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

We proved that the proposers are matched to their top-ranked stable partners. We now want to establish that the algorithm is unfair from the perspective of the proposal recipients. We’d like to do this by showing that each woman is matched to her _lowest-ranked stable partner_. 

A **lowest-ranked stable partner** for a woman is a partner whom she ranks lowest among all the matches for that woman over all stable matchings. 


# The task at hand

Argue that each woman is matched to her lowest-ranked stable partner by the Gale-Shapley algorithm. To prove this by contradiction, begin by making the following assumption:


> **Starting assumption$^\dagger$:** Suppose the Gale-Shapely algorithm pairs a woman $w_0$ with a partner $m_0$, instead of her lowest-ranked stable partner $m_1$.
 

Explore how the Gale-Shapley algorithm is asymmetric from the perspective of proposal recipients.

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: Asymmetry in the Gale-Shapley Algorithm