|Language of Instruction:||Czech|
|Guarantor:||Meduna Alexander, prof. RNDr., CSc. (DIFS)|
|Deputy guarantor:||Křivka Zbyněk, Ing., Ph.D. (DIFS)|
|Lecturer:||Křivka Zbyněk, Ing., Ph.D. (DIFS)|
|Instructor:||Křivka Zbyněk, Ing., Ph.D. (DIFS)|
Tomko Martin, Ing. (FIT)
|Faculty:||Faculty of Information Technology BUT|
|Department:||Department of Information Systems FIT BUT|
|Fri||lecture||lectures||A112 ||08:00||10:50||1MIT 2MIT ||MMM MSK xx |
| || ||Familiarity with graphs and graph algorithms with their complexities.|
| || ||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.|
|Knowledge and skills required for the course:|
| || ||Foundations in discrete mathematics and algorithmic thinking.|
|Learning outcomes and competencies:|
| || ||Fundamental ability to construct an algorithm for a graph problem and to analyze its time and space complexity.||Why is the course taught:|
| || ||First, we recall all important algorithms for systematic graph
exploration including the demonstrations of algorithm correctness. Then,
we proceed to more demanding algorithms for shortest path search and
other advanced graph analysis. We place emphasis on the explanation of
the algorithm principles and implementation discussion including the
discussion of used data structures and their time/space complexities.
Apart from the graph algorithms, the student improves his/her ability to
formally describe an algorithm and estimate its complexity. In project,
the students are usually asked to modify, implement and experiment with
some chosen graph algorithm(s).|
|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.
- Graph coloring, Chromatic polynomial.
- Eulerian graphs and tours, Chinese postman problem, and Hamiltonian cycles.
|Syllabus - others, projects and individual work of students:|
- Solving of selected graph problems and presentation of solutions (principle, complexity, implementation, optimization).
- 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. (More about the book)
- 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.
- Copy of lectures.
- T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to Algorithms, MIT Press, 3rd Edition, 1312 p., 2009.
| || ||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.
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.
| || |
- Mid-term written examination (15 point)
- Evaluated project(s) (25 points)
- Final written examination (60 points)
minimal number of points which can be obtained from the final exam is
25. Otherwise, no points will be assigned to a student.