How to make a faster algorithm

Question

Let = (, ) be a directed graph with edge weights and let be a vertex of . All of the edge weights are integers between 1 and 20. Design an algorithm for finding the shortest paths from . The running time of your algorithm should be asymptotically faster than Dijkstra’s running time.

I know that Dijkstra’s running time is O( e + v log v), and try to find a faster algorithm.

If all weights are 1 or only include 0 and 1, I can use BFS O(e+v) in a directed graph, but how to make a faster algorithm for edge weights are integers between 1 and 20.

sorry, running time just need faster than O( |E| + |V| log V) — Talor, Mar 18 '19 at 06:37
Hmm, the edited question now looks quite blur, any reason why you stripped out the details? Otherwise, I will need to revert it. — Pham Trung, Mar 18 '19 at 07:43
I put back the original description, as the edited one is ambiguous for others, which is not aligned with this community's benefits. — Pham Trung, Mar 18 '19 at 09:45

Photon · Accepted Answer · 2019-03-18T06:45:34.330

5

Well let`s say you have an edge that goes from u to v with weight w
We can insert w-1 extra nodes between nodes u and v
So after modifying each edge (which takes O(20*E)) we have a graph where every edge is 1
And on such graph we can use regular BFS, we will have atmost 20*N new nodes, and 20*E new edges so complexity is still O(V+E)

i.e.

This gets transformed to this:

edited Mar 18 '19 at 06:45

answered Mar 18 '19 at 06:29

Photon

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Thank you for answering my question. I understand this step, and I'm sure it works if this is an undirected graph, but is this works in a directed graph? because All of the edge weights are integers between 1 and 20. – Talor Mar 18 '19 at 06:38
@Talor Yes it works for both directed & unidrected graphs, updated answer to include a visual example. – Photon Mar 18 '19 at 06:46

How to make a faster algorithm

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