There were algorithms called A1 and A2. Later, it was proved that A2 was optimal and in fact also the best algorithm possible, so he gave it the name A* which symbolically includes all possible version numbers. Source: In 1964 Nils Nilsson invented a heuristic based approach to increase the speed of Dijkstra’s algorithm. This … Read more
As others have pointed out, Floyd-Warshall runs in time O(n3) and running a Dijkstra’s search from each node to each other node, assuming you’re using a Fibonacci heap to back your Dijkstra’s implementation, takes O(mn + n2 log n). However, you cannot always safely run Dijkstra’s on an arbitrary graph because Dijkstra’s algorithm does not … Read more
As a caveat, remember that there can be exponentially many shortest paths between two nodes in a graph. Any algorithm for this will potentially take exponential time. That said, there are a few relatively straightforward algorithms that can find all the paths. Here’s two. BFS + Reverse DFS When running a breadth-first search over a … Read more
Lee’s algorithm: http://en.wikipedia.org/wiki/Lee_algorithm It’s essentially a BF search, here’s an example: http://www.oop.rwth-aachen.de/documents/oop-2007/sss-oop-2007.pdf To implement it effectively, check my answer here: Change FloodFill-Algorithm to get Voronoi Territory for two data points? – when I say mark, you mark it with the number on the position you came from + 1. For example, if you have this … Read more
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