Course details

# Graph Algorithms

GAL Acad. year 2023/2024 Winter semester 5 credits

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.

Guarantor

Course coordinator

Language of instruction

Czech, English

Completion

Examination (written)

Time span

• 39 hrs lectures
• 13 hrs projects

Assessment points

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

Department

Lecturer

Instructor

Course Web Pages

Public course webpage (in Czech; if you are looking for English version, visit webpage of GALe)

Learning objectives

Familiarity with graphs and graph algorithms with their complexities.
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 graph algorithm(s).

Prerequisite knowledge and skills

Foundations in discrete mathematics and algorithmic thinking.

Study literature

• T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, 3rd edition. MIT Press, 2009.
• J.A. McHugh, Algorithmic Graph Theory, Prentice-Hall, 1990.
• K. Erciyes: Guide to Graph Algorithms (Sequential, Parallel and Discributed). Springer, 2018.
• A. Mitina: Applied Combinatorics with Graph Theory. NEIU, 2019.

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. Graph coloring, Chromatic polynomial.
13. Eulerian graphs and tours, Chinese postman problem, and Hamiltonian cycles.

Syllabus - others, projects and individual work of students

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

Progress assessment

• 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.

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.

Schedule

DayTypeWeeksRoomStartEndCapacityLect.grpGroupsInfo
Tue exam 2024-01-23 A112 10:0012:50 GAL: 1. opravný termín
Tue exam 2024-01-16 E112 16:0018:50 GAL: řádná
Fri lecture 1., 2., 3., 4., 5., 6., 7., 8., 10., 11., 12., 13. of lectures M104 M105 08:0010:5041 1MIT 2MIT NMAT NNET xx Křivka
Fri exam 2023-11-10 C228 M103 M104 M105 08:3010:45 GAL: Půlsemestrální zkouška
Fri exam 2024-02-02 E105 13:0015:50 GAL: 2. opravný termín

Course inclusion in study plans