Note, that here you do not find detailed explanation of all algorithms. For that purposes, see the original article.
Steiner tree problem in graphs
The Steiner tree problem on unweighted graphs seeks a minimum subtree (i.e. subtree with minimal number of edges), containing a given subset of the vertices (terminals). This problem is NP-complete. This package provides several heuristic and one exact approach for finding Steiner trees, as well as tools for analyzing resultant trees and comparing different algorithms. This R package was originally applied to analyzing biological networks.
Heuristic approaches:
- shortest paths based approximation (SP)
- minimum spanning tree based approximation (KB)
- randomized all shortest paths approximation (RSP)
- all shortest paths between terminals (ASP)
- (SPM)
Before we start, let’s attach several packages.
library(igraph)
library(SteinerNet)
As an example of graph, we are going to take well-known “Cubical” graph. Also let’s randomly pick 4 terminals using generate_st_samples. We also specify prob variable, so at each step of random walk procedure node is accepted with probability prob = 0.1.
g <- graph("Cubical")
set.seed(4)
terminal_nodes <- generate_st_samples(graph = g, ter_number = 2, prob = 0.1)
terminal_nodes
#> [[1]]
#> [1] 5 7
As we can see, there is only one element in a list and it contains ids of selected vertices. Of course, we can generate more sets of terminals (e.g. pass a vector to ter_number variable), but we do not need it now.
V(g)$color <- "yellow"
V(g)[terminal_nodes[[1]]]$color <- "red"
plot(g)
Shortest paths based approximation (SP)
repeattimes, and merge variables are ignored for “SP”.
steinertree(type = "SP", terminals = terminal_nodes[[1]],
graph = g, color = FALSE, merge = FALSE)
#> [[1]]
#> IGRAPH 638effc UN-- 3 2 -- Cubical
#> + attr: name (g/c), color (v/c), name (v/c)
#> + edges from 638effc (vertex names):
#> [1] 5--6 6--7
Note, that in 3.0 version (as well as in 2.0 version) Steiner trees with different randomly chosen start nodes are not repetitively constructed, as it is written in article. This can be done by hand as follows:
tree_list <- c()
for (i in 1:20)
tree_list[[i]] <- steinertree(type = "SP", terminals = terminal_nodes[[1]],
graph = g, color = FALSE, merge = FALSE)
# calculate sizes of trees
tree_list_len <- unlist(lapply(tree_list, function (x) length(E(x[[1]]))))
# select trees with minimal size
index <- which(tree_list_len == min(tree_list_len))
Minimum spanning tree based approximation (KB)
repeattimes, and merge variables are ignored for “KB”.
steinertree(type = "KB", terminals = terminal_nodes[[1]],
graph = g, color = FALSE, merge = FALSE)
#> [[1]]
#> IGRAPH 45fc990 UN-- 3 2 -- Cubical
#> + attr: name (g/c), color (v/c), name (v/c)
#> + edges from 45fc990 (vertex names):
#> [1] 5--8 7--8
All shortest paths between terminals (ASP)
This method is provided only for comparison reasons. In version 2.0 it was hidden, however it is now available for the sake of convenience.
repeattimes, and merge variables are ignored for “ASP”.
steinertree(type = "RSP", terminals = terminal_nodes[[1]],
graph = g, color = FALSE, merge = FALSE)
#> [[1]]
#> IGRAPH af15697 UN-- 3 2 -- Cubical
#> + attr: name (g/c), color (v/c), name (v/c)
#> + edges from af15697 (vertex names):
#> [1] 5--6 6--7
Exact algorithm (EXA)
Note, that this method can find multiple exact solutions, if they exist. In the following example we specify merge variable equals to FALSE, because we want to get a list of steiner trees.
repeattimes and optimize variables are ignored for “EXA”.
steinertree(type = "EXA", terminals = terminal_nodes[[1]],
graph = g, color = FALSE, merge = FALSE)
#> [[1]]
#> IGRAPH ac57654 UN-- 3 2 -- Cubical
#> + attr: name (g/c), color (v/c), name (v/c)
#> + edges from ac57654 (vertex names):
#> [1] 5--6 6--7
#>
#> [[2]]
#> IGRAPH 090209c UN-- 3 2 -- Cubical
#> + attr: name (g/c), color (v/c), name (v/c)
#> + edges from 090209c (vertex names):
#> [1] 5--8 7--8
As we can see, two steiner trees are found.
Sub-graph of merged steiner trees (SPM)
“SPM” algorithm can return multiple graphs. Note, that in article this method is called “STM”.
steinertree(type = "SPM", terminals = terminal_nodes[[1]],
graph = g, color = FALSE, merge = FALSE)
#> [[1]]
#> IGRAPH 506c2c7 UN-- 3 2 -- Cubical
#> + attr: name (g/c), color (v/c), name (v/c)
#> + edges from 506c2c7 (vertex names):
#> [1] 5--8 7--8
#>
#> [[2]]
#> IGRAPH aa108b3 UN-- 3 2 -- Cubical
#> + attr: name (g/c), color (v/c), name (v/c)
#> + edges from aa108b3 (vertex names):
#> [1] 5--6 6--7
Optimization of resultant tree
Optimization means returning the minimum spanning tree on resultant graph and removing all non-terminal nodes of degree one. Note, that in version 2.0 this function was runned by default for several algorithms. Again it is explicitly available in version 3.0 for the sake of convenience.
If you want to reproduce results of the article, you need to specify optimize = TRUE for the following algorithms:
For “EXA” algorithm this option is ignored. Note, that in version 2.0 for “ASP” algorithm optimization was not avaliable at all. It means, that experiments was condacted without further optimization.
Article results reproducibility
For experiments terminal sets of the following size are used:
- 50 sets with 5 randomly selected terminals
- 50 sets with 8 randomly selected terminals
- etc.
All vertices was selected with 0.5 probability. Therefore in generate_st_samples you probably passed the following:
# in version 2.0
# eval = FALSE
listofterminaltest <- c(5, 8, 15, 50, 70)
repetition <- rep(x = 0.5, 50)
However, simultaneous work with the number of terminals selected and size of sets makes an output of some function too wired. So now you need to specify:
ter_number. Each element indicates the number of terminals to be selected and length of vector indecates the number of terminal sets to be picked.prob. prob[i] defines a propability with which each next node accepted or rejected while selecting ter_number[i] terminals. Usually this probability is the same for all terminals, so you may write something like this prob = rep(0.5, #len of ter_number#)
The main difference is that you can now operate with only one value of number of terminal sets. So to reproduce results of the article, firstly, you need, for example, generate 50 terminal sets with 5 terminals in each.
# in version 3.0
# eval = FALSE
generate_st_samples(graph = #your graph#,
ter_number = rep(x = 5, 50),
prob = rep(x = 5, 50))
After running some simulations, secondly, you may want to generate 50 terminal sets with 8 terminals in each.
# in version 3.0
# eval = FALSE
generate_st_samples(graph = #your graph#,
ter_number = rep(x = 8, 50),
prob = rep(x = 8, 50))