This course covers problem-oriented algorithm design and analysis techniques. To this end, the instructor gives an overview of the writing and symbols of algorithms, indicating computational complexity as a measure of the efficiency of algorithms. 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.
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)
complexity, randomized algorithms, online algorithms, algebraic method, probabilistic method
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
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|
|Class 1||Introduction (descriptions of algorithms, notations, etc.)||Foundations on algorithms|
|Class 2||Complexity of algorithms||Formal definition of a basic programming language|
|Class 3||Exponential time algorithms: subexponential time algorithms||An example of subexponential algorithm|
|Class 4||Exponential time algorithms: expected polynomial time algorithms||Analysis of exponential algorithms|
|Class 5||Exponential time algorithms: enumeration algorithms||An example of exponential algorithm|
|Class 6||Randomized algorithms for equality of sequences||Number of zeros of multivariate polynomilas and (total) degree of multivariate polynomilas|
|Class 7||Randomized algorithms for matrix products||Orthogonality of nonzero vectors|
|Class 8||Randomized algorithms for maximum cut||Applications of linearity of expectations|
|Class 9||Derandomization for maximum cut randomized algorithms||Applications of pairwise independence|
|Class 10||Online algorithms for job assignment||An example of online algorithm|
|Class 11||Online algorithms for caching||An example of online algorithm|
|Class 12||Greedy algorithms for minimum spaning trees||An example of greedy algorithm|
|Class 13||Greedy algorithms for Matroids||Characterization of greedy algorithms|
|Class 14||Algebraic method for set systems||An example of algebraic method|
|Class 15||Probabilistic method for maximum independent set||An example of probabilistic method|
All materials are found on OCW-i or are provides during class.
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.
Students' course scores are determined by solutions to several homework assignments given at the end of class and a report given at the final class.
No prerequisites are necessary, but basic knowledge on algorithms is expected.
Osamu Watanabe watanabe[at]is.titech.ac.jp
Toshiya Itoh titoh[at]c.titech.ac.jp
Contact by e-mail in advance for an appointment