fundamentals of informatics introduction to algorithms and...
TRANSCRIPT
![Page 1: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/1.jpg)
Fundamentals of InformaticsIntroduction to Algorithms and Informatics
Lecture 2: Graphs as models
I Prof: David Avis
I Research bldg. No. 7, room404
I www.i-kyoto-u.ca.jp/∼avis
I Check course web page at leastonce a week !
I TA: Yang Cao
![Page 2: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/2.jpg)
How to pass this course smiling!
I Attend the lectures !
I Do all the reading assignments on the lecturesummaries web page
I Do the exercises after each lecture
I 3 reports consisting of exercises from 3-4 lectures
I Report due dates: Wed May 14, Wed June 29, WedJuly 16 in class (or give to Yang Cao)
I Ask questions !
www.i-kyoto-u.ca.jp/∼avis
![Page 3: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/3.jpg)
How to pass this course smiling!
I Attend the lectures !
I Do all the reading assignments on the lecturesummaries web page
I Do the exercises after each lecture
I 3 reports consisting of exercises from 3-4 lectures
I Report due dates: Wed May 14, Wed June 29, WedJuly 16 in class (or give to Yang Cao)
I Ask questions !
www.i-kyoto-u.ca.jp/∼avis
![Page 4: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/4.jpg)
How to pass this course smiling!
I Attend the lectures !
I Do all the reading assignments on the lecturesummaries web page
I Do the exercises after each lecture
I 3 reports consisting of exercises from 3-4 lectures
I Report due dates: Wed May 14, Wed June 29, WedJuly 16 in class (or give to Yang Cao)
I Ask questions !
www.i-kyoto-u.ca.jp/∼avis
![Page 5: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/5.jpg)
How to pass this course smiling!
I Attend the lectures !
I Do all the reading assignments on the lecturesummaries web page
I Do the exercises after each lecture
I 3 reports consisting of exercises from 3-4 lectures
I Report due dates: Wed May 14, Wed June 29, WedJuly 16 in class (or give to Yang Cao)
I Ask questions !
www.i-kyoto-u.ca.jp/∼avis
![Page 6: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/6.jpg)
How to pass this course smiling!
I Attend the lectures !
I Do all the reading assignments on the lecturesummaries web page
I Do the exercises after each lecture
I 3 reports consisting of exercises from 3-4 lectures
I Report due dates: Wed May 14, Wed June 29, WedJuly 16 in class (or give to Yang Cao)
I Ask questions !
www.i-kyoto-u.ca.jp/∼avis
![Page 7: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/7.jpg)
How to pass this course smiling!
I Attend the lectures !
I Do all the reading assignments on the lecturesummaries web page
I Do the exercises after each lecture
I 3 reports consisting of exercises from 3-4 lectures
I Report due dates: Wed May 14, Wed June 29, WedJuly 16 in class (or give to Yang Cao)
I Ask questions !
www.i-kyoto-u.ca.jp/∼avis
![Page 8: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/8.jpg)
How to pass this course smiling!
I Attend the lectures !
I Do all the reading assignments on the lecturesummaries web page
I Do the exercises after each lecture
I 3 reports consisting of exercises from 3-4 lectures
I Report due dates: Wed May 14, Wed June 29, WedJuly 16 in class (or give to Yang Cao)
I Ask questions !
www.i-kyoto-u.ca.jp/∼avis
![Page 9: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/9.jpg)
Kyoto bridge problem
I Is it possible to cross each bridge exactly once?
I If so, can you do so and start and end at the same place?
![Page 10: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/10.jpg)
Kyoto bridge problem
I Is it possible to cross each bridge exactly once?
I If so, can you do so and start and end at the same place?
![Page 11: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/11.jpg)
Kyoto bridge problem
I Is it possible to cross each bridge exactly once?
I If so, can you do so and start and end at the same place?
![Page 12: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/12.jpg)
Konigsberg bridges: 1735 and now
I The problem is to cross each bridge exactly once and end at thestarting point
I This is the Euler circuit problem ...
I ... and can be solved in linear time
I Considered as the birth of graph theory
![Page 13: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/13.jpg)
Konigsberg bridges: 1735 and now
I The problem is to cross each bridge exactly once and end at thestarting point
I This is the Euler circuit problem ...
I ... and can be solved in linear time
I Considered as the birth of graph theory
![Page 14: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/14.jpg)
Konigsberg bridges: 1735 and now
I The problem is to cross each bridge exactly once and end at thestarting point
I This is the Euler circuit problem ...
I ... and can be solved in linear time
I Considered as the birth of graph theory
![Page 15: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/15.jpg)
Konigsberg bridges: 1735 and now
I The problem is to cross each bridge exactly once and end at thestarting point
I This is the Euler circuit problem ...
I ... and can be solved in linear time
I Considered as the birth of graph theory
![Page 16: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/16.jpg)
Konigsberg bridges: 1735 and now
I The problem is to cross each bridge exactly once and end at thestarting point
I This is the Euler circuit problem ...
I ... and can be solved in linear time
I Considered as the birth of graph theory
![Page 17: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/17.jpg)
Euler Circuits: Konigsberg bridge problem (1735)
B
g
e
f
C
c
a b
d
A D
I A graph consists of a set of vertices V and edges E connectingvertices
I Vertices in graph correspond to land, edges to bridges
I Euler’s Theorem: A circuit exists that crosses every bridge if andonly if each vertex in the corresponding graph has even degree
I We can find the circuit in linear time
![Page 18: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/18.jpg)
Euler Circuits: Konigsberg bridge problem (1735)
B
g
e
f
C
c
a b
d
A D
I A graph consists of a set of vertices V and edges E connectingvertices
I Vertices in graph correspond to land, edges to bridges
I Euler’s Theorem: A circuit exists that crosses every bridge if andonly if each vertex in the corresponding graph has even degree
I We can find the circuit in linear time
![Page 19: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/19.jpg)
Euler Circuits: Konigsberg bridge problem (1735)
B
g
e
f
C
c
a b
d
A D
I A graph consists of a set of vertices V and edges E connectingvertices
I Vertices in graph correspond to land, edges to bridges
I Euler’s Theorem: A circuit exists that crosses every bridge if andonly if each vertex in the corresponding graph has even degree
I We can find the circuit in linear time
![Page 20: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/20.jpg)
Euler Circuits: Konigsberg bridge problem (1735)
B
g
e
f
C
c
a b
d
A D
I A graph consists of a set of vertices V and edges E connectingvertices
I Vertices in graph correspond to land, edges to bridges
I Euler’s Theorem: A circuit exists that crosses every bridge if andonly if each vertex in the corresponding graph has even degree
I We can find the circuit in linear time
![Page 21: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/21.jpg)
Extensions: Chinese Postman Problem
I If the graph has exactly two vertices of odd degree we can find apath visiting each edge exactly once
I What if there are more than two vertices of odd degree?
I Find a minimum length path that visits each edge at least once.
I Proposed by Kwan Mei-Ko (1962) solved by Jack Edmonds and EllisJohnson (1973)
I Many applications of this: postal delivery, street cleaning, garbagepickup, snow removal, ...
![Page 22: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/22.jpg)
Extensions: Chinese Postman Problem
I If the graph has exactly two vertices of odd degree we can find apath visiting each edge exactly once
I What if there are more than two vertices of odd degree?
I Find a minimum length path that visits each edge at least once.
I Proposed by Kwan Mei-Ko (1962) solved by Jack Edmonds and EllisJohnson (1973)
I Many applications of this: postal delivery, street cleaning, garbagepickup, snow removal, ...
![Page 23: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/23.jpg)
Extensions: Chinese Postman Problem
I If the graph has exactly two vertices of odd degree we can find apath visiting each edge exactly once
I What if there are more than two vertices of odd degree?
I Find a minimum length path that visits each edge at least once.
I Proposed by Kwan Mei-Ko (1962) solved by Jack Edmonds and EllisJohnson (1973)
I Many applications of this: postal delivery, street cleaning, garbagepickup, snow removal, ...
![Page 24: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/24.jpg)
Extensions: Chinese Postman Problem
I If the graph has exactly two vertices of odd degree we can find apath visiting each edge exactly once
I What if there are more than two vertices of odd degree?
I Find a minimum length path that visits each edge at least once.
I Proposed by Kwan Mei-Ko (1962) solved by Jack Edmonds and EllisJohnson (1973)
I Many applications of this: postal delivery, street cleaning, garbagepickup, snow removal, ...
![Page 25: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/25.jpg)
Extensions: Chinese Postman Problem
I If the graph has exactly two vertices of odd degree we can find apath visiting each edge exactly once
I What if there are more than two vertices of odd degree?
I Find a minimum length path that visits each edge at least once.
I Proposed by Kwan Mei-Ko (1962) solved by Jack Edmonds and EllisJohnson (1973)
I Many applications of this: postal delivery, street cleaning, garbagepickup, snow removal, ...
![Page 26: Fundamentals of Informatics Introduction to Algorithms and ...cgm.cs.mcgill.ca/~avis/Kyoto/courses/ia/2014/notes/slide2.pdf · Extensions: Chinese Postman Problem I If the graph has](https://reader036.vdocuments.us/reader036/viewer/2022071108/5fe388035c8da049a65c868c/html5/thumbnails/26.jpg)
ExercisesI Let G = (V ,E ) be a connected undirected graph with exactly two
vertices, A and B, of odd degree. Prove that any maximal lengthpath with no repeated edges that starts at A must end at B.Conclude that after this path has been removed from G all vertexdegrees must be even.
I Consider the Montreal map at the top of the next slide. Decidewhether or not there is a path (or circuit) that crosses each bridge.If so, give the path (or circuit). (Consider all lines crossing the wateras bridges)
I Consider the new bridge connecting highway 25 shown in yellowbelow. How does this change the answer to the preceding question?
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Montreal bridge problem
Montreal map before 2011