Graph Algorithms

IT-MSC-2MSK1stCompulsory-Elective - group M
Language of Instruction:Czech
Public info:http://www.fit.vutbr.cz/study/courses/GAL/public/
Completion:examination (written)
Type of
Hour/semLecturesSem. ExercisesLab. exercisesComp. exercisesOther
Guarantor:Meduna Alexander, prof. RNDr., CSc., DIFS
Lecturer:Křivka Zbyněk, Ing., Ph.D., DIFS
Instructor:Charvát Lucie, Ing., DIFS
Křivka Zbyněk, Ing., Ph.D., DIFS
Soukup Ondřej, Ing., DIFS
Faculty:Faculty of Information Technology BUT
Department:Department of Information Systems FIT BUT
Learning objectives:
  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 competences:
  Fundamental ability to construct an algorithm for a graph problem and to analyze its time and space complexity.
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).
Fundamental literature:
  • 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.
Study literature:
  • Copy of lectures.
  • T.H. Cormen, C.E. Leiserson, R.L. Rivest, Introduction to Algorithms, McGraw-Hill, 2002.
Controlled instruction:
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.
Progress assessment:
A mid-term exam evaluation (max. 15 points) and an evaluation of projects (max. 25 points).

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