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

Attempt the following questions to convince yourself that the Gale-Shapley algorithm produces a stable matching in which each proposer is paired with his top-ranked *stable* partner. 

> We define a **stable partner** for an individual to be any partner with whom that individual can be matched in _any_ stable matching.


# The proof 

We want to start by assuming the opposite of what we want to prove and see where that leads us. 

> **Starting assumption$^\dagger$:** Assume that there is at least one proposer who is not partnered with his top-ranked stable partner. 

During the execution of the Gale-Shapley algorithm, let $m_0$ be the _first_ man to be rejected by a stable partner $w_1$. This implies that at the time when $m_0$ was rejected by $w_1$, $w_1$ was matched to some other person, say $m_1$. 

Explore the exact sense in which the Gale-Shapley algorithm favors the proposers.

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

Proof that the Gale-Shapley Algorithm Favors the Proposers