Title:  Graph Algorithms 

Code:  GAL 

Ac.Year:  2018/2019 

Sem:  Winter 

Curriculums:  

Language of Instruction:  Czech 

Public info:  http://www.fit.vutbr.cz/study/courses/GAL/public/ 

Credits:  5 

Completion:  examination (written) 

Type of instruction:  Hour/sem  Lectures  Seminar Exercises  Laboratory Exercises  Computer Exercises  Other 

Hours:  39  0  0  0  13 

 Exams  Tests  Exercises  Laboratories  Other 

Points:  60  15  0  0  25 



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. (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.  Description: 

  This course discusses graph representations and graphs algorithms for searching (depthfirst search, breadthfirst search), topological sorting, graph components and strongly connected components, trees and minimal spanning trees, singlesource and allpairs 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 advanced graph algorithm(s).
 Syllabus of lectures: 


 Introduction, algorithmic complexity, basic notions and graph representations.
 Graph searching, depthfirst search, breadthfirst 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.
 Singlesource shortest paths, the BellmanFord algorithm, shortest path in DAGs.
 Dijkstra's algorithm. Allpairs shortest paths.
 Shortest paths and matrix multiplication, the FloydWarshall algorithm.
 Flows and cuts in networks, maximal flow, minimal cut, the FordFulkerson algorithm.
 Matching in bipartite graphs, maximal matching.
 Euler graphs and tours and Hamilton cycles.
 Graph coloring.
 Syllabus  others, projects and individual work of students: 


 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, McGrawHill, 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, SpringerVerlag, Heidelberg, 2000.
 J.A. McHugh, Algorithmic Graph Theory, PrenticeHall, 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, McGrawHill, 2002.
 Controlled instruction: 

  A written midterm 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 midterm exam evaluation (max. 15 points) and an evaluation of projects (max. 25 points).  
