adgeometriam situs pertinentis”
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Euler Trails and Circuits
The Euler Tour
Konigsberg Bridge ProblemEuler’s Approach
Problem Definition
Can one find out whether or not it is possible to crosseach bridge exactly once?
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Euler Trails and Circuits
The Euler Tour
Konigsberg Bridge ProblemConclusion
Solution
It is impossible to travel the bridges in the city ofKonigsberg once and only once.
Generalization
1 If there are more than two landmasses with an odd number ofbridges, then no such journey is possible
2 If the number of bridges is odd for exactly two landmasses,then the journey is possible if it starts in one of the two oddnumbered landmasses
3 If there are no regions with an odd number of landmassesthen the journey can be accomplished starting in any region
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Euler Trails and Circuits
The Euler Tour
Modern Graph Representation of the Bridge Network
The Origin of Graph Theory
Euler accidentally sparked a new branch of mathematics calledgraph theory, where a graph is simply a collection of vertices andedges. It is a new kind of Geometry, which Leibniz referred to asGeometry of Position.
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Euler Trails and Circuits
Euler Trails and Circuits
1 The Euler Tour
2 Euler Trails and Circuits
3 Fleury’s Algorithm
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Euler Trails and Circuits
Euler Trails and Circuits
Euler Path
An Euler path is a path that uses every edge of a graphexactly once
An Euler path starts and ends at different vertices
Euler Circuit
An Euler circuit is a circuit that uses every edge of a graphexactly once
An Euler path starts and ends at same vertices
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Euler Trails and Circuits
Fleury’s Algorithm
1 The Euler Tour
2 Euler Trails and Circuits
3 Fleury’s Algorithm
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Euler Trails and Circuits
Fleury’s Algorithm
Fleury’s Algorithm
a : 4
b : 2
c : 4
d : 4
e : 2 f : 2
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Euler Trails and Circuits
Fleury’s Algorithm
Fleury’s Algorithm
a : 4
b : 2
c : 4
d : 4
e : 2 f : 2f
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Euler Trails and Circuits
Fleury’s Algorithm
Fleury’s Algorithm
a : 4
b : 2
c : 4
d : 4
e : 2 f : 2f
c
f
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Euler Trails and Circuits
Fleury’s Algorithm
Fleury’s Algorithm
a : 4
b : 2
c : 4
d : 4
e : 2 f : 2f
c
d
f
c
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Euler Trails and Circuits
Fleury’s Algorithm
Fleury’s Algorithm
a : 4
b : 2
c : 4
d : 4
e : 2 f : 2f
c
d
a
f
c
d
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Euler Trails and Circuits
Fleury’s Algorithm
Fleury’s Algorithm
a : 4
b : 2
c : 4
d : 4
e : 2 f : 2f
c
d
a
e f
c
d
a
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Euler Trails and Circuits
Fleury’s Algorithm
Fleury’s Algorithm
a : 4
b : 2
c : 4
d : 4
e : 2 f : 2f
c
d
a
e
c
f
c
d
a
e
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Euler Trails and Circuits
Fleury’s Algorithm
Fleury’s Algorithm
a : 4
b : 2
c : 4
d : 4
e : 2 f : 2f
c
d
a
e
c
a
f
c
d
a
e
c
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Euler Trails and Circuits
Fleury’s Algorithm
Fleury’s Algorithm
a : 4
b : 2
c : 4
d : 4
e : 2 f : 2f
c
d
a
e
c
a
b
f
c
d
a
e
c
a
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Euler Trails and Circuits
Fleury’s Algorithm
Fleury’s Algorithm
a : 4
b : 2
c : 4
d : 4
e : 2 f : 2f
c
d
a
e
c
a
b
d
f
c
d
a
e
c
a
b
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Euler Trails and Circuits
Fleury’s Algorithm
Fleury’s Algorithm
a : 4
b : 2
c : 4
d : 4
e : 2 f : 2f
c
d
a
e
c
a
b
d
ff
c
d
a
e
c
a
b
d
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Euler Trails and Circuits
Fleury’s Algorithm
Fleury’s Algorithm
a : 4
b : 2
c : 4
d : 4
e : 2 f : 2f
c
d
a
e
c
a
b
d
ff
c
d
a
e
c
a
b
d
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Euler Trails and Circuits
Fleury’s Algorithm
Direct Application of Euler TourFragment Assembly in DNA Sequence
Find the exact structure by putting all the pieces together
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Euler Trails and Circuits
Fleury’s Algorithm
Today’s KonigsbergNow Kaliningrad
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Euler Trails and Circuits
Fleury’s Algorithm
Today’s KonigsbergThe Seven Bridges
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Euler Trails and Circuits
Fleury’s Algorithm
Thank YouQuestions are Welcome!
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