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Updated in [June 30th, 2023]
Analysis of Algorithms is a course that provides students with the tools to make precise quantitative predictions of large combinatorial structures. It covers topics such as generating functions, real asymptotics, and the symbolic method. These topics are then applied to the analysis of algorithms and basic structures such as permutations, trees, strings, words, and mappings. This course is available for free and does not offer a certificate upon completion.
[Applications]
Upon completion of this course, students should be able to apply the concepts learned to analyze algorithms and basic structures such as permutations, trees, strings, words, and mappings. They should also be able to use the calculus to make precise quantitative predictions of large combinatorial structures. Additionally, students should be able to use the symbolic method to analyze algorithms.
[Career Paths]
[Recommended Career Path]Data Scientist. Data Scientists are responsible for analyzing large amounts of data and using it to make informed decisions. They use a variety of techniques, such as machine learning, statistical analysis, and data mining, to uncover patterns and trends in data. Data Scientists must have a strong understanding of algorithms and be able to apply them to solve complex problems.
The development trend of Data Scientists is very positive. As businesses become increasingly reliant on data-driven decisions, the demand for Data Scientists is expected to grow. Companies are investing more in data-driven strategies, and Data Scientists are becoming increasingly important in helping organizations make informed decisions. Additionally, the development of new technologies, such as artificial intelligence and machine learning, is creating new opportunities for Data Scientists.
[Education Paths]
For learners interested in pursuing a degree in the analysis of algorithms, a Bachelor's degree in Computer Science is recommended. This degree will provide students with a comprehensive understanding of the fundamentals of computer science, including algorithms, data structures, programming languages, operating systems, computer architecture, and software engineering. Students will also learn how to design, develop, and evaluate software systems.
The development trend of this degree is to focus on the application of algorithms in various fields, such as artificial intelligence, machine learning, data mining, and natural language processing. Students will learn how to use algorithms to solve complex problems and develop innovative solutions. Additionally, the degree will also cover topics such as computer networks, distributed systems, and security. As technology advances, the degree will continue to evolve to keep up with the latest trends and developments in the field.
Course Syllabus
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.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.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.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.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.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.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.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.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.
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