Math 118, Topics in combinatorics, Mathematical Analysis of Algorithms
“People who analyze algorithms have double happiness. First of all
they experience the sheer beauty of elegant mathematical patterns
that surround elegant computational procedures. Then they receive a
practical payoff when their theories make it possible to get other jobs
done more quickly and more economically.”
– Donald Ervin Knuth
Instructor: Nadia Lafrenière (firstname.lastname@example.org)
Class schedule: MWF 10:20 - 11:25 AM, Zoom # 925 0515 7790, password: 818910
x-hour: Th 12:30-1:20 PM, will likely not be used
Office hours: Monday 1:30-3:30 on Zoom (same meeting as lectures), and by appointment.
Analysis of algorithms is a field of research on the edge of computer science and mathematics.
The approach for this class is the one introduced in the 60’s by Donald Ervin Knuth. Namely, we look at best-case, worst-case and average-case performance analysis, with emphasis on the latter. This study needs an understanding of combinatorial structures and probabilistic arguments, which will be detailed in this course.
Part of our exploration will imply asymptotic methods. Most arguments in the analysis of algorithms assume that we repeat the procedure often enough, so it is worth analyzing its cost. Therefore, analysis of algorithms goes along with asymptotic combinatorics, as we count the cost of running an algorithm in relationship to the size
of its input.
In this class, we study algorithms independently of their implementation. This means that we look at theoretical limitations, regardless of the functionalities offered by any given programming language.
This class is remote with synchronous components. Class time will be devoted to live lectures on Zoom, and students are expected to work on problem sets outside of class. On some days, student presentations will replace the instructor’s lecture.
Despite the focus of the class being on algorithms, students are not expected to implement them. For students who want to code, there will be opportunities to replace a few homework problems over the term by implementation and bench marking of algorithms.
Attendance Policy and x-hour
Class attendance and participation are expected. If you know that you will not be able to deliver an evaluation on time for a serious reason, let me know as soon as possible.
The x-hour will not be used, except for one lecture that is moved to the x-hour due to Martin Luther King Jr. Day.
Content’s Schedule (tentative)
This schedule is tentative, and will be updated over the term. Homework assignments are due on Wednesdays.
|Introduction to the analysis of algorithms
|First example: merge sort, optimal sorting algorithms
|Divide-and-conquer algorithms, part 1
|No class, MLK day
|Divide-and-conquer algorithms, part 2 (Master’s theorem, Strassen algorithm for matrix multiplication)
|January 21 (x-hour)
|The symbolic method, ordinary generating functions
|Labeled classes and exponential generating functions
|Averages and moments
|Asymptotic expansions, part 1
|Asymptotic expansions, part 2
|Compositions and partitions
|Trees for searching
|Q&A: The symbolic method
|Permutation statistics: Ascents, descents, runs
|Permutation statistics: Cycle lengths
|Permutation statistics: Increasing sequences
|Sorting: Bubble sort, Quicksort
|Randomized algorithms, Quicksort
Dartmouth hosts a combinatorics seminar every other Friday at 1:15. You are encouraged to attend it, and I will advertise it in class before each talk. More details can be found on this webpage: https://math.dartmouth.edu/~comb/.
The homework assignments will be available on Fridays, and will be due each Wednesday. There might be one or two problems that you will not be able to solve on the first days the problem set is available, but these will be marked.2:45-3:45
|Weekly on Wednesdays
|2 student presentations
|Weeks of February 5 and March 5
|Throughout the term
Finding the right answer is not equivalent to solving a problem, and therefore your solution will be evaluated as a whole. Please, keep in mind while writing the assignments that I will not be with you when I will read it, so it must be complete.
The student presentations can be about the analysis of any standard algorithm, or about techniques for analyzing or creating algorithms. Early in the term, I will provide the class with a list of topics that can be explored, but students will be allowed to choose topics not listed there. Students must notify me when they choose a topic.
I encourage students to do their presentations in teams of two students. The goal is twofold: to invite students to collaborate and socialize, because most math is done in collaboration, and to allow for some harder or larger topics to be presented.
There is no required textbook for the class. Instead, course notes will be shared with students, and references will be indicated on them. A list of references that I am using to design the course material follows in this syllabus. All books listed here are either freely available online (at least partially) or are available (in both paper and electronic format) through the Dartmouth library. The main reference for the course will be [AofA]. Depending on students’ interests, a list of papers will be added to this list of books.
- [AC] P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge University Press, 2009. Freely available online at https://ac.cs.princeton.edu/home/.
- [Algo] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms, 3rd Edition. MIT Press, 2009. Available online through Dartmouth Libraries.
- [AoCP] D. E. Knuth. The Art of Computer Programming, Volume I: Fundamental Algorithms. Addison-Wesley, 1968. Available online through Dartmouth Libraries.
- [AofA] R. Sedgewick and P. Flajolet. An introduction to the analysis of algorithms. Addison-Wesley-Longman, 1996. Available online through Dartmouth Libraries.
- [EC1] R. P. Stanley. Enumerative combinatorics. Volume 1, volume 49 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2012. Freely available online at http://www-math.mit.edu/~rstan/ec/ec1.pdf.
- [EC2] R. P. Stanley. Enumerative combinatorics. Volume 2, volume 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. Available online through Dartmouth Libraries.
- [Wilf] H. S. Wilf. generatingfunctionology. Academic Press, Inc., Boston, MA, second edition, 1994. Freely available at https://www.math.upenn.edu/~wilf/gfologyLinked2.pdf. Only the first edition is physically available at the library.
Consent to Record
(1) Consent to recording of course and group office hours
a) I affirm my understanding that this course and any associated group meetings involving students and the instructor, including but not limited to scheduled and ad hoc office hours and other consultations, may be recorded within any digital platform used to offer remote instruction for this course;
b) I further affirm that the instructor owns the copyright to their instructional materials, of which these recordings constitute a part, and distribution of any of these recordings in whole or in part without prior written consent of the instructor may be subject to discipline by Dartmouth up to and including expulsion;
(2) Requirement of consent to one-on-one recordings
By enrolling in this course, I hereby affirm that I will not under any circumstance make a recording in any medium of any one-on-one meeting with the instructor without obtaining the prior written consent of all those participating, and I understand that if I violate this prohibition, I will be subject to discipline by Dartmouth up to and including expulsion, as well as any other civil or criminal penalties under applicable law.
Assignments and examinations serve both the purpose of fixing your ideas on a given subject and, for the evaluator, to assess the comprehension you have of the class and how the class material has been understood. None of these goals can be achieved if you hand out a solution that is taken from someone else or from a publicly available source.
Nevertheless, students are allowed to work together on problems to explore their solutions. The homework you submit must reflect your own understanding, and you thus need to write your own copy of the solution, in your own words.
More information on the Academic Honor Principle can be found on Dartmouth’s website: https://students.dartmouth.edu/judicial-affairs/policy/academic-honor-principle
Student Accessibility and Accommodations
Students with disabilities who may need disability-related academic adjustments and services for this course are encouraged to see me privately as early in the term as possible. Students requiring disability-related academic adjustments and services must consult the Student Accessibility Services office by phone: 646-9900 or email: Stu-
Once SAS has authorized services, students must show the originally signed SAS Services and Consent Form and/or a letter on SAS letterhead to me. As a first step, if you have questions about whether you qualify to receive academic adjustments and services, you should contact the SAS office. All inquiries and discussions will remain
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Some students may wish to take part in religious observances that occur during this academic term. If you have a religious observance that conflicts with your participation in the course, please meet with me as soon as possible, preferably before the end of the second week of the term to discuss appropriate accommodations.
All material for this class is aimed to be freely available, either at the library or online. If you encounter financial challenges related to this class (because of the workload for example), please let me know.
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