Advanced Algorithms and Complexity

Go to Course: https://www.coursera.org/learn/advanced-algorithms-and-complexity

Introduction

### Course Review: Advanced Algorithms and Complexity on Coursera If you have a foundational understanding of algorithms and are ready to dive deeper into the world of computing, **Coursera's Advanced Algorithms and Complexity** course promises to elevate your skills significantly. Offered as part of an online specialization, this course is tailored for learners who have already grasped basic algorithms and are keen to tackle more complex problems. #### Course Overview In this course, you will explore advanced algorithms, building upon the basic ones you've previously learned. The curriculum begins with **Network Flows**, a foundational concept that has broad applications, from optimizing supply chains to managing internet traffic. As the course develops, you will encounter a wide range of sophisticated topics such as **Linear Programming**, NP-complete problems, and strategies for coping with NP-Complexity. One of the standout aspects of this course is its real-world applicability; the skills you acquire will not only be theoretical but will also be relevant in fields such as computer vision and data analysis. #### Detailed Syllabus 1. **Flows in Networks** - This unit introduces the concept of network flows, crucial for understanding how goods or information traverse complex networks. It combines theoretical foundations with practical algorithms, showcasing surprising connections to various real-world problems. 2. **Linear Programming** - Here, you'll learn to optimize linear functions under specific constraints. This module is particularly engaging as it extends beyond flow problems, addressing a wide range of applications, from production optimization to dietary planning. The unit emphasizes efficient algorithms for solving these linear programming challenges. 3. **NP-complete Problems** - A significant highlight of the course, this unit provides insights into some of the most challenging problems in computer science. You'll delve into the infamous P vs NP problem and learn about classical NP-complete problems and their reductions. Practical exercises will showcase effective methods for handling large instances despite their inherent complexity. 4. **Coping with NP-completeness** - This unit lifts your spirits after grappling with NP-complete problems by presenting strategies to tackle them in practice. You will discover that certain special cases can be solved efficiently, explore exact algorithms, and learn about approximation algorithms that yield near-optimal solutions quickly. 5. **Streaming Algorithms (Optional)** - For those interested in big data, this optional module is a treasure trove. It addresses scenarios where data cannot be stored in memory but instead comes in streams. You’ll explore streaming algorithms, gaining insight into how to process vast amounts of data efficiently while maintaining a compact summary. #### Learning Experience This course is well-structured, blending theoretical concepts with hands-on practice. Each module encourages learners to engage with the material actively, making use of quizzes and assignments to reinforce understanding. The clarity of the instructional content ensures that even complex ideas are accessible. The instructors bring a wealth of expertise, guiding you through intricate topics with patience and insight. Additionally, the online platform allows for flexible learning, enabling you to progress at your own pace. #### Recommendation **Who Should Enroll?** - This course is highly recommended for students and professionals who have a solid grasp of basic algorithms and wish to expand their knowledge into more intricate areas of algorithm design and complexity theory. If you’re considering a career in software engineering, data science, or any field that relies on algorithm optimization, this course will be invaluable. **Final Thoughts** The **Advanced Algorithms and Complexity** course on Coursera is an essential next step for anyone serious about mastering the art of algorithms. It bridges the gap between theoretical knowledge and practical application, empowering learners with the tools needed to tackle complex real-world challenges. Whether you aim to enhance your resume or gain deeper insights into algorithms, this course is a worthwhile investment in your education. Enroll today to unlock the full potential of your algorithmic capabilities!

Syllabus

Flows in Networks

Network flows show up in many real world situations in which a good needs to be transported across a network with limited capacity. You can see it when shipping goods across highways and routing packets across the internet. In this unit, we will discuss the mathematical underpinnings of network flows and some important flow algorithms. We will also give some surprising examples on seemingly unrelated problems that can be solved with our knowledge of network flows.

Linear programming is a very powerful algorithmic tool. Essentially, a linear programming problem asks you to optimize a linear function of real variables constrained by some system of linear inequalities. This is an extremely versatile framework that immediately generalizes flow problems, but can also be used to discuss a wide variety of other problems from optimizing production procedures to finding the cheapest way to attain a healthy diet. Surprisingly, this very general framework admits efficient algorithms. In this unit, we will discuss some of the importance of linear programming problems along with some of the tools used to solve them.

Although many of the algorithms you've learned so far are applied in practice a lot, it turns out that the world is dominated by real-world problems without a known provably efficient algorithm. Many of these problems can be reduced to one of the classical problems called NP-complete problems which either cannot be solved by a polynomial algorithm or solving any one of them would win you a million dollars (see Millenium Prize Problems) and eternal worldwide fame for solving the main problem of computer science called P vs NP. It's good to know this before trying to solve a problem before the tomorrow's deadline :) Although these problems are very unlikely to be solvable efficiently in the nearest future, people always come up with various workarounds. In this module you will study the classical NP-complete problems and the reductions between them. You will also practice solving large instances of some of these problems despite their hardness using very efficient specialized software based on tons of research in the area of NP-complete problems.

After the previous module you might be sad: you've just went through 5 courses in Algorithms only to learn that they are not suitable for most real-world problems. However, don't give up yet! People are creative, and they need to solve these problems anyway, so in practice there are often ways to cope with an NP-complete problem at hand. We first show that some special cases on NP-complete problems can, in fact, be solved in polynomial time. We then consider exact algorithms that find a solution much faster than the brute force algorithm. We conclude with approximation algorithms that work in polynomial time and find a solution that is close to being optimal.

In most previous lectures we were interested in designing algorithms with fast (e.g. small polynomial) runtime, and assumed that the algorithm has random access to its input, which is loaded into memory. In many modern applications in big data analysis, however, the input is so large that it cannot be stored in memory. Instead, the input is presented as a stream of updates, which the algorithm scans while maintaining a small summary of the stream seen so far. This is precisely the setting of the streaming model of computation, which we study in this lecture. The streaming model is well-suited for designing and reasoning about small space algorithms. It has received a lot of attention in the literature, and several powerful algorithmic primitives for computing basic stream statistics in this model have been designed, several of them impacting the practice of big data analysis. In this lecture we will see one such algorithm (CountSketch), a small space algorithm for finding the top k most frequent items in a data stream.