| Title: | Complexity |
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| Code: | SLOa |
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| Ac.Year: | 2017/2018 |
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| Term: | Summer |
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| Curriculums: | |
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| Language of Instruction: | English |
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| News: | This course is instructed in English, and it is intended for incoming Erasmus+ students too.
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| Private info: | http://www.fit.vutbr.cz/study/courses/SLOa/private/ |
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| Credits: | 5 |
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| Completion: | examination (written&verbal) |
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Type of
instruction: | | Hour/sem | Lectures | Sem. Exercises | Lab. exercises | Comp. exercises | Other |
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| Hours: | 26 | 0 | 0 | 0 | 26 |
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| | Examination | Tests | Exercises | Laboratories | Other |
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| Points: | 68 | 0 | 0 | 0 | 32 |
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| Guarantor: | Vojnar Tomáš, prof. Ing., Ph.D., DITS |
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| Lecturer: | Rogalewicz Adam, doc. Mgr., Ph.D., DITS |
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| Instructor: | Rogalewicz Adam, doc. Mgr., Ph.D., DITS |
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| Faculty: | Faculty of Information Technology BUT |
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| Department: | Department of Intelligent Systems FIT BUT |
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| Prerequisites: | |
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| Follow-ups: | |
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| Substitute for: | |
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| | | Learning objectives: |
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| | | Familiarize students with the complexity theory, which is necessary to
understand practical limits of algorithmic problem solving on physical
computational systems.
Familiarize students with a selected methods for solving hard computational problems. | | Description: |
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| | | Turing machines as a basic computational model for computational
complexity analysis, time and space complexity on Turing machines.
Alternative models of computation, RAM and RASP machines and their
relation to Turing machines in the context of complexity. Asymptotic
complexity estimations, complexity classes based on time- and
space-constructive functions, typical examples of complexity classes.
Properties of complexity classes: importance of determinism and
non-determinism in the area of computational complexity, Savitch
theorem, relation between non-determinism and determinism, closure
w.r.t. complement of complexity classes, proper inclusion between
complexity classes. Selected advanced properties of complexity classes:
Blum theorem, gap theorem. Reduction in the context of complexity and
the notion of complete classes. Examples of complete problems for
different complexity classes. Deeper discussion of P and NP classes with
a special attention on NP-complete problems (SAT problem, etc.).
Relationship between decision and optimization problems. Methods for
computational solving of hard optimization problems: deterministic
approaches, approximation, probabilistic algorithms. Relation between
complexity and cryptography. Deeper discussion of PSPACE complete
problems, complexity of formal verification methods. | | Knowledge and skills required for the course: |
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| | | Formal language theory and theory of computability on master level. | | Learning outcomes and competences: |
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| | | Understanding theoretical and practical limits of arbitrary
computational systems. Ability to use a selected methods for
computationally hard problems. | | Syllabus of lectures: |
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| | - Introduction. Complexity, time and space complexity.
- Matematical models of computation, RAM, RASP machines and their relation with Turing machines.
- Asymptotic estimations, complexity classes, determinism and non-determinism from the point of view of complexity.
- Relation between time and space, closure of complexity classes w.r.t. complementation, proper inclusion of complexity classes.
- Blum theorem. Gap theorem.
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Reduction, notion of complete problems, well known examples of complete problems.
- Classes P and NP. NP-complete problems. SAT problem.
- Travelling salesman problem, Knapsack problem and other important NP-complete problems
- NP optimization problems and their deterministic solution: pseudo-polynomial algorithms, parametric complexity
- Approximation algorithms.
- Probabilistic algorithms, probabilistic complexity classes.
- Complexity and cryptography
- PSPACE-complete problems. Complexity and formal verification.
| | Syllabus - others, projects and individual work of students: |
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| | 4 projects dedicated on different aspects of the complexity theory. | | Fundamental literature: |
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- Gruska, J.: Foundations of Computing, International Thomson Computer Press, 1997, ISBN 1-85032-243-0
- Hopcroft, J.E. et al: Introduction to Automata Theory, Languages, and Computation, Addison Wesley, 2001, ISBN 0-201-44124-1
| | Study literature: |
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- Gruska, J.: Foundations of Computing, International Thomson Computer Press, 1997, ISBN 1-85032-243-0
- Hopcroft, J.E. et al: Introduction to Automata Theory, Languages, and Computation, Addison Wesley, 2001, ISBN 0-201-44124-1
| | Progress assessment: |
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| | | - 4 projects - 8 points each (recommended minimal gain is 15 points).
- Final exam: max. 68 points
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Faculty of Information Technology
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