course - discrete mathematics

NATIONAL RESEARCH OGAREV MORDOVIA STATE UNIVERSITY

Course: Discrete MathematicsMajor profiles:

Informatics and computer science (ICS), Programming Engineering (PE)

CONTENT – 1

The discipline “Discrete Mathematics” (hereinafter referred to as DM) is studied at the 2st (spring) semester of the 1st (2nd) year of tuition.The content is divided into 4 blocks / modules:• “Set theory and combinatorics”.• “Graph theory”.• “Algebraic structures”.• “Coding theory”.The volumes of the 1st and the 2nd modules (in academic hours) are approximately the same, the 3rd and the 4th volumes are somewhat smaller.At the end of studying the module 1, 2 and 3–4 every student must write and defend an essay (will be discussed later). So there are 3 essays to be written during the semester.The studying of DM ends with exam that takes place in June.

OMSU – Discrete MathematicsPE & ICS

CONTENT – 2

Set theory and combinatoricsThe definition of set. Universal and empty sets. Set operations and their properties. Venn diagrams.Cartesian product of sets. Binary relations. Application to relational databases. Ordering and equivalence relationships. Cardinality of a set. Mappings. Ordered and disordered sets. Sorting algorythms.Problems of combinatorics. Principles of sum and of product. Permutations, arrangements and combinations (with and without repeating elements). The number of subsets in n-element set. Binomial theorem and its generalization. The Pascal triangle.Coverings and partitions. The cardinality of sets’ union. Stirling (of the 2nd kind) and Bell numbers.


CONTENT – 3

Graph theoryGraphs. adjacency and incidence. The ways of graph representation: adjacency and incidence matrices, list of adjacency. Basic operations upon graphs.Graph connectivity. Distances in graphs. Tree graphs and circuits. Cyclomatic number. Euler (unicursal) and Hamilton graphs. Spanning trees. Algorithms of Prim and Kruskal.Depth-first search and breadth-first search. Modification of algorithms for searching of graph’s connectivity components.Transport networks. Dijkstra algorithm. The problems of maximum flow and Ford-Fulkerson algorithm.


CONTENT – 4

Algebraic structuresOperation upon the elements of set. Algebra as some structure (elements + operations). Cayley table.Sets with one operation. Groupoids and groups. Examples. Subgroup. Abel (commutative) groups. The group of permutations. Cyclic groups.Sets with two operations. Rings and fields.The ring of integers. Divisibility, GCD and LCM. Euclidean algorithm due to integers. Comparisons by integer module. Rings and fields of residues.The ring of one-variable polynomials. The polynomial roots. Divisibility and GCD of polynomials. Fundamental algebra theorem. Factorization of polynomials. Horner’s rule. Euclidean algorithm due to polynomials.


CONTENT – 5

Coding theoryCoding as mapping, its purposes and types. Uniquely decodable codes. Kraft-McMillan theorem. Prefix codes.Message redundancy and the problem of optimal coding. Example: Morse code. Shannon-Fano and Huffman codes.Elementary mistakes in messages: symbol drop-out, inserting of symbol and substitution error. Noise combating codes. The code distance. Hamming code.Coding for data compression. LZW algorithm. The idea of compressing data with information losses (JPEG and MP3).Coding for information security. Ciphers. Public key encryption. Diffie-Hellman key exchange and RSA algorithm.


Pedagogical METHODS – 1

Auditorial• Lections – for all 1st (2nd)-year students of the profile.• Practice / trainings – for each academic group separately (if there are many).

On practice / trainings students solve typical computational problems of the discipline proposed by the teacher. The topics of practice / trainings must be in harmony with the lections’ topics.

Subgroup (5– 6 students) work is done from time to time at the trainings. It may be a collective solving of a problem that allows “parallelization” (e.g. searching for spanning tree using Prim and Kruskal algorithms). Students enjoy this kind of command work as it is a competition between subgroups.

It’s quite effective, too, because students explain the learning material to each other while fulfilling the collective task.


Extraauditorial• Essays / tests – mandatory

For DM essay is a sequence of theoretical questions and computational problems that must be solved by a student. 3 essays must be written: in the middle of March, April and May, respectively.

• Computer programming – additionalThe PE and ICS-students are the future programmers, so they must be able

to realize some typical algorithms of DM using high-level programming language. The student may choose the language as he/she wants, dialects of C and Pascal are the common choices.


All the extraauditorial work is controlled by the teacher. Students must defend their essays and programs.



Example of computational problems in essay “Graph theory”1. Two graphs are described by their matrices of adjacency. Check if the graphs are isomorphic. Then make a picture of both graphs. (Each graph has 7 vertices.)2. Two graphs are given in a picture. Check if the graphs are homeomorphic. (The number of vertices is some like 10–12 and may not be same for both graphs.)3. In one of graphs mentioned above (see #2) find the Euler subgraph with maximal number of edges.4. For a graph on the picture find diameter, centers, radius, articulation nodes and bridges. Write down the incidence matrix for this graph.5. 10-vertice graph is given by its matrix of adjacency. Find the way from the 3rd vertex to the 10th vertex using depth-first search. Find the way from the 10th vertex to the 1st vertex using breadth-first search. Don’t make a picture of the graph while fulfilling the task.

…to be continued on the next slide


… the beginning is on the previous slide6. 10-vertice graph is given by its list of adjacency. Find all the connectivity components of the graph by using depth-first search and breadth-first search. As in the previous task, you should not make a picture of the graph.7. Find the minimal spanning tree of weighted graph using algorithms of Prim and Kruskal.8. Weighted graph from #7 represents transport network. Find the minimal-cost way from vertex 0 to vertex 10 using Dijkstra algorithm.9. Find the maximal flow in a transport network with the help of Ford-Fulkerson algorithm.

Pedagogical METHODS – 4 OMSU – Discrete MathematicsPE & ICS


Examples of programming tasks (module “Graph theory”)(The ways of graph representation are defined by a programmer; the choice of algorithm relies on a programmer, too).

1. Find the way from vertex A to vertex B in a connected graph (A and B are chosen by user).2. Find the minimal spanning tree in a weighted graph.3. In a transport network find the minimal-cost way from the source to the sink.4. Find out the maximal flow in the transport network.


ASSESSMENTS

Rating system is used: semester (3 essays + auditorial work) + exam100 points is the possible maximum• 86 – 100 – “Excellent” (approx. “A” or “B” in ECTS grading scale)• 71 – 85 – “Good” (approx. “C” or “D” in ECTS grading scale)• 51 – 70 – “Satisfactory” (approx. “D” or “E” in ECTS grading scale)• 0 – 50 – “Bad” (approx. “Fx” or “F” in ECTS grading scale)


Example of examination taskTheoretical questions1. Binary relations and their properties. Ordering relationship.2. Transport networks. Dijkstra algorythm.

Computational problemCorrect the substitution error in a message which is coded by (15, 11)

Hamming code, if there is any: 001010111010011.

course - discrete mathematics

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