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 授業科目
 Course Title
グラフ理論特論
Special Topics in Graph Theory
 担当者
 Instructor
准教授 ボサール アントワーヌ  後学期 金曜日2時限
 単 位
 Credit
2

到達目標 Target to be Reached
Precise knowledge of graph theory definitions and notations. For instance, understanding what is the degree and diameter of a graph, a spanning tree, a graph morphism. Ability to program graph traversal algorithms and path selection (routing) algorithms. Also, concrete understanding of the definition of network topologies such as hypercubes.
 
授業内容 Course Content
Graph theory is ubiquitous in computer and information science. Data structures, optimisation, parallel computing, etc. are only a few examples of areas strongly relying on graph theory. It is thus very important for computer scientists to be knowledgeable in this field.

The objective of this course is to first present to students important and various aspects of graph theory, ranging from paths, cycles and diameter to extremal and random graphs. The second objective is to describe concrete applications of graph theory, especially in the field of interconnection networks of parallel systems. Network topologies, routing algorithms, fault-tolerance, etc. will be discussed in the second part of this course.
 
授業計画 Course Planning
Note that the following lecture plan may be slightly adjusted upon needs.
As homework, in preparation for the next lecture, students are advised to carefully read and study the materials presented and discussed during each lecture.

Part 1: A few important topics of graph theory.

1. Guidance, introduction, motivation
The syllabus will be reviewed and discussed.

2. Definitions and notations
Formally recalling trees and graphs.

3. Remarkable graph properties
On degrees, connectivity, spanning trees, Euler tours.

4. Paths and cycles, distance and diameter.
Hamilton cycle, Hamilton path.

5. Graphs algorithms (1)
Depth-first search, Breadth-first search.

6. Graphs algorithms (2)
Dijkstra algorithm, and others.

7. r-partite graphs, matchings (1)
Bi-partite graphs, matching definition.

8. Matchings (2)
The marriage condition and Hall's theorem.

9. Graph decycling
Cycle detection, and general approach to graph decycling.

10. Mid-term examination
Test (60min) and after-test discussion (30min).


Part 2: Application to networks.

1. Hypercubes
Definition, topological properties, simple routing.

2. Hypercubes: algorithms
Gray code and Hamiltonian cycles, about fault-tolerance, fault-tolerant point-to-point routing, the container problem.

3. Hierarchical networks
They are popular in supercomputing and massively parallel systems.

4. Routing in hierarchical networks
Discussing main strategies.

5. Final examination
Test (60min) and after-test discussion (30min).
 
授業運営 Course Management
As usual for graduate lectures, this course will be given in English.
 
評価方法 Evaluation Method
Mid-term examination: 40%.
Final examination: 60%.
Absence to 5 lectures (or more) will result in an automatic 0 (zero) score to the course evaluation (no credit given).
 
オフィスアワー Office Hour (s)
Friday 10am~11am, 6-231.
 
使用書 Textbook (s)
R. Diestel『Graph Theory』4th[Springer-Verlag Heidelberg, New York]2010
(electronic edition available online at no charge: http://diestel-graph-theory.com/basic.html)

 
 
 
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