Consider a graph like this: A—(3)—–B | | \-(1)-C–(1)/ The shortest path from A to B is via C (with a total weight of 2). A normal BFS will take the path directly from A to B, marking B as seen, and A to C, marking C as seen. At the next stage, propagating from … Read more
Breadth-first search is just Dijkstra’s algorithm with all edge weights equal to 1. Dijkstra’s algorithm is conceptually breadth-first search that respects edge costs. The process for exploring the graph is structurally the same in both cases.
Bellman-Ford algorithm is a single-source shortest path algorithm, so when you have negative edge weight then it can detect negative cycles in a graph. The only difference between the two is that Bellman-Ford is also capable of handling negative weights whereas Dijkstra Algorithm can only handle positives. From wiki However, Dijkstra’s algorithm greedily selects the … Read more
You should be able to make it much faster by trading off optimality. See Admissibility and optimality on wikipedia. The idea is to use an epsilon value which will lead to a solution no worse than 1 + epsilon times the optimal path, but which will cause fewer nodes to be considered by the algorithm. … Read more
EDIT: See simon’s answer, where he fixed the formula presented here. Actually there is an O(1) formula This is an image that I’ve made to visualize it ( Squares a knight can reach on Nth move are painted with same color ). Can you notice the pattern here? Although we can see the pattern, it … Read more
The algorithm you have suggested will indeed find the shortest path in this graph, but not all graphs in general. For example, consider this graph: Let’s trace through the execution of your algorithm. First, you set d(A) to 0 and the other distances to ∞. You then expand out node A, setting d(B) to 1, … Read more
Technically, Breadth-first search (BFS) by itself does not let you find the shortest path, simply because BFS is not looking for a shortest path: BFS describes a strategy for searching a graph, but it does not say that you must search for anything in particular. Dijkstra’s algorithm adapts BFS to let you find single-source shortest … Read more
Recall that in Dijkstra’s algorithm, once a vertex is marked as “closed” (and out of the open set) – the algorithm found the shortest path to it, and will never have to develop this node again – it assumes the path developed to this path is the shortest. But with negative weights – it might … Read more