Course detail

Graph Algorithms

FIT-GALAcad. year: 2019/2020

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.

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, MIT Press, 3. vydání, 1312 s., 2009.
T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to Algorithms (http://www.introductiontoalgorithms.com), McGraw-Hill, 2002.
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

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.

Language of instruction

Czech

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.

Classification of course in study plans

  • Programme IT-MGR-2 Master's

    branch MBI , any year of study, winter semester, 5 credits, optional
    branch MPV , any year of study, winter semester, 5 credits, optional
    branch MGM , any year of study, winter semester, 5 credits, optional
    branch MIS , any year of study, winter semester, 5 credits, optional
    branch MBS , any year of study, winter semester, 5 credits, optional
    branch MIN , any year of study, winter semester, 5 credits, optional
    branch MMI , any year of study, winter semester, 5 credits, optional
    branch MMM , any year of study, winter semester, 5 credits, compulsory

  • Programme MITAI Master's

    specialization NADE , any year of study, winter semester, 5 credits, optional
    specialization NBIO , any year of study, winter semester, 5 credits, optional
    specialization NGRI , any year of study, winter semester, 5 credits, optional
    specialization NNET , any year of study, winter semester, 5 credits, compulsory
    specialization NVIZ , any year of study, winter semester, 5 credits, optional
    specialization NCPS , any year of study, winter semester, 5 credits, optional
    specialization NSEC , any year of study, winter semester, 5 credits, optional
    specialization NEMB , any year of study, winter semester, 5 credits, optional
    specialization NHPC , any year of study, winter semester, 5 credits, optional
    specialization NISD , any year of study, winter semester, 5 credits, optional
    specialization NIDE , any year of study, winter semester, 5 credits, optional
    specialization NISY , any year of study, winter semester, 5 credits, optional
    specialization NMAL , any year of study, winter semester, 5 credits, optional
    specialization NMAT , any year of study, winter semester, 5 credits, compulsory
    specialization NSEN , any year of study, winter semester, 5 credits, optional
    specialization NVER , any year of study, winter semester, 5 credits, optional
    specialization NSPE , any year of study, winter semester, 5 credits, optional

  • Programme IT-MGR-2 Master's

    branch MSK , 1. year of study, winter semester, 5 credits, compulsory

Type of course unit

 

Lecture

39 hours, 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.

Projects

13 hours, compulsory

Teacher / Lecturer

Syllabus


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

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