14,053

1

## Introduction - Basic Objects in Discrete Mathematics

This module gives the learner a first impression of what discrete mathematics is about, and in which ways its "flavor" differs from other fields of mathematics. It introduces basic objects like sets, relations, functions, which form the foundation of discrete mathematics....
2 個視頻 （總計 27 分鐘）, 3 個測驗
2 個視頻
Sets, Relations, Functions10分鐘
1 個練習
Sets, relations, and functions30分鐘
2

## Partial Orders

Even without knowing, the learner has seen some orderings in the past. Numbers are ordered by <=. Integers can be partially ordered by the "divisible by" relation. In genealogy, people are ordered by the "A is an ancestor of B" relation. This module formally introduces partial orders and proves some fundamental and non-trivial facts about them....
2 個視頻 （總計 28 分鐘）, 2 個測驗
2 個視頻
Mirsky's and Dilworth's Theorem14分鐘
1 個練習
Partial orders, maximal and minimal elements, chains, antichains
3

## Enumerative Combinatorics

A big part of discrete mathematics is about counting things. A classic example asks how many different words can be obtained by re-ordering the letters in the word Mississippi. Counting problems of this flavor abound in discrete mathematics discrete probability and also in the analysis of algorithms....
3 個視頻 （總計 35 分鐘）, 2 個測驗
3 個視頻
Evaluating Simple Sums8分鐘
Pascal's Triangle14分鐘
1 個練習
Counting Basic Objects
4

## The Binomial Coefficient

The binomial coefficient (n choose k) counts the number of ways to select k elements from a set of size n. It appears all the time in enumerative combinatorics. A good understanding of (n choose k) is also extremely helpful for analysis of algorithms....
3 個視頻 （總計 55 分鐘）, 3 個測驗
3 個視頻
Estimating the Binomial Coefficient22分鐘
Excursion to Discrete Probability: Computing the Expected Minimum of k Random Elements from {1,...,n}18分鐘
1 個練習
An Eagle's View of Pascal's Triangle8分鐘
5

## Asymptotics and the O-Notation

...
1 個視頻 （總計 14 分鐘）, 3 個測驗
1 個視頻
1 個練習
The Big-O-Notation18分鐘
6

## Introduction to Graph Theory

Graphs are arguably the most important object in discrete mathematics. A huge number of problems from computer science and combinatorics can be modelled in the language of graphs. This module introduces the basic notions of graph theory - graphs, cycles, paths, degree, isomorphism....
3 個視頻 （總計 41 分鐘）, 3 個測驗
3 個視頻
Graph Isomorphism, Degree, Graph Score13分鐘
Graph Score Theorem16分鐘
1 個練習
Graphs, isomorphisms, and the sliding tile puzzle30分鐘
7

## Connectivity, Trees, Cycles

We continue with graph theory basics. In this module, we introduce trees, an important class of graphs, and several equivalent characterizations of trees. Finally, we present an efficient algorithm for detecting whether two trees are isomorphic....
3 個視頻 （總計 36 分鐘）, 3 個測驗
3 個視頻
Cycles and Trees15分鐘
An Efficient Algorithm for Isomorphism of Trees12分鐘
1 個練習
Cycles and Trees30分鐘
8

## Eulerian and Hamiltonian Cycles

Starting with the well-known "Bridges of Königsberg" riddle, we prove the well-known characterization of Eulerian graphs. We discuss Hamiltonian paths and give sufficient criteria for their existence with Dirac's and Ore's theorem....
2 個視頻 （總計 27 分鐘）, 2 個測驗
2 個視頻
Hamilton Cycles - Ore's and Dirac's Theorem16分鐘
1 個練習
Hamiltonian Cycles and Paths
9

## Spanning Trees

We discuss spanning trees of graphs. In particular we present Kruskal's algorithm for finding the minimum spanning tree of a graph with edge costs. We prove Cayley's formula, stating that the complete graph on n vertices has n^(n-2) spanning trees....
2 個視頻 （總計 29 分鐘）, 3 個測驗
2 個視頻
The Number of Trees on n Vertices15分鐘
1 個練習
Spanning Trees40分鐘
10

## Maximum flow and minimum cut

This module is about flow networks and has a distinctively algorithmic flavor. We prove the maximum flow minimum cut duality theorem....
2 個視頻 （總計 29 分鐘）, 2 個測驗
2 個視頻
Flow Networks: The Maxflow - Mincut Theorem15分鐘
1 個練習
Network flow8分鐘
11

## Matchings in Bipartite Graphs

We prove Hall's Theorem and Kőnig's Theorem, two important results on matchings in bipartite graphs. With the machinery from flow networks, both have quite direct proofs. Finally, partial orderings have their comeback with Dilworth's Theorem, which has a surprising proof using Kőnig's Theorem....
3 個視頻 （總計 46 分鐘）, 1 個測驗
3 個視頻
Matchings in Bipartite Graphs: Hall's and König's Theorem16分鐘
Partial Orders: Dilworth's Theorem on Chains and Antichains15分鐘
3.5
32 個審閱

### 熱門審閱

Fantastic course. Fascinating material, presented at a reasonably fast pace, and some really challenging assignments.

This course is good to comprehend relation, function and combinations.

## 講師

### Dominik Scheder

Assistant Professor
The Department of Computer Science and Engineering

## 關於 上海交通大学

Shanghai Jiao Tong University, a leading research university located in Shanghai, China, has been regarded as the fastest developing university in the country for the last decade. With special strengths in engineering, science, medicine and business, it now offers a comprehensive range of disciplines in 27 schools with more than 41,000 enrolled students from more than one hundred countries....

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