Graph Algorithms

• 39 hrs lectures
• 13 hrs projects

• 60 pts final exam (written part)
• 15 pts mid-term test (written part)
• 25 pts projects

Fundamental ability to construct an algorithm for a graph problem and to analyze its time and space complexity.

Familiarity with graphs and graph algorithms with their complexities.

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).

Foundations in discrete mathematics and algorithmic thinking.

• Copy of lectures.
• 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 (http://www.introductiontoalgorithms.com), McGraw-Hill, 2002.
• J. Demel, Grafy, SNTL Praha, 1988.
• J. Demel, Grafy a jejich aplikace, Academia, 2002. (More about the book (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.

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.

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

A mid-term exam evaluation (max. 15 points) and an evaluation of projects (max. 25 points).

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.