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

# The task at hand

Grab a paper and a pen, and figure out how to use Johnson’s algorithm to find the lengths of shortest paths between all pairs of vertices in the given digraph.

You may verify your solution by clicking the “Run Johnson’s Algorithm” button. 

> **Note:** 
> - You can also experiment on a different digraph using these #concept=instructions#instructions.#concept# 
> - If you choose to add a new edge, its weight will be generated randomly. 
> - If the automatically generated labels overlap with vertices or edges, you may view them more clearly by dragging the vertices of the given digraph (after selecting the “Drag vertices” checkbox).




Test your understanding of Johnson’s algorithm.

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

Exercise: Johnson’s Algorithm

The task at hand