问题
Given a directed, connected graph with only positive edge weights, are there faster algorithms for finding the shortest path between two vertices, than Dijkstra using a fibonacci heap?
Wikipedia says, that Dijkstra is in O(|E| + |V| * log(|V|)) (using a fibonacci heap).
I'm not looking for optimizations that, for example, half the execution time, but rather algorithms that are in a different time complexity (like going from O(n * log n) to O(n)).
Further, I would like to know your opinion on the following approach:
- Determine the GCD of all edge weights.
- Transform the graph into a graph with uniform edge weights.
- Use BFS to find the shortest path between two given vertices.
Example for point 2:
Imagine the GCD to be 1. Then I would transform the edge
A--->B (edge weight 3)
into
A->A'->A''->B (3 times edge weight 1)
This transformation costs constant time and would have to be done once for every edge. So I expect this algorithm to be in O(|E|) (transformation) + O(|E| + |V|) (BFS) = O(2 * |E| + |V|) = O(|E| + |V|)
Thanks for taking the time to read my question and I hope not having waisted your time^^. Have a nice day.
回答1:
The big oh analysis you did for your algorithm is deeply flawed. Assume all edges are prime numbers. The number of edges in the new graph will be equal to sum of all weights. Thus O(|E| + |V|) of the new graph is actually O(W x |E| + |V|) in the original graph which can be much larger than O(|E| + |V| log |V|).
回答2:
Are there faster algorithms than Dijkstra?
Yes. The question isn't qualified so as to require better performance in all cases, or even in most cases. An algorithm with better performance in a single case is sufficient to establish an affirmative answer.
Despite the generally larger number of iterations required by the Bellman-Ford method over Dijkstra’s method, in practice the Bellman-Ford method can be superior because of the smaller overhead per iteration [Golden, B., 1976. “Shortest-Path Algorithms: A Comparison,” Operations Research, Vol. 44, pp. 1164-1168].
The quote above is from Dimitri P. Bertsekas (March 1992). "A Simple and Fast Label Correcting Algorithm for Shortest Paths" (PDF). Networks, Vol. 23, pp. 703-709, 1993. http://www.mit.edu/people/dimitrib/SLF.pdf. Retrieved 2008-10-01.
In short, my claim is based on Bertsekas' interpretation of Golden. Whether my conclusion stands up or not, you may find Bertsekas interesting for its classification of the Dijkstra algorithm as a label setting method, as contrasted with label correcting methods.
回答3:
There is an algorithm which has O(1): Turn the weights into chain lengths and use key rings for nodes (real key rings as those in your pocket). Connect the key rings with the right chains. Select the two nodes and pull them away from each other.
Follow the taut chains from one node to the other. This is the shortest path.
To implement this as a computer program, you'll need two industrial robots :)
For a more real world example, use the Ant colony optimization which gives very good results in a short time. Since you can specify the number of runs in this algorithm, you can decide how much time it spent (i.e. the runtime is only dependent on the number of nodes) which gives you O(n) but not a guaranteed perfect result.
回答4:
There's always A*, and it's derivates like Hierarchical A* and A* JPS.
来源：https://stackoverflow.com/questions/1701221/are-there-faster-algorithms-than-dijkstra