Unpaired set-to-set disjoint path routings in recursive match networks

Abstract

The recursive match networks represent a family of networks, encompassing various types of network structures. Among these network structures, the bijective connection networks and BCube are all special cases of recursive match networks. On the other hand, the bijective connection networks also stand for a family of networks, encompassing well-known hypercubes, twisted cubes, Möbius cubes, and crossed cubes. The BCube, a promising candidate for the data center network model, contains as many as thousands (even millions) of servers. Recursive match networks integrate diverse known networks as well as potentially other future ones, underscoring the significance of exploring their study. One of the key topics is finding vertex-disjoint paths in recursive match networks. An unpaired set-to-set disjoint paths problem is as follows: given a set of source vertices $S=\{s_1,s_2,\dots ,s_p\}$ and a set of sink vertices $T=\{t_1,t_2,\dots ,t_q\}$ in an r-connected graph $G=\left(V\right(G),E(G\left)\right)$ with $m\le min\{p,q,r\}$, construct m vertex-disjoint paths $P_i$ from source $s_{a_i}$ to sink $t_{b_i}$ ($1\le i\le m$) such that $\{a_1,a_2,\dots ,a_m\}\subseteq \{1,2,\dots ,p\}$ and $\{b_1,b_2,\dots ,b_m\}\subseteq \{1,2,\dots ,q\}$. In this paper, we give a proof of existence of unpaired set-to-set disjoint paths in a k-order, n-dimensional recursive match network $X_{k,n}$, where the length of each path does not exceed $2n-1$. Then, we propose an $O\left(N{k}^4{\left({\text{log}}_{k}N\right)}^3\right)$ algorithm to construct nk vertex-disjoint paths between any pair of nk-vertex sets in $X_{k,n}$, where N is the vertex number of $X_{k,n}$. Furthermore, we randomly generate multiple $X_{k,n}$ with different parameters k and n, and apply the algorithm to simulate experiments on them. Finally, we evaluate the algorithm by comparing the maximum length of the obtained vertex-disjoint paths with the upper limit of diameter of $X_{k,n}$. The experimental results show that the maximum length is close to the upper limit, with a deviation not exceeding 2.