Course details

# Graph Algorithms (in English)

GALe 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, searching of 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

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

Learning objectives

Introduction to graph theory with focus on graph representations, graph algorithms and their complexities.
Fundamental ability to construct an algorithm for a graph problem and to analyze its time and space complexity.

Prerequisite knowledge and skills

Foundations in discrete mathematics and algorithmic thinking.

Study literature

• Electronic copy of lectures.
• 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 Distributed). 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, Bellman-Ford algorithm, shortest path in DAGs.
8. Dijkstra algorithm. All-pairs shortest paths.
9. Shortest paths and matrix multiplication, Floyd-Warshall algorithm.
10. Flows and cuts in networks, maximal flow, minimal cut, Ford-Fulkerson algorithm.
11. Matching in bipartite graphs, maximal matching.
12. Graph coloring.
13. Eulerian graphs and tours, Hamiltonian graphs and 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 exam - 15 points.
• Projects - 25 points.
• Final exam - 60 points. The minimal number of points which can be obtained from the final exam is 25. Otherwise, no points from the final exam will be assigned to a student.

Schedule

DayTypeWeeksRoomStartEndCapacityLect.grpGroupsInfo
Mon exam 2023-10-30 M104 M105 11:0013:00 GALe: Midterm
Mon lecture 2., 3., 4., 5., 7., 8., 9., 10., 11., 12., 13. of lectures M104 M105 11:0013:5041 1EIT 2EIT INTE xx Křivka
Mon lecture 2023-09-18 M104 M105 11:0013:5041 1EIT 2EIT INTE xx Kocman
Mon lecture 2023-10-23 M104 M105 11:0013:5041 1EIT 2EIT INTE xx Havel
Mon exam 2024-01-08 A113 12:0014:50 GALe: 1st term
Fri exam 2024-01-19 D105 11:1013:40 GALe: 2nd term
Fri exam 2024-02-02 E105 13:0015:50 GALe: 3rd term

Course inclusion in study plans