The Bridges of Konigsberg In Konigsberg Germany there was a river that ran through the city with an island in the middle of the river Seven bridges were built on the river so people could easily get from one part to another The people had wondered whether or not it was possible to get around ID: 653164
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Slide1
By: Mike Basham,
Math in SchedulingSlide2
The Bridges of KonigsbergSlide3
In Konigsberg, Germany there was a river that ran through the city with an island in the middle of the river. Seven bridges were built on the river so people could easily get from one part to another. The people had wondered whether or not it was possible to get around the city in a way that would allow them to cross each of the seven bridges only once. Slide4
Euler’s Solution
Euler approaches this problem by labeling the sections of land separated by the water with capital letters. Mathematicians call this process, “labeling the vertices.” Vertices are the plural form of vertex, which is a point where two or more straight lines meet. In the figure shown below, the vertices are labeled by red dots and the edges are represented by the black lines. Euler solves this problem by drawing a figure similar to the one below without retracing any line, and making sure to keep his writing utensil on the paper. Slide5
Definition: A network is a figure made up of points (vertices) connected by non-intersecting curves (arcs).
Definition: A vertex is called odd if it has an odd number of arcs leading to it, other wise it is called even.
Definition: An Euler path is a continuous path that passes through every arc once and only once.
Graph TheorySlide6
Graph Theory Continued…
Theorem:
If a network has more than two odd vertices, it does not have an Euler path.
Theorem:
If a network has two or zero odd vertices, it has at least one Euler path. In particular, if a network has exactly two odd vertices, then its Euler paths can only start on one of the odd vertices, and end on the other.
Euler also provides these two rules.Slide7
What is a circuit?
What is the difference between
euler
and
hamilton’s
circuits?
Is the
konigsberg
problem unsolvable?
CircuitsSlide8
What is a Circuit?
Circuit-
a circuit is a closed, simple path.
Paths-
a
path
in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence.
Edges- An edge is a line segment that joins two vertices.Vertices- A vertex (plural: vertices) is a point where two or more straight lines meet. Basically it’s a corner. Slide9
Euler’s Circuit
A
Eulerian circuit
or
Eulerian cycle
is a path on a graph that starts and ends of the same vertex.
A
Eulerian Circuit uses each edge exactly once.Leonhard Euler first explored this idea in 1736 with his Seven Bridges of Konigsberg Problem.It is not a Eulerian Circuit unless all of it’s vertices have an even degree.Slide10
Fleury and Hierholzer’s Algorithm
Leonhard Euler never developed an algorithm for his mathematical theory however there were two mathematicians that did.
Hierholzer established his algorithm first in 1873. It is said to be the more efficient of the two.
Fleury developed his algorithm in 1883, and although it recieves criticism, it is still a method that works. Slide11
Fleury’s Algorithm
Consider a graph known to have all edges in the same component and at most two vertices of odd degree. We start with a vertex of odd degree—if the graph has none, then start with any vertex. At each step we move across an edge whose deletion would not disconnect the graph, unless we have no choice, then we delete that edge. At the end of the algorithm there are no edges left, and the sequence of edges we moved across forms an
Eulerian
cycle if the graph has no vertices of odd degree; or an
Eulerian
trail if there are exactly two vertices of odd degree.Slide12
Hierholzer’s Algorithm
Choose any starting vertex
v
, and follow a trail of edges from that vertex until returning to
v
. It is not possible to get stuck at any vertex other than
v
, because the even degree of all vertices ensures that, when the trail enters another vertex w
there must be an unused edge leaving
w. The tour formed in this way is a closed tour, but may not cover all the vertices and edges of the initial graph.As long as there exists a vertex v that belongs to the current tour but that has adjacent edges not part of the tour, start another trail from
v, following unused edges until returning to
v
, and join the tour formed in this way to the previous tour.
By using a data structure such as a
doubly linked list
to maintain the set of unused edges incident to each vertex, to maintain the list of vertices on the current tour that have unused edges, and to maintain the tour itself, the individual operations of the algorithm (finding unused edges exiting each vertex, finding a new starting vertex for a tour, and connecting two tours that share a vertex) may be performed in constant time each, so the overall algorithm takes
linear timeSlide13
Hamilton Circuit
Hamilton circuits are named after the renowned Irish mathematician (and
astonomer
) Sir William Rowan Hamilton who lived from 1805 to 1865. Hamilton was a child prodigy. He could read English, Hebrew, Greek, and Latin by time he was four years old. In addition, he wrote poetry and maintained close friendships with Wordsworth and Coleridge. Hamilton became Professor of Astronomy at Trinity College in Dublin, Ireland when he was twenty-three years old.
A circuit that starts at a vertex of a graph, passes through every vertex exactly once, and returns to the starting vertex is a HAMILTON CIRCUIT.
If a graph contains a Hamiltonian circuit is it considered a Hamiltonian graph.Slide14
Dodecahedron (Hamiltonian Circuit)Slide15
Euler Circuits
Hamiltonian Circuits
All edges of graph are traversed exactly once.
All vertices HAVE to be of even degree.
In these circuits you only have to pass through every vertex once and end on the same vertex on which you started.
The vertices do not need to be of even degree.
The “Optimal Hamiltonian Path,” is that in which the least slides are used.
The Difference Between Euler Circuits and Hamiltonian CircuitsSlide16
Quick Examples to PracticeSlide17
Can you take a walk through the town, visiting each part of the town
and
crossing each bridge only once?Slide18
To "visit each part of the town" the person taking the walk should visit the points
A, B, C and D
.
And they should cross each bridge
p, q, r, s, t, u and v
just once.Slide19
It can be further simplified in a graph like this. See if you can solve it.Slide20
Why is the Konigsberg Problem Unsolvable?Slide21
Vertices
A
,
B
and
D
have degree 3 and vertex
C has degree 5, so this graph has four vertices of odd degree. Therefore, it does
not have an Euler Path
. However, it does have a Hamiltonian Path.