Course detail

Graph Algorithms

FIT-GALAcad. year: 2017/2018

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, 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

Study evaluation is based on marks obtained for specified items. Minimimum number of marks to pass is 50.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

    Syllabus of lectures:
    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. Euler graphs and tours and Hamilton cycles.
    13. Graph coloring.

    Syllabus - others, projects and individual work of students:
    1. Solving of selected graph problems and presentation of solutions (principle, complexity, implementation, optimization).

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


  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. Euler graphs and tours and Hamilton cycles.
  13. Graph coloring.

Project

13 hours, optionally

Teacher / Lecturer