Explore the basics of graph theory, learn to represent graphs in C++, and master essential algorithms like DFS and Dijkstra to solve complex optimization problems, including matching and network flow.

Graph algorithms are the core of many real-world applications of computer science,  such as automotive navigation or routing in computer networks. They’re also a common subject in coding interviews at top-tier tech companies.
 
In this course, we’ll learn about the basic concepts of graph theory and how to represent graphs as data structures in code. We’ll study essential graph algorithms such as depth-first search or Dijkstra's algorithm to traverse graphs and find shortest paths. Finally, we’ll learn to solve more complex optimization problems on graphs, such as matching and network flow problems.

Learn Graph Algorithms in C++

In this Challenge Lesson, we’ll implement the **Bellmann-Ford** algorithm, an alternative algorithm for the SSSP. While it runs slower than Dijkstra, it is able to correctly handle graphs with negative edge weights as long as there are no negative cycles.

# The Bellmann-Ford algorithm

The Bellmann-Ford algorithm computes distances by repeatedly relaxing all edges of the graphs. Recall that relaxing an edge $(u,v)$ means checking whether 
$$d(u) + w(u, v) < d(v),$$

in which case we can update the tentative distance $d(v)$.

To run Bellmann-Ford from a starting node $u$, we 
1. Initialize $d(u) = 0$ and $d(v) = \infty$ for $v \neq u$.
2. Try to relax each edge of the graph.
3. Repeat step (2) until the distance estimates do not change anymore.

Finally, the value of $d(v)$ corresponds to the distance of node $v$ from node $u$. As usual, we can also keep track of predecessor nodes when relaxing edges to construct the shortest paths themselves.

## Example execution

Let's run Bellmann-Ford on this small example graph, starting from node $3$:


Implement the Bellmann-Ford algorithm to solve SSSP problems in graphs with negative edge weights.

Introduction

Graph Representations

Graph Traversal

Shortest Paths

Spanning Trees

Flow Problems

Conclusion

Appendix

Challenge: Bellmann-Ford

The Bellmann-Ford algorithm