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Mastering Graph Algorithms

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.

Beginner

91 Lessons

15h

Certificate of Completion

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.

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This course includes

32 Playgrounds

50 Quizzes

Course Overview

What You'll Learn

Course Content

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Course Overview

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...Show More

What You'll Learn

An understanding of classical graph algorithms

The ability to think and visualize in terms of graph structures

An inside-out understanding of what makes each algorithm work the way it does

Cultivation of a broader perspective that informs other computing disciplines

An understanding of classical graph algorithms

Course Content

Introduction

1 Lessons

Get familiar with the fundamentals and applications of graph algorithms across disciplines.

What Is This Course About?

Review: Asymptotic Notation and Math Prerequisites

11 Lessons

Grasp the fundamentals of logarithms, running time, and big-O notation for algorithm analysis.

Getting Started with Logarithms

Food for Thought: Running Time

Running Time as a Function of Input Size

Comparing Running times

Big-O Notation

Ruminating on Constants in Big-O Notation

Some Useful Facts

A Big-O Drill

Leveraging the Proof by Contradiction Technique

Exercise: Order of Growth

Quiz: Big-O Notation

Working with Graphs

9 Lessons

Work your way through understanding and representing graphs, including adjacency matrices, lists, and special graph types.

Subgraphs, Paths, Cycles and Components

Trees

Bipartite Graphs

Quiz: Working with Graphs

Depth-First Search and Applications

16 Lessons

Grasp the fundamentals of Depth-First Search and its applications in graph traversal and analysis.

Depth-First Search

Edge Classification in Directed Graphs

Exercise: Classify Edges of a Directed Graph

Edge Classification in Undirected Graphs

Exercise: Classify Edges of an Undirected Graph

Identifying Cut-Vertices

Exercise: Identify Cut-Vertices

Topological Sorting

Exercise: Sort the Vertices in Topological Order

Thought Exercise: Detecting Cycles in Directed Graphs

Characterizing Strongly Connected Components

Tarjan’s Algorithm

Exercise: Practice Tarjan’s Algorithm

The Kosaraju-Sharir Algorithm

Exercise: Practice the Kosaraju-Sharir Algorithm

Quiz: DFS and Applications

Prelude: Shortest Paths

7 Lessons

Explore shortest-path strategies, BFS for unweighted digraphs, and apply graph traversal concepts.

The Shortest-Paths Problem

Reflections on Shortest Paths

Quiz: Shortest Paths Terminology

Breadth-First Search

Shortest Paths in Unweighted Digraphs

Exercise: Print Shortest Path

Quiz: BFS and Shortest Paths

Single-source Shortest Paths in Weighted Digraphs

12 Lessons

Focus on solving the single-source shortest paths problem using various algorithms in weighted digraphs.

Single-Source Shortest-Paths Problem

Exercise: Path-Relaxation

An Immediate Algorithm for DAGs

Thought Exercise: Longest Paths in DAG

Exercise: Shortest Paths in a DAG

Dijkstra’s Algorithm

Exercise: Dijkstra’s Algorithm

Thought Exercise: Negative Weights

The Bellman-Ford Algorithm

Exercise: The Bellman-Ford Algorithm

Thought Exercise: The Bellman-Ford Algorithm

Quiz: Single-Source Shortest-Paths

All Pairs Shortest Paths

5 Lessons

Practice using Johnson's and Floyd-Warshall algorithms for efficient all-pairs shortest paths.

Johnson’s Algorithm

Exercise: Johnson’s Algorithm

The Floyd-Warshall Algorithm

Exercise: Floyd-Warshall Algorithm

Quiz: All-Pairs Shortest Paths

Minimum Spanning Tree

12 Lessons

Learn how to use Minimum Spanning Trees via Prim's and Kruskal's algorithms effectively.

What’s an MST?

Quiz: Minimum Spanning Trees

Thought Exercise: Uniqueness

Prim’s Algorithm

Exercise: Select Edges as Specified by Prim’s Algorithm

Exercise: Prim’s Algorithm

Kruskal’s Algorithm

Implementation of Disjoint-Set Data Structure

Exercise: Select Edges as Specified by Kruskal’s Algorithm

Exercise: Kruskal’s Algorithm

Thought Exercise: MSTs and Shortest Paths

Quiz: Minimum Spanning Trees

Flows

6 Lessons

Walk through network flows, the Ford-Fulkerson algorithm, and practical exercises.

Network Flows

The Essence of the Ford-Fulkerson Algorithm

Ford-Fulkerson Algorithm: The Fine Print

Thought Exercise: Rational Capacities

Exercise: The Ford-Fulkerson Algorithm

Quiz: Network Flows

10.

Matchings and Vertex Covers

11 Lessons

Examine matchings, vertex covers, the Ford-Fulkerson algorithm, and the Gale-Shapley algorithm.

Matchings

Maximum Matchings in Bipartite Graphs

Exercise: Maximum Matchings in Bipartite Graphs

The Dual Problem of Minimum Vertex Cover

Minimum Vertex Cover in Bipartite Graphs

Exercise: Minimum Vertex Cover in Bipartite Graphs

The Gale-Shapley Algorithm

Proof that the Gale-Shapley Algorithm Favors the Proposers

Thought Exercise: Asymmetry in the Gale-Shapley Algorithm

Exercise: Find Stable Matching

Quiz: Matchings and Vertex Covers

11.

Conclusion

1 Lessons

Grasp the fundamentals of graph algorithms' applications, utility, and practical significance in various fields.

Final Remarks

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