Algorithms and Graphs 1

Garant předmětu:

Dušan Knop

Lecturer:

Dušan Knop, Michal Opler

Tutor:

Suzan Catay, Michal Dvořák, Radek Hušek, Dušan Knop, Jitka Mertlová, Xuan Thang Nguyen, Michal Opler, Josef Erik Sedláček, Martin Slávik, José Gaspar Smutný, Ondřej Suchý, Ondřej Šofr, Tomáš Valla

Supervisor:

Department of Theoretical Computer Science

Synopsis:

The course covers the basics of efficient algorithm design, data structures, and graph theory, belonging to the core knowledge of every computing curriculum.

It links and partially develops the knowledge from the course BI-DML.21, in which students acquire the knowledge and skills in combinatorics necessary for evaluating the time and space complexity of algorithms. The course also follows up knowledge from BI-MA1.21, the practical usage of asymptotic mathematics, in particular, the asymptotic notation.

Requirements:

Active algorithmic skills for solving basic types of computational tasks, programming skills in C++ (e.g., the level needed for passing BIE-PA1.21 and BIE-PA2.21) , and knowledge of basic notions from mathematical analysis and combinatorics are expected (e.g., by passing BIE-DML.21 a BIE-MA1.21). Students are expected to take the concurrent course BIE-AAG.21 and BIE-MA2.21.

Syllabus of lectures:

1. Motivation, graph definition, important types of graphs, undirected graphs, graph representation, subgraphs.

2. Connectivity, connected components, DFS, directed graphs, trees.

3. Spanning trees, distances in graphs, BFS, topological ordering.

4. Basic sorting algorithms with the quadratic time complexity. Binary heap as a partially ordered structure, HeapSort.

5. Extendable array, amortized complexity. Binomial Heaps.

6. Operations and properties of binary search trees, balancing strategies, and AVL trees.

7. Randomized algorithms. Introduction to probability theory. Hash tables and strategies of collision resolving.

8. Recursive algorithms and Divide and Conquer algorithms.

9. QuickSort. Lower bound of complexity for sorting problem in the comparison model. Special sorting algorithms.

10. Dynamic programming.

11. Minimum spanning trees of edge-labelled graphs. Jarník’s algorithm and Kruskal’s algorithm and their implementations.

12. Shortest paths algorithms on edge-labeled graphs.

Syllabus of tutorials:

1. Motivation and Elements of Graph Theory I.

2. Elements of Graph Theory II.

3. Elements of Graph Theory III.

4. Sorting Algorithms O(n^2). Binary Heaps.

5. Extendable Array, Amortized Complexity, Binomial Heaps.

6. Search Trees and Balance Strategies.

7. Hashing and Hash tables.

8. Recursive Algorithms and Divide et Impera Method.

9. Probabilistic Algorithms and their Complexity. QuickSort.

10. Semestral test.

11. Dynamic Programming.

13. Minimum Spanning Trees, Shortest Paths.

Study Objective:

Students learn basic techniques for proving the correctness of algorithms and techniques of asymptotic mathematics for estimation of their complexity in the best, worst, or average case.

Study materials:

1. Cormen T.H., Leiserson C.E., Rivest R.L., Stein C.: Introduction to Algorithms (4th Edition). MIT Press, 2022. ISBN 978-0262033848.

2. J. Matoušek, J. Nešetřil: Invitation to Discrete Mathematics, 2008, 2th edition, Oxford University Press. (Available online in English.)

3. R. Diestel: Graph Theory, 2010, 4th edition, Springer-Verlag, Berlin. (Available online, new edition released in 2017.)

Note:

Further information:

https://courses.fit.cvut.cz/BI-AG1

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Bachelor specialization Information Security, in Czech, 2021 (compulsory course in the program)
• Bachelor specialization Management Informatics, in Czech, 2021 (compulsory course in the program)
• Bachelor specialization Computer Graphics, in Czech, 2021 (compulsory course in the program)
• Bachelor specialization Computer Engineering, in Czech, 2021 (compulsory course in the program)
• Bachelor program, unspecified specialization, in Czech, 2021 (compulsory course in the program)
• Bachelor specialization Web Engineering, in Czech, 2021 (compulsory course in the program)
• Bachelor specialization Artificial Intelligence, in Czech, 2021 (compulsory course in the program)
• Bachelor specialization Computer Science, in Czech, 2021 (compulsory course in the program)
• Bachelor specialization Software Engineering, in Czech, 2021 (compulsory course in the program)
• Bachelor specialization Computer Systems and Virtualization, in Czech, 2021 (compulsory course in the program)
• Bachelor specialization Computer Networks and Internet, in Czech, 2021 (compulsory course in the program)