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

Abstract

The bijective connection graph encompasses a family of cube-based topologies, and <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-dimensional bijective connection graphs include the hypercube and almost all of its variants with the order <inline-formula> <tex-math notation="LaTeX">$2^{n}$ </tex-math></inline-formula> and the degree <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">$S=\{ \boldsymbol s _{1}, \boldsymbol s _{2},\ldots, \boldsymbol s _{p}\}$ </tex-math></inline-formula> and a set of destination nodes <inline-formula> <tex-math notation="LaTeX">$D=\{ \boldsymbol d _{1}, \boldsymbol d _{2},\ldots, \boldsymbol d _{p}\}$ </tex-math></inline-formula> in a <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-connected graph <inline-formula> <tex-math notation="LaTeX">$G=(V,E)$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$p\le k$ </tex-math></inline-formula>, construct <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> paths <inline-formula> <tex-math notation="LaTeX">$P_{i}$ </tex-math></inline-formula>: <inline-formula> <tex-math notation="LaTeX">$\boldsymbol s _{i}\leadsto \boldsymbol d _{j_{i}}$ </tex-math></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$1\le i\le p$ </tex-math></inline-formula>) such that <inline-formula> <tex-math notation="LaTeX">$\{j_{1},j_{2},\ldots,j_{p}\}=\{1,2,\ldots,p\}$ </tex-math></inline-formula> and the paths <inline-formula> <tex-math notation="LaTeX">$P_{i}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$p~(\le n)$ </tex-math></inline-formula> disjoint paths between any pair of node sets in <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-dimensional bijective connection graphs in polynomial-order time of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. We give a proof of correctness of the algorithm as well as the estimates of the time complexity <inline-formula> <tex-math notation="LaTeX">$O(n^{3}p^{4})$ </tex-math></inline-formula> and the maximum path length <inline-formula> <tex-math notation="LaTeX">$n+p-1$ </tex-math></inline-formula>. According to a computer experiment in a locally twisted cube as an example of a bijective connection graph to construct <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> disjoint paths, the average time complexity of the algorithm is <inline-formula> <tex-math notation="LaTeX">$O(n^{2})$ </tex-math></inline-formula>, and the average maximum path is <inline-formula> <tex-math notation="LaTeX">$0.6333n-0.266$ </tex-math></inline-formula>.