Gain insights into key graph algorithms, including depth-first search, shortest paths, and flow networks. Explore their applications and foundational role in advanced computing disciplines.

This technology-agnostic course will teach you about many classical graph algorithms essential for a well-rounded software engineer. 

You will begin with an introduction to the running-time analysis of algorithms and the representation and structure of graphs. You’ll cover the depth-first search algorithm and its many applications, like detecting cycles, topological sorting, and identifying cut-vertices and strongly connected components. You will then cover algorithms for finding shortest paths and minimum spanning trees. You’ll also learn about the Ford-Fulkerson algorithm and its use in finding a maximum flow and minimum cut in networks. Moreover, you will adapt it to find maximum matchings and minimum vertex covers in bipartite graphs. Finally, you will learn about the Gale-Shapley algorithm for finding stable matchings.

The material in this course serves as a basis for other advanced algorithms, like parallel and distributed algorithms, and has many diverse applications in other computing disciplines.

Mastering Graph Algorithms

# Matchings and related terminology

A **matching** $M$ of an undirected graph $G$ is any collection of its edges that have no end-vertex in common. In other words, each edge in $M$ uniquely pairs or matches two vertices. We refer to the number of edges in a matching $M$ as the **size of the matching** $M$.
 
If a vertex appears as an end-vertex of an edge in $M$, we say it is **matched** by $M$. Otherwise, we say it is **unmatched** by $M$. Some examples of matchings in graphs are shown in the set of slides below.

 


Introduction

Review: Asymptotic Notation and Math Prerequisites

Working with Graphs

Depth-First Search and Applications

Prelude: Shortest Paths

Single-source Shortest Paths in Weighted Digraphs

All Pairs Shortest Paths

Minimum Spanning Tree

Flows

Matchings and Vertex Covers

Conclusion

Use Simulated Annealing for the Traveling Salesman Problem

Matchings

Matchings and related terminology