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

Since Dijkstra’s algorithm fails for negative edge weights, we’re tempted to devise a way to add a constant to all edge weights of a digraph such that the following objectives are met:

1. The new edge weights are nonnegative.

2. The shortest paths with respect to the new weights are also the shortest paths with respect to the old weights.

> If this idea works, we can use Dijkstra’s algorithm for  graphs with negative edge weights by simply tweaking the weights (assuming  the digraph is known to have no negative weight cycles.)





Reflect on whether it’s possible to adapt Dijkstra’s algorithm for negative edge weights.

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

Thought Exercise: Negative Weights