Kou's algorithm to find Steiner Tree using igraph - r

Question

I am trying to implement the Kou's algorithm to identify Steiner Tree(s) in R using igraph.
The Kou's algorithm can be described like this:
Find the complete distance graph G' (G' has V' = S (steiner nodes) , and for each pair of nodes (u,v) in VxV there is an edge with weight equal to the weight of the min-cost path between these nodes p_(u,v) in G)
Find a minimum spanning tree T' in G'
Construct the subgraph Gs, of G by substituting every edge of T', which is an edge of G' with the corresponding shortest path of G (it there are several shortest paths, pick an arbitrary one).
Find the minimal spanning tree, Ts, of Gs (If there are several minimal spanning trees, pick an arbitrary one)
Construct a Steiner tree, Th, from Ts by deleting edges in Ts, if necessary, to that all the leaves in Th are Steiner nodes.
The first 2 steps are easy:
g <- erdos.renyi.game(100, 1/10) # graph
V(g)$name <- 1:100
# Some steiner nodes
steiner.points <- sample(1:100, 5)
# Complete distance graph G'
Gi <- graph.full(5)
V(Gi)$name <- steiner.points
# Find a minimum spanning tree T' in G'
mst <- minimum.spanning.tree(Gi)
However, I don't know how to replace the edges in T' for the shortest path in G. I know that with get.shortest.paths I can get the vpath from a pair of nodes, but how I replace and edge in T' with the shortest.path in G?
Many thanks in advance

Answer 1

If I'm understanding the algorithm as you've written it, I think this gets you through step 3, but please clarify if that's not the case:
library(igraph)
set.seed(2002)
g <- erdos.renyi.game(100, 1/10) # graph
V(g)$name <- as.character(1:100)
## Some steiner nodes:
steiner.points <- sample(1:100, 5)
## Complete distance graph G'
Gi <- graph.full(5)
V(Gi)$name <- steiner.points
## Find a minimum spanning tree T' in G'
mst <- minimum.spanning.tree(Gi)
## For each edge in mst, replace with shortest path:
edge_list <- get.edgelist(mst)
Gs <- mst
for (n in 1:nrow(edge_list)) {
i <- edge_list[n,2]
j <- edge_list[n,1]
## If the edge of T' mst is shared by Gi, then remove the edge from T'
## and replace with the shortest path between the nodes of g:
if (length(E(Gi)[which(V(mst)$name==i) %--% which(V(mst)$name==j)]) == 1) {
## If edge is present then remove existing edge from the
## minimum spanning tree:
Gs <- Gs - E(Gs)[which(V(mst)$name==i) %--% which(V(mst)$name==j)]
## Next extract the sub-graph from g corresponding to the
## shortest path and union it with the mst graph:
g_sub <- induced.subgraph(g, (get.shortest.paths(g, from=V(g)[i], to=V(g)[j])$vpath[[1]]))
Gs <- graph.union(Gs, g_sub, byname=T)
}
}
par(mfrow=c(1,2))
plot(mst)
plot(Gs)
Plot of minimum spanning tree on the left, replaced with shortest paths on right: