### 2020　Theory of Algorithms

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Graduate major in Mathematical and Computing Science
Instructor(s)
Ito Toshiya
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Tue3-4(Zoom)  Fri3-4(Zoom)
Group
-
Course number
MCS.T405
Credits
2
2020
Offered quarter
3Q
Syllabus updated
2020/10/2
Lecture notes updated
-
Language used
Japanese
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### Course description and aims

This course covers problem-oriented algorithm design and analysis techniques. To this end, the instructor gives an overview of computational modell, complexity classes, polynomial-time reduction, and compete set of complexity classes. In addition to using computational complexity as a criterion, quality (accuracy and error rate) of output obtained from algorithms is also used as a criteria for designing algorithms for a variety of problems, while also performing theoretical analysis of the quality. The instructor will specifically show exponential time algorithms important for enumeration, typical randomized algorithms as examples of efficient algorithms, online algorithms for searching for good output from partial information, and greedy algorithms for problems with a special structure (general concept of independence), as well as perform theoretical analysis on them.

### Student learning outcomes

At the end of this course, students will be able to:
1) design and analyze algorithms
2) understand efficiency measure of algorithms (time complexity and space complexity)
3) understand accuracy measure of algorithms (approximation ratio and competitive ratio)

### Keywords

complexity, randomized algorithms, online algorithms, approximation algorithms, algebraic method, probabilistic method

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

Exercise problems are assigned (due next class) for homework every few classes to review the lesson content. The material is explained in the next lecture.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Computation model: Turing machine Foundations on computation model
Class 2 Complexity Classes and polynomial-time reduction Definition of complexity classes, Nodeterministic computation
Class 3 Natural NP-complete languages Examples of natural NP complete languages
Class 4 [1] Randomized algorithms for equality of sequences [2] Randomized algorithms for matrix products [1] Number of zeros of multivariate polynomials and (total) degree of multivariate polynomials [2] Orthogonality of nonzero vectors
Class 5 Randomized algorithms for maximum cut Applications of linearity of expectations
Class 6 Derandomization for maximum cut Applications of pairwise independence
Class 7 [1] Online algorithms for job assignment [2] Online algorithms for caching Examples of online algorithms
Class 8 Greedy algorithms for minimum spaning trees An example of greedy algorithm
Class 9 Greedy algorithms for Matroids Characterization of greedy algoritthm
Class 10 Approximation algorithms and approximation classes Approximation ration and performance ration
Class 11 Metric traveling salesperson problem Application of minimum spanning trees and Euler cycle
Class 12 Maximum knapsack problem Polynomial-time approximation scheme
Class 13 Inapproximablity Separation among approximation classes
Class 14 Mathematical Tools for Analysis Examples of the algebraic method and the probabilistic method

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

### Textbook(s)

All materials are found on OCW-i or are provides during class.

### Reference books, course materials, etc.

1. Fedor V. Fomin and Dieter Kratsch, Exact Exponential Algorithms, Springer, 2010
2. Stasys Jukna, External Combinatorics, Springer, 2001.
3. Allan Borodin and Ran El-Yaniv, Online Computation and Competitive Analysis, Cambridge Univ. Press, 1998.
4. Noga Alon and Joel H. Spencer, The Probabilistic Method, 3rd eds, Wiley, 2008.

### Assessment criteria and methods

Students' course scores are determined by solutions to several homework assignments.

### Related courses

• MCS.T213 ： Introduction to Algorithms and Data Structures
• CSC.T271 ： Data Structures and Algorithms
• ZUS.F302 ： Discrete Structures and Algorithms
• MCS.T322 ： Combinatorial Algorithms
• MCS.T411 ： Computational Complexity Theory

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

No prerequisites are necessary, but basic knowledge on algorithms is expected.

### Contact information (e-mail and phone)    Notice : Please replace from "[at]" to "@"(half-width character).

Toshiya Itoh titoh[at]c.titech.ac.jp

### Office hours

Contact by e-mail in advance for an appointment