Summary of 12. Bellman-Ford

00:00:00 - 00:55:00

The Bellman-Ford algorithm is a pathfinding algorithm that is used to find the shortest path between two points in a graph. This algorithm is faster than the other two algorithms, which use more memory.

• 00:00:00 In this video, the lecturer discusses the Beltman Ford algorithm, which is a general algorithm for solving shortest paths in graphs that could contain cycles and/or negative weight. First, they warm up with two exercises to get used to the theory. Then, they discuss Beltman Ford, which is a algorithm for computing shortest path distances and shortest path weights for every vertex in a graph. If there is a negative weight cycle in the graph, then the weights for the vertices in that cycle are set to minus infinity.
• 00:05:00 The video discusses the Bellman-Ford algorithm, which is used to find single source shortest paths in directed graphs. The video shows an algorithm that solves single source shortest paths in v times e time, asymptotically no bigger than e.
• 00:10:00 In this video, we discuss the shortest path problem, and how negative weights can affect the solution. We show that, even with negative weights, a shortest path exists that is simple and does not repeat vertices.
• 00:15:00 The video goes over a problem where the shortest path between two points may not be a simple path. The video discusses a method to find a shortest path that uses at most v minus one edges, or a witness, and if this is the case, the shortest path between the two points must be minus infinity.
• 00:20:00 The video discusses the concept of a witness, which is a vertex in a graph that has a property that guarantees that there is a negative weight cycle in the graph. The video then claims that every vertex with this property is reachable from a negative weight cycle by definition. If this is proven, it would prove the claim that a vertex with minus infinite shortest path weight is also reachable from a witness.
• 00:25:00 The Bellman-Ford algorithm is a shortest pathfinding algorithm that uses a weight function to find the shortest path between two vertices. The original Bellman-Ford algorithm does something different, and a modification is shown that is easier to analyze. If a path is not a witness, then the path is bigger than the original path.
• 00:30:00 The video discusses the concept of graph duplication, which is a common technique for solving graph-related problems. The video shows how to find an algorithm that runs in linear time on a graph that has been duplicated at least once.
• 00:35:00 The video explains a directed graph that contains a negative weight cycle. The author creates a grid of vertices and connects every edge with a negative weight edge, simulating the cycle at most k edges. He then proves that any path in the graph using at most k edges corresponds to a path from 0 to a vertex in that level.
• 00:40:00 The Bellman-Ford algorithm constructs the graph G prime, which has v vertices and v edges. The algorithm is done in polynomial time, and it achieves a optimal solution for each vertex.
• 00:45:00 The video discusses a proof that the shortest path distance from a given vertex to all other vertices is the same as the k-edge shortest path distance between that vertex and the given vertex's k nearest neighbors. The video then inducts on k and shows that the base case is true, leading to the conclusion that the proof is valid for all k.
• 00:50:00 The video explains Bowman Ford, a pathfinding algorithm that runs in polynomial time. The algorithm uses a minimum weight edge in the adjacencies of a vertex to find the shortest path between the vertex and any other vertex in the graph.
• 00:55:00 The Bellman-Ford algorithm finds the shortest path between two points in a graph. This algorithm is faster than the other two algorithms, which use more memory.