street-routing problems - lecture 29 sections...
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Street-Routing ProblemsLecture 29
Sections 5.2
Robb T. Koether
Hampden-Sydney College
Mon, Apr 6, 2015
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1 Street-Routing ProblemsThe Security Guard ProblemThe Mail Carrier ProblemThe Königsberg ProblemThe Bridges of Madison County Problem
2 Examples
3 Euler Paths and Circuits
4 Definitions
5 Euler’s Theorems
6 Assignment
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Outline
1 Street-Routing ProblemsThe Security Guard ProblemThe Mail Carrier ProblemThe Königsberg ProblemThe Bridges of Madison County Problem
2 Examples
3 Euler Paths and Circuits
4 Definitions
5 Euler’s Theorems
6 Assignment
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Street-Routing Problems
We have already considered the Traveling Salesman Problem.There are many other problems that would fall under the heading“Street-Routing Problems.”We will consider four of them.
The Security Guard ProblemThe Mail Carrier ProblemThe Königsberg Bridge ProblemThe Bridges of Madison County
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Street-Routing Problems
We have already considered the Traveling Salesman Problem.There are many other problems that would fall under the heading“Street-Routing Problems.”We will consider four of them.
The Security Guard Problem
The Mail Carrier ProblemThe Königsberg Bridge ProblemThe Bridges of Madison County
Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Mon, Apr 6, 2015 4 / 29
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Street-Routing Problems
We have already considered the Traveling Salesman Problem.There are many other problems that would fall under the heading“Street-Routing Problems.”We will consider four of them.
The Security Guard ProblemThe Mail Carrier Problem
The Königsberg Bridge ProblemThe Bridges of Madison County
Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Mon, Apr 6, 2015 4 / 29
![Page 7: Street-Routing Problems - Lecture 29 Sections 5people.hsc.edu/faculty-staff/robbk/Math111/Lectures/Spring 2015... · Street-Routing Problems Lecture 29 Sections 5.2 Robb T. Koether](https://reader031.vdocuments.net/reader031/viewer/2022021900/5b53922c7f8b9add3a8bf855/html5/thumbnails/7.jpg)
Street-Routing Problems
We have already considered the Traveling Salesman Problem.There are many other problems that would fall under the heading“Street-Routing Problems.”We will consider four of them.
The Security Guard ProblemThe Mail Carrier ProblemThe Königsberg Bridge Problem
The Bridges of Madison County
Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Mon, Apr 6, 2015 4 / 29
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Street-Routing Problems
We have already considered the Traveling Salesman Problem.There are many other problems that would fall under the heading“Street-Routing Problems.”We will consider four of them.
The Security Guard ProblemThe Mail Carrier ProblemThe Königsberg Bridge ProblemThe Bridges of Madison County
Robb T. Koether (Hampden-Sydney College) Street-Routing Problems Mon, Apr 6, 2015 4 / 29
![Page 9: Street-Routing Problems - Lecture 29 Sections 5people.hsc.edu/faculty-staff/robbk/Math111/Lectures/Spring 2015... · Street-Routing Problems Lecture 29 Sections 5.2 Robb T. Koether](https://reader031.vdocuments.net/reader031/viewer/2022021900/5b53922c7f8b9add3a8bf855/html5/thumbnails/9.jpg)
Outline
1 Street-Routing ProblemsThe Security Guard ProblemThe Mail Carrier ProblemThe Königsberg ProblemThe Bridges of Madison County Problem
2 Examples
3 Euler Paths and Circuits
4 Definitions
5 Euler’s Theorems
6 Assignment
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The Security Guard Problem
Example (The Security Guard Problem)
A security guard must patrol every street of the neighborhood.What route should he follow to minimize the total distance?Must he walk some streets twice?
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Outline
1 Street-Routing ProblemsThe Security Guard ProblemThe Mail Carrier ProblemThe Königsberg ProblemThe Bridges of Madison County Problem
2 Examples
3 Euler Paths and Circuits
4 Definitions
5 Euler’s Theorems
6 Assignment
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The Mail Carrier Problem
Example (The Mail Carrier Problem)
A mail carrier must deliver mail to both sides of every street of theneighborhood, except the boundary.What route should he follow to minimize the total distance?Must he walk some streets twice?
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Outline
1 Street-Routing ProblemsThe Security Guard ProblemThe Mail Carrier ProblemThe Königsberg ProblemThe Bridges of Madison County Problem
2 Examples
3 Euler Paths and Circuits
4 Definitions
5 Euler’s Theorems
6 Assignment
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The Königsberg Problem
Example (The Königsberg Problem)
A Königsberger would like to take a stroll across the sevenbridges.Can it be done without ever crossing the same bridge twice?Does it matter where the stroller starts?
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Outline
1 Street-Routing ProblemsThe Security Guard ProblemThe Mail Carrier ProblemThe Königsberg ProblemThe Bridges of Madison County Problem
2 Examples
3 Euler Paths and Circuits
4 Definitions
5 Euler’s Theorems
6 Assignment
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The Bridges of Madison County Problem
Example (The Bridges of Madison County Problem)
A photographer wants to photograph each of the 11 bridges ofMadison County.He must cross the bridge to photograph it and each bridge has a$5 toll.What route will minimize the total cost?
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The Bridges of Madison County Problem
Example (The Bridges of Madison County Problem)
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A photographer wants to photograph each of the 11 bridges ofMadison County.He must cross the bridge to photograph it and each bridge has a$5 toll.What route will minimize the total cost?
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Outline
1 Street-Routing ProblemsThe Security Guard ProblemThe Mail Carrier ProblemThe Königsberg ProblemThe Bridges of Madison County Problem
2 Examples
3 Euler Paths and Circuits
4 Definitions
5 Euler’s Theorems
6 Assignment
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Example
Example (The Security Guard Problem)
In the Security Guard Problem, we want a path that traversesevery edge at least once and has the minimal total length.
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Example
Example (The Mail Carrier Problem)
In the Mail Carrier Problem, we want a path that traverses everyedge at least twice (except the boundary) and has the minimaltotal length.With the duplicated edges, the Mail Carrier Problem is the sameas the Security Guard Problem.
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Example
Example (The Bridges of Königsberg Problem)
In the Bridges of Königsberg Problem, we want a circuit thattraverses each edge exactly once.
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Example
Example (The Bridges of Königsberg Problem)
We draw a graph that shows only the relevant parts.
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Example
Example (The Bridges of Madison County Problem)
In the Bridges of Madison County Problem, we want a circuit thattraverses each edge at least once and has the minimal totallength.This is the same as the Security Guard Problem.
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Outline
1 Street-Routing ProblemsThe Security Guard ProblemThe Mail Carrier ProblemThe Königsberg ProblemThe Bridges of Madison County Problem
2 Examples
3 Euler Paths and Circuits
4 Definitions
5 Euler’s Theorems
6 Assignment
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Euler Paths and Circuits
Definition (Euler Path)An Euler path (pronounced "oiler") is a path that traverses each edgeexactly once.
Definition (Euler Circuit)An Euler circuit is an Euler path that is a circuit.
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Euler Paths and Circuits
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In the Bridges of Königsberg Problem, we seek an Euler path andan Euler circuit.
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Outline
1 Street-Routing ProblemsThe Security Guard ProblemThe Mail Carrier ProblemThe Königsberg ProblemThe Bridges of Madison County Problem
2 Examples
3 Euler Paths and Circuits
4 Definitions
5 Euler’s Theorems
6 Assignment
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Definitions
Definition (Connected)A graph is connected if, for any two vertices, there is a path from oneto the other.
Definition (Even and Odd Vertices)A vertex is even if an even number of edges emanate from it.Otherwise, it is odd.
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Outline
1 Street-Routing ProblemsThe Security Guard ProblemThe Mail Carrier ProblemThe Königsberg ProblemThe Bridges of Madison County Problem
2 Examples
3 Euler Paths and Circuits
4 Definitions
5 Euler’s Theorems
6 Assignment
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Euler’s Theorems
Theorem (Euler Circuits)If a graph is connected and every vertex is even, then it has an Eulercircuit. Otherwise, it does not have an Euler circuit.
Theorem (Euler Paths)If a graph is connected and it has exactly 2 odd vertices, then it has anEuler path. If it has more than 2 odd vertices, then it does not have anEuler path.
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Connected and Disconnected
Connected Disconnected
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Outline
1 Street-Routing ProblemsThe Security Guard ProblemThe Mail Carrier ProblemThe Königsberg ProblemThe Bridges of Madison County Problem
2 Examples
3 Euler Paths and Circuits
4 Definitions
5 Euler’s Theorems
6 Assignment
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Assignment
AssignmentChapter 5: Exercises 19, 20, 21, 22, 27.
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