关于此课程： 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.
制作方： Princeton University
教学方： Robert Sedgewick, William O. Baker *39 Professor of Computer Science
Computer Science
| 级别 | 高级设置 |
| 语言 | English |
| 如何通过 | 通过所有计分作业以完成课程。 |
| 用户评分 |
授课大纲
第 1 周
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 个视频, 2 个阅读材料
- Reading: Getting Started
- 视频: A Scientific Approach
- 视频: Example: Quicksort
- 视频: Resources
- Reading: Exercises from Lecture 1
- Discussion Prompt: Exercises from Lecture 1
已评分: Analysis of Algorithms
第 2 周
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 个视频, 1 个阅读材料, 2 个练习测验
- 视频: Telescoping
- Practice Quiz: Pop Quiz on Telescoping
- 视频: Types of Recurrences
- 视频: Mergesort
- 视频: Master Theorem
- Practice Quiz: Pop Quiz on the Master Theorem
- Reading: Exercises from Lecture 2
- Discussion Prompt: Exercises from Lecture 2
已评分: Recurrences
第 3 周
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 个视频, 1 个阅读材料
- 视频: Counting with Generating Functions
- 视频: Catalan Numbers
- 视频: Solving Recurrences
- 视频: Exponential Generating Functions
- Reading: Exercises from Lecture 3
- Discussion Prompt: Exercises from Lecture 3
已评分: Generating Functions
第 4 周
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 个视频, 1 个阅读材料
- 视频: Manipulating Expansions
- 视频: Asymptotics of Finite Sums
- 视频: Bivariate Asymptotics
- Reading: Exercises from Lecture 4
- Discussion Prompt: Exercises from Lecture 4
已评分: Asymptotics
第 5 周
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 个视频, 2 个阅读材料
- 视频: Labelled Objects
- 视频: Coefficient Asymptotics
- Reading: Errata
- 视频: Perspective
- Reading: Exercises from Lecture 5
- Discussion Prompt: Exercises from Lecture 5
已评分: Analytic Combinatorics
第 6 周
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 个视频, 1 个阅读材料
- 视频: Binary Search Trees
- 视频: Path Length
- 视频: Other Types of Trees
- Reading: Exercises from Lecture 6
- Discussion Prompt: Exercises from Lecture 6
已评分: Trees
第 7 周
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 个视频, 1 个阅读材料
- 视频: Sets of Cycles
- 视频: Left-Right-Minima
- 视频: Other Parameters
- 视频: BGFs and Distributions
- Reading: Exercises from Lecture 7
- Discussion Prompt: Exercises from Lecture 7
已评分: Permutations
第 8 周
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 个视频, 1 个阅读材料
- 视频: Languages
- 视频: Tries
- 视频: Trie Parameters
- 视频: Exercises
- Reading: Exercises from Lecture 8
- Discussion Prompt: Exercises from Lecture 8
已评分: Strings and Tries
第 9 周
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 个视频, 1 个阅读材料
- 视频: Birthday Problem
- 视频: Coupon Collector Problem
- 视频: Hash Tables
- 视频: Mappings
- 视频: Exercises
- Reading: Exercises from Lecture 9
- Discussion Prompt: Exercises from Lecture 9
已评分: Strings and Words
常见问题解答
运作方式
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制作方
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.
评分和审阅
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.
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.