About this course: This course teaches a calculus that enables precise quantitative predictions of large combinatorial structures. In addition, this course covers generating functions and real asymptotics and then introduces the symbolic method in the context of applications in the analysis of algorithms and basic structures such as permutations, trees, strings, words, and mappings.
Created by: Princeton University
Taught by: Robert Sedgewick, William O. Baker *39 Professor of Computer Science
Computer Science
| Level | Advanced |
| Language | English |
| How To Pass | Pass all graded assignments to complete the course. |
| User Ratings |
Syllabus
WEEK 1
Analysis of Algorithms
We begin by considering historical context and motivation for the scientific study of algorithm performance. Then we consider a classic example that illustrates the key ingredients of the process: the analysis of Quicksort. The lecture concludes with a discussion of some resources that you might find useful during this course.
4 videos, 2 readings
- Материал для самостоятельного изучения: Getting Started
- Video: A Scientific Approach
- Video: Example: Quicksort
- Video: Resources
- Материал для самостоятельного изучения: Exercises from Lecture 1
- Вопрос для обсуждения: Exercises from Lecture 1
Graded: Analysis of Algorithms
WEEK 2
Recurrences
We begin this lecture with an overview of recurrence relations, which provides us with a direct mathematical model for the analysis of algorithms. We finish by examining the fascinating oscillatory behavior of the divide-and-conquer recurrence corresponding to the mergesort algorithm and the general "master theorem" for related recurrences.
5 videos, 1 reading, 2 practice quizzes
- Video: Telescoping
- Тренировочный тест: Pop Quiz on Telescoping
- Video: Types of Recurrences
- Video: Mergesort
- Video: Master Theorem
- Тренировочный тест: Pop Quiz on the Master Theorem
- Материал для самостоятельного изучения: Exercises from Lecture 2
- Вопрос для обсуждения: Exercises from Lecture 2
Graded: Recurrences
WEEK 3
Generating Functions
Since the 17th century, scientists have been using generating functions to solve recurrences, so we continue with an overview of generating functions, emphasizing their utility in solving problems like counting the number of binary trees with N nodes.
5 videos, 1 reading
- Video: Counting with Generating Functions
- Video: Catalan Numbers
- Video: Solving Recurrences
- Video: Exponential Generating Functions
- Материал для самостоятельного изучения: Exercises from Lecture 3
- Вопрос для обсуждения: Exercises from Lecture 3
Graded: Generating Functions
WEEK 4
Asymptotics
Exact answers are often cumbersome, so we next consider a scientific approach to developing approximate answers that, again, mathematicians and scientists have used for centuries.
4 videos, 1 reading
- Video: Manipulating Expansions
- Video: Asymptotics of Finite Sums
- Video: Bivariate Asymptotics
- Материал для самостоятельного изучения: Exercises from Lecture 4
- Вопрос для обсуждения: Exercises from Lecture 4
Graded: Asymptotics
WEEK 5
Analytic Combinatorics
Analytic Combinatorics. With a basic knowledge of recurrences, generating functions, and asymptotics, you are ready to learn and appreciate the basic features of analytic combinatorics, a systematic approach that avoids much of the detail of the classical methods that we have been considering. We introduce unlabeled and labelled combinatorial classes and motivate our basic approach to studying them, with numerous examples.
4 videos, 2 readings
- Video: Labelled Objects
- Video: Coefficient Asymptotics
- Материал для самостоятельного изучения: Errata
- Video: Perspective
- Материал для самостоятельного изучения: Exercises from Lecture 5
- Вопрос для обсуждения: Exercises from Lecture 5
Graded: Analytic Combinatorics
WEEK 6
Trees
The quintessential recursive structure, trees of various sorts are ubiquitous in scientific enquiry, and they arise explicitly in countless computing applications. You can find broad coverage in the textbook, but the lecture focuses on the use of analytic combinatorics to enumerate various types of trees and study parameters.
4 videos, 1 reading
- Video: Binary Search Trees
- Video: Path Length
- Video: Other Types of Trees
- Материал для самостоятельного изучения: Exercises from Lecture 6
- Вопрос для обсуждения: Exercises from Lecture 6
Graded: Trees
WEEK 7
Permutations
The study of sorting algorithms is the study of properties of permutations. We introduce analytic-combinatoric approaches to studying permutations in the context of this relationship.
5 videos, 1 reading
- Video: Sets of Cycles
- Video: Left-Right-Minima
- Video: Other Parameters
- Video: BGFs and Distributions
- Материал для самостоятельного изучения: Exercises from Lecture 7
- Вопрос для обсуждения: Exercises from Lecture 7
Graded: Permutations
WEEK 8
Strings and Tries
From DNA sequences to web indices, strings (sequences of characters) are ubiquitous in modern computing applications, so we use analytic combinatorics to study their basic properties and then introduce the trie, an essential and fundamental structure not found in classical combinatorics.
5 videos, 1 reading
- Video: Languages
- Video: Tries
- Video: Trie Parameters
- Video: Exercises
- Материал для самостоятельного изучения: Exercises from Lecture 8
- Вопрос для обсуждения: Exercises from Lecture 8
Graded: Strings and Tries
WEEK 9
Words and Mappings
We view strings as sets of characters or as functions from [1..N] to [1..M] to study classical occupancy problems and their application to fundamental hashing algorithms. Functions from [1..N] to [1..N] are mappings, which have an interesting and intricate structure that we can study with analytic combinatorics.
6 videos, 1 reading
- Video: Birthday Problem
- Video: Coupon Collector Problem
- Video: Hash Tables
- Video: Mappings
- Video: Exercises
- Материал для самостоятельного изучения: Exercises from Lecture 9
- Вопрос для обсуждения: Exercises from Lecture 9
Graded: Strings and Words
FAQs
How It Works
Задания курса
Каждый курс — это интерактивный учебник, который содержит видеоматериалы, тесты и проекты.
Помощь сокурсников
Общайтесь с тысячами других учащихся: обсуждайте идеи, материалы курса и помогайте друг другу осваивать новые понятия.
Creators
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. It is one of the eight universities of the Ivy League, and one of the nine Colonial Colleges founded before the American Revolution.
Ratings and Reviews
This course is more about mathematic than algorithms, it teaches how to solve tricky combinatorial problems
This is great course if you already done some algorithms courses and want to go deeper.
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IF
This course offers an extensive coverage of mathematical material like tools and techniques, putting special emphasis on simple principles and never losing its focus on the general perspective of the topic.
Developing insight on analytic functions together with combinatorics and the practical applications of theory into code gives a significant advantage over using naive approaches.
A special appreciation to Prof. Sedgewick's endless efforts for disseminating mathematical knowledge among programmers, over this platform and even outside!
Thank you very much.