Graph Algorithms
Code  Completion  Credits  Range  Language 

BIGRA  Z,ZK  5  2P+2C  Czech 
 Lecturer:
 Tutor:
 Supervisor:
 Department of Theoretical Computer Science
 Synopsis:

Students get an overview of typical usages of graph models in computing. They learn algorithmic methods of solution of graph problems, using the programming techniques presented in the BIEFA module. They understand algorithms for the key application domains of graph theory (flows in networks, heuristic search, approximation of complex problems, matching problems). Students get basic competence in computer science background: they understand Turing machine models and issues of NPcompleteness and NPhardness.
 Requirements:

We assume a working knowledge of basic abstract data types, its efficient implementation and application to solve graph problems. Students are assumed to have a passive knowledge of basic proof techniques used in mathematics (proof by induction, by contradiction, constructive) and apart from being able to design a new or modify an existing algorithm, they should be able to analyse its complexity.
 Syllabus of lectures:

1. Graph models, graphs, isomorphism.
2. Degree, connectedness, components, digraphs.
3. Strong connectedness, acyclic graphs, graph representation and traversal.
4. Depthfirst traversal, topologic sort, strong components.
5. Euler graphs, dominating and independent sets, graph coloring, distance.
6. Trees, spanning trees, circuits, minimal spanning trees.
7. Shortest paths, algorithms for 1 ? n shortest paths.
8. Algorithms for 1 ? n shortest paths, planarity.
9. Flows in networks, maximum flow algorithm.
10. Cheapest circulation, matchings, assignment problem.
11. Problem state space, heuristic search.
12. Turing machines.
13. Complexity classes, NPcomplete problems. Approximation algorithms.
 Syllabus of tutorials:

1. Induction, recursion, recurrence.
2. Trees and their computer representation, tree traversals.
3. Graph properties (isomorfism, connectedness, components, node degrees).
4. Strong connectedness, decomposition into strong components, topological sort.
5. Computer graph representations and traversals.
6. Eulerian trail, Chinese postman problem, semestral work selection.
7. Independence, dominance, chromatic number, cyclomatic number.
8. Trees, spanning trees and minimal spanning trees.
9. Shortest paths.
10. Shortest paths, planarity, semestral work (consultation).
11. Flows in networks, circulations, matchings.
12. Heuristic search algorithms.
13. Turing machines.
 Study Objective:

Graph models and related graph algorithms belong to a basic theoretical framework that has a number of applications both in computing and other domains. The aim of this course is to develop the abilities needed in order to recognize which graph model is appropriate and which algorithm should be used when solving a specific problem. It also builds some useful habits as how to find or estimate bounds of algorithmic complexity or limits of practical solvability of some types of problems.
 Study materials:

1. Kolář, J. Theoretical Computer Science. Praha: ČVUT, 1998. ISBN 8001017885.
2. Cormen, T. H., Leiserson, C. E., Rivest, R. L. Introduction to Algorithms. The MIT Press, 2001. ISBN 0262032937.
 Note:
 Further information:
 https://courses.fit.cvut.cz/BIGRA/
 No timetable has been prepared for this course
 The course is a part of the following study plans: