Pairwise Disjoint Paths Routing in Tori

Abstract

The pairwise disjoint paths problem is to construct c disjoint paths s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> → d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> (1 ≤ i ≤ c) between given pairs of nodes (s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ) (1 ≤ i ≤ c) in a graph whose connectivity is no less than 2c. It is a major problem for interconnection networks, together with the node-to-node disjoint paths problem, the node-to-set disjoint paths problem, and the set-to-set disjoint paths problem. In this paper, we propose an algorithm that solves this problem for a torus. In a previous work, Bossard and Kaneko have developed an algorithm that constructs disjoint paths between c pairs of nodes in a k-ary n-dimensional torus, where c ≤ n. The time complexity of their algorithm is O(c <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> n+kcn), and the maximum path length is [k/2jn+2k(c-1). However, the algorithm proposed in this paper achieves a time complexity of O(c <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> n+kcn), and its maximum path length is [k/2jn + ([3k/21 - 2)(c - 1), which are both improvements over the previous algorithm. We also conducted an evaluation experiment to show that the average path lengths are proportional to k if n is fixed, and ton if k is fixed. The theoretical maximum path lengths were not attained by the paths constructed by our algorithm, and the average execution time was proportional to k if n is fixed, and to n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> if k is fixed. Additional experimental results show that, compared to the previous algorithm by Bossard and Kaneko, our algorithm achieves a better performance with respect to the maximum path lengths, but both algorithms achieve a similar level of performance with respect to the average maximum path lengths. Also, it is shown that the average execution time of our algorithm is about O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ), which is better than the average execution time of the previous algorithm O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) in the experimental framework.