Question

I have a undirected, weighted graph with objects of an arbitrary type as nodes. The weight of an edge between two nodes A and B is the similarity of these two nodes in the interval (0, 1]. A similarity of 0 leads to no connection between to nodes, so the graph may be partitioned.

Given a target weight w and a start-node S, which is an algorithm to find all nodes that have a weight > w. Subnodes (seen from S) should have the product of all weights on the path. I.e:

S --(0.9)-- N1 --(0.9)-- N2 --(0.6) -- N3

Starting with S the nodes will have the following similarity values:

N1: 0.9 
N2: 0.9 * 0.9 = 0.81
N3: 0.9 * 0.9 * 0.6 = 0.486

So given S and the target weight 0.5 the search should return N1 and N3. Wheres a search starting from N2 would return S, N1 and N3.

Are their any algorithms that fit my needs?

Answer 1

note the following:

1. log(p1 * p2) = log(p1) + log(p2)
2. if p1 < p2 then log(p1) < log(p2) and thus: -log(p1) > -log(p2)

Claim [based on the 1,2 mentioned above]: finding the most similar route from s to t, is actually finding the minimum path from s to t, where weights are -log of original.

I suggest the following algorithm, based on Dijkstra’s algorithm and the above claim.

1. define for each edge e: w'(e) = -log(w(e)) //well defined because w(e) > 0
2. run Dijkstra's algorithm on the graph with the modified weights.
3. return all vertices v that their weight is dijkstra(v) < -log(needed)