Euler trails and circuit

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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The Euler Tour

1 The Euler Tour

2 Euler Trails and Circuits

3 Fleury’s Algorithm

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The Euler Tour

Let’s Try to Clean Our City

Garbage Collector Truck

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The Euler Tour

The Story Begins ...

The bridge over the river Pregolya, Kaliningrad, Russia

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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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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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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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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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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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1 The Euler Tour



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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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Fleury’s Algorithm

1 The Euler Tour



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a : 4

b : 2

c : 4

d : 4

e : 2 f : 2

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a : 4

b : 2

c : 4

d : 4

e : 2 f : 2f

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a : 4

b : 2

c : 4

d : 4

e : 2 f : 2f

c

f

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a : 4

b : 2

c : 4

d : 4

e : 2 f : 2f

c

d

f

c

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a : 4

b : 2

c : 4

d : 4

e : 2 f : 2f

c

d

a

f

c

d

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a : 4

b : 2

c : 4

d : 4

e : 2 f : 2f

c

d

a

e f

c

d

a

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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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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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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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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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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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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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Direct Application of Euler TourFragment Assembly in DNA Sequence

Find the exact structure by putting all the pieces together

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Today’s KonigsbergNow Kaliningrad

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Today’s KonigsbergThe Seven Bridges

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Thank YouQuestions are Welcome!

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Education

Euler trails and circuit