This course is part of the Introduction to Discrete Mathematics for Computer Science Specialization

Offered By

University of California San Diego

National Research University Higher School of Economics

Introduction to Discrete Mathematics for Computer Science Specialization

University of California San Diego

About this Course

4.6

133 ratings

•

29 reviews

We invite you to a fascinating journey into Graph Theory — an area which connects the elegance of painting and the rigor of mathematics; is simple, but not unsophisticated. Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them.
In this course, among other intriguing applications, we will see how GPS systems find shortest routes, how engineers design integrated circuits, how biologists assemble genomes, why a political map can always be colored using a few colors. We will study Ramsey Theory which proves that in a large system, complete disorder is impossible!
By the end of the course, we will implement an algorithm which finds an optimal assignment of students to schools. This algorithm, developed by David Gale and Lloyd S. Shapley, was later recognized by the conferral of Nobel Prize in Economics.
As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students.

Start instantly and learn at your own schedule.

Reset deadlines in accordance to your schedule.

Suggested: 5 weeks, 3-5 hours/week ...

Subtitles: English...

Start instantly and learn at your own schedule.

Reset deadlines in accordance to your schedule.

Suggested: 5 weeks, 3-5 hours/week ...

Subtitles: English...

Week

1What are graphs? What do we need them for? This week we'll see that a graph is a simple pictorial way to represent almost any relations between objects. We'll see that we use graph applications daily! We'll learn what graphs are, when and how to use them, how to draw graphs, and we'll also see the most important graph classes. We start off with two interactive puzzles. While they may be hard, they demonstrate the power of graph theory very well! If you don't find these puzzles easy, please see the videos and reading materials after them....

14 videos (Total 52 min), 5 readings, 5 quizzes

Knight Transposition2m

Seven Bridges of Königsberg4m

What is a Graph?7m

Graph Examples2m

Graph Applications3m

Vertex Degree3m

Paths5m

Connectivity2m

Directed Graphs3m

Weighted Graphs2m

Paths, Cycles and Complete Graphs2m

Trees6m

Bipartite Graphs4m

Slides1m

Slides1m

Slides1m

Slides1m

Glossary10m

Definitions10m

Graph Types10m

Week

2We’ll consider connected components of a graph and how they can be used to implement a simple program for solving the Guarini puzzle and for proving optimality of a certain protocol. We’ll see how to find a valid ordering of a to-do list or project dependency graph. Finally, we’ll figure out the dramatic difference between seemingly similar Eulerian cycles and Hamiltonian cycles, and we’ll see how they are used in genome assembly! ...

12 videos (Total 89 min), 4 readings, 6 quizzes

Total Degree5m

Connected Components7m

Guarini Puzzle: Code6m

Lower Bound5m

The Heaviest Stone6m

Directed Acyclic Graphs10m

Strongly Connected Components7m

Eulerian Cycles4m

Eulerian Cycles: Criteria11m

Hamiltonian Cycles4m

Genome Assembly12m

Slides1m

Slides1m

Slides1m

Glossary10m

Computing the Number of Edges10m

Number of Connected Components10m

Number of Strongly Connected Components10m

Eulerian Cycles2m

Week

3This week we will study three main graph classes: trees, bipartite graphs, and planar graphs. We'll define minimum spanning trees, and then develop an algorithm which finds the cheapest way to connect arbitrary cities. We'll study matchings in bipartite graphs, and see when a set of jobs can be filled by applicants. We'll also learn what planar graphs are, and see when subway stations can be connected without intersections. Stay tuned for more interactive puzzles!...

11 videos (Total 55 min), 4 readings, 6 quizzes

Trees8m

Minimum Spanning Tree6m

Job Assignment3m

Bipartite Graphs5m

Matchings3m

Hall's Theorem7m

Subway Lines1m

Planar Graphs3m

Euler's Formula4m

Applications of Euler's Formula7m

Slides1m

Slides1m

Slides1m

Glossary10m

Trees10m

Bipartite Graphs10m

Planar Graphs10m

Week

4We'll focus on the graph parameters and related problems. First, we'll define graph colorings, and see why political maps can be colored in just four colors. Then we will see how cliques and independent sets are related in graphs. Using these notions, we'll prove Ramsey Theorem which states that in a large system, complete disorder is impossible! Finally, we'll study vertex covers, and learn how to find the minimum number of computers which control all network connections....

14 videos (Total 52 min), 5 readings, 8 quizzes

Graph Coloring3m

Bounds on the Chromatic Number3m

Applications3m

Graph Cliques3m

Cliques and Independent Sets3m

Connections to Coloring1m

Mantel's Theorem5m

Balanced Graphs2m

Ramsey Numbers2m

Existence of Ramsey Numbers5m

Antivirus System2m

Vertex Covers3m

König's Theorem8m

Slides1m

Slides1m

Slides1m

Slides1m

Glossary10m

Graph Coloring10m

Cliques and Independent Sets10m

Ramsey Numbers10m

Vertex Covers10m

4.6

29 ReviewsBy RH•Nov 17th 2017

Was pretty fun and gave a good intro to graph theory. Definitely felt inspired to go deeper and understood the most basic proof ideas. The later lectures can spike in difficulty though. Very nice!

By DN•Nov 12th 2017

I like this course. Very basic, but teachers are really great and explanations are perfect! Highly recommended for all who wants to begin with Graph Theory.

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....

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...

Discrete Math is needed to see mathematical structures in the object you work with, and understand their properties. This ability is important for software engineers, data scientists, security and financial analysts (it is not a coincidence that math puzzles are often used for interviews). We cover the basic notions and results (combinatorics, graphs, probability, number theory) that are universally needed. To deliver techniques and ideas in discrete mathematics to the learner we extensively use interactive puzzles specially created for this specialization. To bring the learners experience closer to IT-applications we incorporate programming examples, problems and projects in our courses....

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