课程信息
4.6
225 个评分
49 个审阅

100% 在线

立即开始,按照自己的计划学习。

可灵活调整截止日期

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初级

完成时间大约为15 小时

建议:5 weeks, 3-5 hours/week ...

英语(English)

字幕:英语(English)

100% 在线

立即开始,按照自己的计划学习。

可灵活调整截止日期

根据您的日程表重置截止日期。

初级

完成时间大约为15 小时

建议:5 weeks, 3-5 hours/week ...

英语(English)

字幕:英语(English)

教学大纲 - 您将从这门课程中学到什么

1
完成时间为 3 小时

What is a Graph?

What 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 个视频 (总计 52 分钟), 5 个阅读材料, 5 个测验
14 个视频
Knight Transposition2分钟
Seven Bridges of Königsberg4分钟
What is a Graph?7分钟
Graph Examples2分钟
Graph Applications3分钟
Vertex Degree3分钟
Paths5分钟
Connectivity2分钟
Directed Graphs3分钟
Weighted Graphs2分钟
Paths, Cycles and Complete Graphs2分钟
Trees6分钟
Bipartite Graphs4分钟
5 个阅读材料
Slides1分钟
Slides1分钟
Slides1分钟
Slides1分钟
Glossary10分钟
2 个练习
Definitions10分钟
Graph Types10分钟
2
完成时间为 5 小时

CYCLES

We’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 个视频 (总计 89 分钟), 4 个阅读材料, 6 个测验
12 个视频
Total Degree5分钟
Connected Components7分钟
Guarini Puzzle: Code6分钟
Lower Bound5分钟
The Heaviest Stone6分钟
Directed Acyclic Graphs10分钟
Strongly Connected Components7分钟
Eulerian Cycles4分钟
Eulerian Cycles: Criteria11分钟
Hamiltonian Cycles4分钟
Genome Assembly12分钟
4 个阅读材料
Slides1分钟
Slides1分钟
Slides1分钟
Glossary10分钟
4 个练习
Computing the Number of Edges10分钟
Number of Connected Components10分钟
Number of Strongly Connected Components10分钟
Eulerian Cycles2分钟
3
完成时间为 3 小时

Graph Classes

This 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 个视频 (总计 55 分钟), 4 个阅读材料, 6 个测验
11 个视频
Trees8分钟
Minimum Spanning Tree6分钟
Job Assignment3分钟
Bipartite Graphs5分钟
Matchings3分钟
Hall's Theorem7分钟
Subway Lines1分钟
Planar Graphs3分钟
Euler's Formula4分钟
Applications of Euler's Formula7分钟
4 个阅读材料
Slides1分钟
Slides1分钟
Slides1分钟
Glossary10分钟
3 个练习
Trees10分钟
Bipartite Graphs10分钟
Planar Graphs10分钟
4
完成时间为 4 小时

Graph Parameters

We'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 个视频 (总计 52 分钟), 5 个阅读材料, 8 个测验
14 个视频
Graph Coloring3分钟
Bounds on the Chromatic Number3分钟
Applications3分钟
Graph Cliques3分钟
Cliques and Independent Sets3分钟
Connections to Coloring1分钟
Mantel's Theorem5分钟
Balanced Graphs2分钟
Ramsey Numbers2分钟
Existence of Ramsey Numbers5分钟
Antivirus System2分钟
Vertex Covers3分钟
König's Theorem8分钟
5 个阅读材料
Slides1分钟
Slides1分钟
Slides1分钟
Slides1分钟
Glossary10分钟
4 个练习
Graph Coloring10分钟
Cliques and Independent Sets10分钟
Ramsey Numbers10分钟
Vertex Covers10分钟
5
完成时间为 4 小时

Flows and Matchings

This week we'll develop an algorithm that finds the maximum amount of water which can be routed in a given water supply network. This algorithm is also used in practice for optimization of road traffic and airline scheduling. We'll see how flows in networks are related to matchings in bipartite graphs. We'll then develop an algorithm which finds stable matchings in bipartite graphs. This algorithm solves the problem of matching students with schools, doctors with hospitals, and organ donors with patients. By the end of this week, we'll implement an algorithm which won the Nobel Prize in Economics!...
13 个视频 (总计 99 分钟), 6 个阅读材料, 4 个测验
13 个视频
An Example6分钟
The Framework8分钟
Ford and Fulkerson: Proof11分钟
Hall's theorem10分钟
What Else?8分钟
Why Stable Matchings?6分钟
Mathematics and Real Life4分钟
Basic Examples6分钟
Looking For a Stable Matching6分钟
Gale-Shapley Algorithm6分钟
Correctness Proof6分钟
Why The Algorithm Is Unfair7分钟
Why the Algorithm is Very Unfair9分钟
6 个阅读材料
Slides1分钟
Slides1分钟
The algorithm and its properties (alternative exposition)10分钟
Gale-Shapley Algorithm2分钟
Project Description10分钟
Glossary10分钟
4 个练习
Choose an Augmenting Path Carefully20分钟
Constant Degree Bipartite Graphs30分钟
Base Cases10分钟
Algorithm
4.6
49 个审阅Chevron Right

33%

完成这些课程后已开始新的职业生涯

50%

通过此课程获得实实在在的工作福利

33%

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热门审阅

创建者 SUFeb 28th 2019

Appreciate the structure and the explanations with examples. The practice tool before every lesson not makes it fun to learn but also sets the student in the context and can anticipate the concept.

创建者 RHNov 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!

讲师

Avatar

Alexander S. Kulikov

Visiting Professor
Department of Computer Science and Engineering

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关于 Introduction to Discrete Mathematics for Computer Science 专项课程

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....
Introduction to Discrete Mathematics for Computer Science

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