Graph Algorithms (in English)
FIT-GALeAcad. year: 2019/2020
This course discusses graph representations and graphs algorithms for
searching (depth-first search, breadth-first search), topological
sorting, searching of 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.
Learning outcomes of the course unit
Fundamental ability to construct an algorithm for a graph problem and to analyze its time and space complexity.
Foundations in discrete mathematics and algorithmic thinking.
Recommended optional programme components
Recommended or required reading
T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to Algorithms, McGraw-Hill, 2002.
T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to Algorithms (http://www.introductiontoalgorithms.com), McGraw-Hill, 2002.
A. Gibbons, Algorithmic Graph Theory, Cambridge University Press, 1985.
J. Demel, Grafy, SNTL Praha, 1988.
J. Demel, Grafy a jejich aplikace, Academia, 2002. (Více o knize (http://kix.fsv.cvut.cz/~demel/grafy/))
R. Diestel, Graph Theory, Third Edition (http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/), Springer-Verlag, Heidelberg, 2000.
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: Graph Theory and Its Applications, Second Edition, Chapman & Hall/CRC, 2005.
J.L. Gross, J. Yellen: Handbook of Graph Theory (Discrete Mathematics and Its Applications), CRC Press, 2003.
Planned learning activities and teaching methods
Assesment methods and criteria linked to learning outcomes
A mid-term exam evaluation (max. 15 points) and an evaluation of projects (max. 25 points).
Language of instruction
Introduction to graph theory with focus on graph representations, graph algorithms and their complexities.
Specification of controlled education, way of implementation and compensation for absences
A written mid-term exam, an evaluation of projects, and a final exam. The
minimal number of points which can be obtained from the final exam is
25. Otherwise, no points from the final exam will be assigned to a student.
Type of course unit
39 hours, optionally
Teacher / Lecturer
- Introduction, algorithmic complexity, basic notions and graph representations.
- Graph searching, depth-first search, breadth-first search.
- Topological sort, acyclic graphs.
- Graph components, strongly connected components, examples.
- Trees, minimal spanning trees, algorithms of Jarník and Borůvka.
- Growing a minimal spanning tree, algorithms of Kruskal and Prim.
- Single-source shortest paths, Bellman-Ford algorithm, shortest path in DAGs.
- Dijkstra algorithm. All-pairs shortest paths.
- Shortest paths and matrix multiplication, Floyd-Warshall algorithm.
- Flows and cuts in networks, maximal flow, minimal cut, Ford-Fulkerson algorithm.
- Matching in bipartite graphs, maximal matching.
- Graph coloring.
- Eulerian graphs and tours, Hamiltonian graphs and cycles.
13 hours, compulsory
Teacher / Lecturer
- Solving of selected graph problems and presentation of solutions (principle, complexity, implementation, optimization).
eLearning: currently opened course