Acerca de este curso: 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.
Creada por: Princeton University
Enseñado por: Robert Sedgewick, William O. Baker *39 Professor of Computer Science
Computer Science
| Nivel | Avanzado |
| Idioma | English |
| Cómo aprobar | Aprueba todas las tareas calificadas para completar el curso. |
| Calificaciones del usuario |
Programa
SEMANA 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 lecturas
- Leyendo: Getting Started
- Vídeo: A Scientific Approach
- Vídeo: Example: Quicksort
- Vídeo: Resources
- Leyendo: Exercises from Lecture 1
- Cuadro de aviso de la discusión: Exercises from Lecture 1
Calificado: Analysis of Algorithms
SEMANA 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 lectura, 2 cuestionarios de práctica
- Vídeo: Telescoping
- Cuestionario de práctica: Pop Quiz on Telescoping
- Vídeo: Types of Recurrences
- Vídeo: Mergesort
- Vídeo: Master Theorem
- Cuestionario de práctica: Pop Quiz on the Master Theorem
- Leyendo: Exercises from Lecture 2
- Cuadro de aviso de la discusión: Exercises from Lecture 2
Calificado: Recurrences
SEMANA 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 lectura
- Vídeo: Counting with Generating Functions
- Vídeo: Catalan Numbers
- Vídeo: Solving Recurrences
- Vídeo: Exponential Generating Functions
- Leyendo: Exercises from Lecture 3
- Cuadro de aviso de la discusión: Exercises from Lecture 3
Calificado: Generating Functions
SEMANA 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 lectura
- Vídeo: Manipulating Expansions
- Vídeo: Asymptotics of Finite Sums
- Vídeo: Bivariate Asymptotics
- Leyendo: Exercises from Lecture 4
- Cuadro de aviso de la discusión: Exercises from Lecture 4
Calificado: Asymptotics
SEMANA 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 lecturas
- Vídeo: Labelled Objects
- Vídeo: Coefficient Asymptotics
- Leyendo: Errata
- Vídeo: Perspective
- Leyendo: Exercises from Lecture 5
- Cuadro de aviso de la discusión: Exercises from Lecture 5
Calificado: Analytic Combinatorics
SEMANA 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 lectura
- Vídeo: Binary Search Trees
- Vídeo: Path Length
- Vídeo: Other Types of Trees
- Leyendo: Exercises from Lecture 6
- Cuadro de aviso de la discusión: Exercises from Lecture 6
Calificado: Trees
SEMANA 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 lectura
- Vídeo: Sets of Cycles
- Vídeo: Left-Right-Minima
- Vídeo: Other Parameters
- Vídeo: BGFs and Distributions
- Leyendo: Exercises from Lecture 7
- Cuadro de aviso de la discusión: Exercises from Lecture 7
Calificado: Permutations
SEMANA 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 lectura
- Vídeo: Languages
- Vídeo: Tries
- Vídeo: Trie Parameters
- Vídeo: Exercises
- Leyendo: Exercises from Lecture 8
- Cuadro de aviso de la discusión: Exercises from Lecture 8
Calificado: Strings and Tries
SEMANA 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 lectura
- Vídeo: Birthday Problem
- Vídeo: Coupon Collector Problem
- Vídeo: Hash Tables
- Vídeo: Mappings
- Vídeo: Exercises
- Leyendo: Exercises from Lecture 9
- Cuadro de aviso de la discusión: Exercises from Lecture 9
Calificado: Strings and Words
Preguntas Frecuentes
Cómo funciona
Trabajo del curso
Cada curso es como un libro de texto interactivo, con videos pregrabados, cuestionarios y proyectos.
Ayuda de tus compañeros
Conéctate con miles de estudiantes y debate ideas y materiales del curso, y obtén ayuda para dominar los conceptos.
Creadores
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.
Calificaciones y revisiones
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.