Advanced Algorithms and Complexity

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About this course: You've learned the basic algorithms now and are ready to step into the area of more complex problems and algorithms to solve them. Advanced algorithms build upon basic ones and use new ideas. We will start with networks flows which are used in more typical applications such as optimal matchings, finding disjoint paths and flight scheduling as well as more surprising ones like image segmentation in computer vision. We then proceed to linear programming with applications in optimizing budget allocation, portfolio optimization, finding the cheapest diet satisfying all requirements and many others. Next we discuss inherently hard problems for which no exact good solutions…

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When you enroll for courses through Coursera you get to choose for a paid plan or for a free plan

  • Free plan: No certicification and/or audit only. You will have access to all course materials except graded items.
  • Paid plan: Commit to earning a Certificate—it's a trusted, shareable way to showcase your new skills.

About this course: You've learned the basic algorithms now and are ready to step into the area of more complex problems and algorithms to solve them. Advanced algorithms build upon basic ones and use new ideas. We will start with networks flows which are used in more typical applications such as optimal matchings, finding disjoint paths and flight scheduling as well as more surprising ones like image segmentation in computer vision. We then proceed to linear programming with applications in optimizing budget allocation, portfolio optimization, finding the cheapest diet satisfying all requirements and many others. Next we discuss inherently hard problems for which no exact good solutions are known (and not likely to be found) and how to solve them in practice. We finish with a soft introduction to streaming algorithms that are heavily used in Big Data processing. Such algorithms are usually designed to be able to process huge datasets without being able even to store a dataset.

Who is this class for: Programmers with basic experience looking to understand the practical and conceptual underpinnings of algorithms, with the goal of becoming more effective software engineers. Computer science students and researchers as well as interdisciplinary students (studying electrical engineering, mathematics, bioinformatics, etc.) aiming to get more profound understanding of algorithms and hands-on experience implementing them and applying for real-world problems. Applicants who want to prepare for an interview in a high-tech company.

Created by:  University of California, San Diego, Higher School of Economics
  • Taught by:  Alexander S. Kulikov, Visiting Professor

    Department of Computer Science and Engineering
  • Taught by:  Michael Levin, Lecturer

    Computer Science
  • Taught by:  Daniel M Kane, Assistant Professor

    Department of Computer Science and Engineering / Department of Mathematics
  • Taught by:  Neil Rhodes, Adjunct Faculty

    Computer Science and Engineering
Basic Info Course 5 of 6 in the Data Structures and Algorithms Specialization Level Advanced Commitment 4 weeks of study, 4-8 hours/week Language English How To Pass Pass all graded assignments to complete the course. User Ratings 4.5 stars Average User Rating 4.5See what learners said Coursework

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University of California, San Diego UC San Diego is an academic powerhouse and economic engine, recognized as one of the top 10 public universities by U.S. News and World Report. Innovation is central to who we are and what we do. Here, students learn that knowledge isn't just acquired in the classroom—life is their laboratory. Higher School of Economics National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communications, IT, mathematics, engineering, and more. Learn more on www.hse.ru

Syllabus


WEEK 1


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.


9 videos, 3 readings expand


  1. Reading: Slides and Resources on Flows in Networks
  2. Video: Introduction
  3. Video: Network Flows
  4. Video: Residual Networks
  5. Video: Maxflow-Mincut
  6. Video: The Ford–Fulkerson Algorithm
  7. Video: Slow Example
  8. Video: The Edmonds–Karp Algorithm
  9. Video: Bipartite Matching
  10. Video: Image Segmentation
  11. Reading: Available Programming Languages
  12. Reading: FAQ on Programming Assignments

Graded: Flow Algorithms
Graded: Programming Assignment 1

WEEK 2


Linear Programming



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.


10 videos, 1 reading expand


  1. Reading: Slides and Resources on Linear Programming
  2. Video: Introduction
  3. Video: Linear Programming
  4. Video: Linear Algebra: Method of Substitution
  5. Video: Linear Algebra: Gaussian Elimination
  6. Video: Convexity
  7. Video: Duality
  8. Video: (Optional) Duality Proofs
  9. Video: Linear Programming Formulations
  10. Video: The Simplex Algorithm
  11. Video: (Optional) The Ellipsoid Algorithm

Graded: Linear Programming Quiz
Graded: Programming Assignment 2

WEEK 3


NP-complete Problems



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.


16 videos, 1 reading expand


  1. Reading: Slides and Resources on NP-complete Problems
  2. Video: Brute Force Search
  3. Video: Search Problems
  4. Video: Traveling Salesman Problem
  5. Video: Hamiltonian Cycle Problem
  6. Video: Longest Path Problem
  7. Video: Integer Linear Programming Problem
  8. Video: Independent Set Problem
  9. Video: P and NP
  10. Video: Reductions
  11. Video: Showing NP-completeness
  12. Video: Independent Set to Vertex Cover
  13. Video: 3-SAT to Independent Set
  14. Video: SAT to 3-SAT
  15. Video: Circuit SAT to SAT
  16. Video: All of NP to Circuit SAT
  17. Video: Using SAT-solvers

Graded: NP-complete Problems
Graded: Programming Assignment 3

WEEK 4


Coping with NP-completeness



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.


11 videos, 1 reading expand


  1. Reading: Slides and Resources on Coping with NP-completeness
  2. Video: Introduction
  3. Video: 2-SAT
  4. Video: 2-SAT: Algorithm
  5. Video: Independent Sets in Trees
  6. Video: 3-SAT: Backtracking
  7. Video: 3-SAT: Local Search
  8. Video: TSP: Dynamic Programming
  9. Video: TSP: Branch and Bound
  10. Video: Vertex Cover
  11. Video: Metric TSP
  12. Video: TSP: Local Search

Graded: Coping with NP-completeness
Graded: Programming Assignment 4

WEEK 5


Streaming Algorithms (Optional)



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.


10 videos, 1 practice quiz expand


  1. Video: Introduction
  2. Video: Heavy Hitters Problem
  3. Video: Reduction 1
  4. Video: Reduction 2
  5. Video: Basic Estimate 1
  6. Video: Basic Estimate 2
  7. Video: Final Algorithm 1
  8. Video: Final Algorithm 2
  9. Video: Proofs 1
  10. Video: Proofs 2
  11. Practice Quiz: Quiz: Heavy Hitters
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