Course detail

# Graph Algorithms

This course discusses graph representations and graphs algorithms for searching (depth-first search, breadth-first search), topological sorting, graph components and strongly connected components, trees and minimal spanning trees, single-source and all-pairs shortest paths, maximal flows and minimal cuts, maximal bipartite matching, Euler graphs, and graph coloring. The principles and complexities of all presented algorithms are discussed.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Learning outcomes of the course unit

Fundamental ability to construct an algorithm for a graph problem and to analyze its time and space complexity.

Prerequisites

Foundations in discrete mathematics and algorithmic thinking.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

• Mid-term written examination (15 point)
• Evaluated project(s) (25 points)
• Final written examination (60 points)
• The minimal number of points which can be obtained from the final exam is 25. Otherwise, no points will be assigned to a student.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

Familiarity with graphs and graph algorithms with their complexities.

Specification of controlled education, way of implementation and compensation for absences

In case of illness or another serious obstacle, the student should inform the faculty about that and subsequently provide the evidence of such an obstacle. Then, it can be taken into account within evaluation:
• The student can ask the responsible teacher to extend the time for the project assignment.
• If a student cannot attend the mid-term exam, (s)he can ask to derive points from the evaluation of his/her first attempt of the final exam.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to Algorithms (http://www.introductiontoalgorithms.com), McGraw-Hill, 2002.
J.A. McHugh, Algorithmic Graph Theory, Prentice-Hall, 1990.
J.A. Bondy, U.S.R. Murty: Graph Theory, Graduate text in mathematics, Springer, 2008.
J.L. Gross, J. Yellen: Handbook of Graph Theory (Discrete Mathematics and Its Applications), CRC Press, 2003.
T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to Algorithms, MIT Press, 3rd Edition, 1312 p., 2009.
J. Demel, Grafy a jejich aplikace, Academia, 2002. (More about the book (http://kix.fsv.cvut.cz/~demel/grafy/))
Copy of lectures.
J. Demel, Grafy, SNTL Praha, 1988.
R. Diestel, Graph Theory, Third Edition (http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/), Springer-Verlag, Heidelberg, 2000.
J.L. Gross, J. Yellen: Graph Theory and Its Applications, Second Edition, Chapman & Hall/CRC, 2005.

Classification of course in study plans

• Programme IT-MSC-2 Master's

branch MMI , 0 year of study, winter semester, elective
branch MBI , 0 year of study, winter semester, elective
branch MSK , 1 year of study, winter semester, compulsory
branch MMM , 0 year of study, winter semester, compulsory
branch MBS , 0 year of study, winter semester, elective
branch MPV , 0 year of study, winter semester, elective
branch MIS , 0 year of study, winter semester, elective
branch MIN , 0 year of study, winter semester, elective
branch MGM , 0 year of study, winter semester, elective

• Programme MITAI Master's

specialization NNET , 0 year of study, winter semester, compulsory
specialization NMAT , 0 year of study, winter semester, compulsory
specialization NBIO , 0 year of study, winter semester, elective
specialization NSEN , 0 year of study, winter semester, elective
specialization NVIZ , 0 year of study, winter semester, elective
specialization NGRI , 0 year of study, winter semester, elective
specialization NISD , 0 year of study, winter semester, elective
specialization NSEC , 0 year of study, winter semester, elective
specialization NCPS , 0 year of study, winter semester, elective
specialization NHPC , 0 year of study, winter semester, elective
specialization NMAL , 0 year of study, winter semester, elective
specialization NVER , 0 year of study, winter semester, elective
specialization NIDE , 0 year of study, winter semester, elective
specialization NEMB , 0 year of study, winter semester, elective
specialization NSPE , 0 year of study, winter semester, elective
specialization NADE , 0 year of study, winter semester, elective
specialization NISY , 0 year of study, winter semester, elective

#### Type of course unit

Lecture

39 hod., optionally

Teacher / Lecturer

Syllabus

1. Introduction, algorithmic complexity, basic notions and graph representations.
2. Graph searching, depth-first search, breadth-first search.
3. Topological sort, acyclic graphs.
4. Graph components, strongly connected components, examples.
5. Trees, minimal spanning trees, algorithms of Jarník and Borůvka.
6. Growing a minimal spanning tree, algorithms of Kruskal and Prim.
7. Single-source shortest paths, the Bellman-Ford algorithm, shortest path in DAGs.
8. Dijkstra's algorithm. All-pairs shortest paths.
9. Shortest paths and matrix multiplication, the Floyd-Warshall algorithm.
10. Flows and cuts in networks, maximal flow, minimal cut, the Ford-Fulkerson algorithm.
11. Matching in bipartite graphs, maximal matching.
12. Graph coloring, Chromatic polynomial.
13. Eulerian graphs and tours, Chinese postman problem, and Hamiltonian cycles.

Project

13 hod., compulsory

Teacher / Lecturer

Syllabus

1. Solving of selected graph problems and presentation of solutions (principle, complexity, implementation, optimization).