shortest path between 2 vertices in undirected weighted graph

Question

I am trying to find shortest path between 2 vertices in undirected weighted graph. It is also known that weights are integers less than log(log|V|), where |V| is amount of vertices. It is easy to solve using Bellman-Ford or Dijkstra algorithms, but is there any algorithm which can do it faster?

So far, I have been thinking of using BFS and dividing edges with weight greater than 1 into couple of them with weight 1, but it is not really good idea if |V| is large number. No, it is not my homework, I am just wondering.

Answer 1

One way to think of this question is to improve the running time of using Dijkstra's algorithm to find the shortest path between two vertices in the undirected weighted graph. So in this case, you can use a binary heap as the data structure. A heap is a complete binary tree with the heap property that every parent node is smaller (greater) than its children nodes in the tree in a min heap (a max heap). Here you can use the min heap to store the cost to each node from the starting node.

More information about heap can be found here: https://courses.csail.mit.edu/6.006/fall10/handouts/recitation10-8.pdf

With a heap, the running time of Dijkstra's algorithm can be reduced from O(V^2) to O(E log E), because selecting the minimum distance from the heap takes O(log V) (removing the minimum distance is O(1) and fixing the heap takes O(log V)) and updating distances to vertices takes O(E log V) in total (fixing heap takes O(log V) and it takes E times to examine neighbors and change costs).

Hope this help.