Great job today. We've covered two of the most important algorithms in all of graph theory.
- Pathfinding is a critical, real-world problem with applications in routing and scheduling.
- Dijkstra's Algorithm finds the shortest path in any graph with non-negative weights using a greedy, priority-queue-based approach.
- Topological Sort (Kahn's) finds a valid linear ordering for a Directed Acyclic Graph (DAG) based on in-degrees.
- We can use a Topological Sort to find the shortest path in a DAG even with negative weights in highly efficient $O(V+E)$ time.
- Our work isn't done! We must address the problem of graphs containing negative cycles and the need for all-pairs shortest paths.
- In the next lecture, we will explore advanced algorithms designed specifically to handle these complex pathfinding challenges.
Algorithm Comparison
| Algorithm | Use Case |
|---|---|
| Dijkstra's | Fastest for SSSP, non-negative weights |
| Topo Sort (for paths) | Fastest for SSSP in DAGs (negatives OK) |