Course detail
Graph Algorithms
FIT-GALAcad. year: 2018/2019
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.
Supervisor
Department
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.
Recommended optional programme components
Not applicable.
Recommended or required reading
- Text přednášek.
- 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, McGraw-Hill, 2002.
- J. Demel, Grafy, SNTL Praha, 1988.
- J. Demel, Grafy a jejich aplikace, Academia, 2002. (Více o knize)
- R. Diestel, Graph Theory, Third Edition, 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
Not applicable.
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
Czech, English
Work placements
Not applicable.
Course curriculum
- Syllabus of lectures:
- 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, the Bellman-Ford algorithm, shortest path in DAGs.
- Dijkstra's algorithm. All-pairs shortest paths.
- Shortest paths and matrix multiplication, the Floyd-Warshall algorithm.
- Flows and cuts in networks, maximal flow, minimal cut, the Ford-Fulkerson algorithm.
- Matching in bipartite graphs, maximal matching.
- Euler graphs and tours and Hamilton cycles.
- Graph coloring.
- Solving of selected graph problems and presentation of solutions (principle, complexity, implementation, optimization).
Syllabus - others, projects and individual work of students:
Aims
Familiarity with graphs and graph algorithms with 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 will be assigned to a student.
Classification of course in study plans
- Programme IT-MGR-2 Master's
branch MBI , any year of study, winter semester, 5 credits, elective
branch MPV , any year of study, winter semester, 5 credits, elective
branch MGM , any year of study, winter semester, 5 credits, elective
branch MIS , any year of study, winter semester, 5 credits, elective
branch MBS , any year of study, winter semester, 5 credits, elective
branch MIN , any year of study, winter semester, 5 credits, elective
branch MMI , any year of study, winter semester, 5 credits, elective
branch MMM , any year of study, winter semester, 5 credits, compulsory
branch MSK , 1. year of study, winter semester, 5 credits, compulsory
Type of course unit
Lecture
39 hours, optionally
Teacher / Lecturer
Syllabus
- 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, the Bellman-Ford algorithm, shortest path in DAGs.
- Dijkstra's algorithm. All-pairs shortest paths.
- Shortest paths and matrix multiplication, the Floyd-Warshall algorithm.
- Flows and cuts in networks, maximal flow, minimal cut, the Ford-Fulkerson algorithm.
- Matching in bipartite graphs, maximal matching.
- Euler graphs and tours and Hamilton cycles.
- Graph coloring.
Project
13 hours, compulsory
Teacher / Lecturer
Syllabus
- Solving of selected graph problems and presentation of solutions (principle, complexity, implementation, optimization).