Course detail

Graph Algorithms

FIT-GALAcad. year: 2024/2025

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.

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).

Links

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

prof. RNDr. Alexandr Meduna, CSc.

Department

Department of Information Systems (UIFS)

Entry knowledge

Foundations in discrete mathematics and algorithmic thinking.

Rules for evaluation and completion of the course

  • 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.

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.

Aims

Familiarity with graphs and graph algorithms with their complexities.
Fundamental ability to construct an algorithm for a graph problem and to analyze its time and space complexity.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Not applicable.

Recommended reading

J. Demel, Grafy a jejich aplikace, Academia, 2002. (Více o knize) (CS)
J.A. McHugh, Algorithmic Graph Theory, Prentice-Hall, 1990.
K. Erciyes: Guide to Graph Algorithms (Sequential, Parallel and Discributed). Springer, 2018.
A. Mitina: Applied Combinatorics with Graph Theory. NEIU, 2019.
Text přednášek v elektronické podobě. (CS)
T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, MIT Press, 3. vydání, 1312 s., 2009. (CS)
T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, 3rd edition. MIT Press, 2009. (EN)

Elearning

eLearning: currently opened course

Classification of course in study plans

  • Programme MITAI Master's

    specialization NGRI , 0 year of study, winter semester, elective
    specialization NADE , 0 year of study, winter semester, elective
    specialization NISD , 0 year of study, winter semester, elective
    specialization NMAT , 0 year of study, winter semester, compulsory
    specialization NSEC , 0 year of study, winter semester, elective
    specialization NISY up to 2020/21 , 0 year of study, winter semester, elective
    specialization NNET , 0 year of study, winter semester, compulsory
    specialization NMAL , 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 NVER , 0 year of study, winter semester, elective
    specialization NIDE , 0 year of study, winter semester, elective
    specialization NISY , 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 NEMB , 0 year of study, winter semester, elective
    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

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

Ing. Zbyněk Křivka, Ph.D.

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

Ing. Tomáš Kožár
Ing. Zbyněk Křivka, Ph.D.

Syllabus

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

Elearning

eLearning: currently opened course