euler trails and circuit
TRANSCRIPT
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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