Analysis of Algorithms

Go to Course: https://www.coursera.org/learn/analysis-of-algorithms

Introduction

### Course Review: Analysis of Algorithms on Coursera If you're interested in deepening your understanding of algorithms and their efficiencies, the *Analysis of Algorithms* course on Coursera is an excellent opportunity. This course offers a comprehensive look at techniques used to analyze algorithm performance based on mathematical principles. It's designed not only for computer science students but also for any curious minds looking to enhance their quantitative skills in algorithm analysis. #### Overview The *Analysis of Algorithms* course stands out for its detailed approach to understanding the scientific study of algorithm performance. The curriculum is structured in a way that begins by providing historical context, followed by a focus on real-world applications. The course emphasizes a methodical understanding of different combinatorial structures, including permutations, trees, strings, and mappings. One of the best parts? All features of this course are offered for free, making it accessible for anyone eager to learn. However, it's worth noting that no certificate is provided upon completion, but the knowledge gained is invaluable. #### Syllabus Breakdown The syllabus is well-organized and covers a wide range of topics crucial to algorithm analysis: 1. **Historical Context and Key Ingredients**: The course kicks off by exploring the motivation behind algorithm performance analysis, using the well-known Quicksort algorithm as a primary example. This foundational lecture sets the stage for what's to come. 2. **Recurrences**: The study of recurrence relations is crucial as it provides a mathematical model for analyzing algorithms. The oscillatory behavior of the divide-and-conquer recurrences, particularly with mergesort, is fascinating and insightful. 3. **Generating Functions**: This section dives into the utility of generating functions, which have been used for centuries to solve recurrences, exemplifying their application in enumerating structures like binary trees. 4. **Asymptotics**: The course then transitions to exploring asymptotic analysis, allowing students to develop approximate solutions to complex problems, a vital skill in algorithm analysis. 5. **Analytic Combinatorics**: With the foundational knowledge in place, this section introduces the systematic study of combinatorial classes, simplifying aspects previously covered in classical methods. 6. **Trees**: Focused on the recursive structure of trees, this part highlights their importance in computing applications, along with insights into enumerating various tree types. 7. **Permutations**: The relationship between sorting algorithms and permutations is examined, offering analytic-combinatoric techniques for better understanding their properties. 8. **Strings and Tries**: This segment underscores the prevalence of strings in modern computing, from DNA sequences to web indices, and presents tries as crucial structures. 9. **Words and Mappings**: Finally, the course investigates strings as sets of characters, delving into hashing algorithms and the complex structure of mappings. #### Final Thoughts The *Analysis of Algorithms* course on Coursera provides a robust foundation for anyone interested in the theoretical aspects of algorithms. The in-depth analysis supported by mathematical principles equips learners with practical skills applicable to real-world computing problems. While the absence of a completion certificate might deter some, the course’s high-quality content is a gain for learners at any level. It’s particularly beneficial for individuals who appreciate a structured, analytical approach to understanding algorithms without the pressure of formal recognition. ### Recommendation I highly recommend the *Analysis of Algorithms* course for anyone looking to enhance their understanding of algorithms from a quantitative perspective. Whether you are a beginner or a seasoned computer scientist, the insights gained will undoubtedly elevate your capacity to analyze and design efficient algorithms. Dive into this course—you won't regret it!

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.

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.

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.

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. 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.

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.

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.

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.

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.