Set-to-Set Disjoint Path Routing in Bijective Connection Graphs

The bijective connection graph encompasses a family of cube-based topologies, and
$n$
-dimensional bijective connection graphs include the hypercube and almost all of its variants with the order
$2^{n}$
and the degree
$n$
. Hence, it is important to design and implement algorithms that work in bijective connection graphs. The set-to-set disjoint paths problem is as follows: given a set of source nodes
$S=\{ \boldsymbol s _{1}, \boldsymbol s _{2},\ldots, \boldsymbol s _{p}\}$
and a set of destination nodes
$D=\{ \boldsymbol d _{1}, \boldsymbol d _{2},\ldots, \boldsymbol d _{p}\}$
in a
$k$
-connected graph
$G=(V,E)$
with
$p\le k$
, construct
$p$
paths
$P_{i}$
:
$\boldsymbol s _{i}\leadsto \boldsymbol d _{j_{i}}$
(
$1\le i\le p$
) such that
$\{j_{1},j_{2},\ldots,j_{p}\}=\{1,2,\ldots,p\}$
and the paths
$P_{i}$
are node-disjoint. Finding a solution to this problem is an important issue in parallel and distributed computation as well as the node-to-node disjoint paths problem and the node-to-set disjoint paths problem. In this paper we propose an algorithm that constructs
$p~(\le n)$
disjoint paths between any pair of node sets in
$n$
-dimensional bijective connection graphs in polynomial-order time of
$n$
. We give a proof of correctness of the algorithm as well as the estimates of the time complexity
$O(n^{3}p^{4})$
and the maximum path length
$n+p-1$
. According to a computer experiment in a locally twisted cube as an example of a bijective connection graph to construct
$n$
disjoint paths, the average time complexity of the algorithm is
$O(n^{2})$
, and the average maximum path is
$0.6333n-0.266$
.