Fault diameter of k-ary n-cube networks

Abstract

We obtain the fault diameter of k-ary n-cube interconnection networks (also known as n-dimensional k-torus networks). We start by constructing a complete set of node-disjoint paths (i.e., as many paths as the degree) between any two nodes of a k-ary n-cube. Each of the obtained paths is of length zero, two, or four plus the minimum length except for one path in a special case (when the Hamming distance between the two nodes is one) where the increase over the minimum length may attain eight. These results improve those obtained in [8] where the length of some of the paths has a variable increase (which can be arbitrarily large) over the minimum length. These results are then used to derive the fault diameter of the k-ary n-cube, which is shown to be D + 1 where D is the fault free diameter.

GARP : A Highly Reliable Grid Based Adaptive Routing Protocol for Underwater Wireless Sensor Networks

“…It is shown in [24] how to construct a set of 2n node-disjoint paths of minimum or near minimum lengths between any two nodes in a k-ary n-cube network topology. We make use of the results of [24] to construct cell-disjoint paths of minimum or near minimum length in the 3D grid from any source cell SC of grid coordinates (x SC , y SC , z SC ) to a destination cell DC of grid coordinates (x DC , y DC , z DC ).…”

“…We make use of the results of [24] to construct cell-disjoint paths of minimum or near minimum length in the 3D grid from any source cell SC of grid coordinates (x SC , y SC , z SC ) to a destination cell DC of grid coordinates (x DC , y DC , z DC ). The length of a path is the number of moves between neighbouring grid cells along this path.…”

Construction of vertex-disjoint paths in alternating group networks

“…In particular, if a cycle contains the symbol 3, we always assume the cycle is C 1 , and normalize C 1 's representation via rotations, so that the symbol 3 is the tail (last) symbol r 1,k1 and p 3 is the head symbol r 1,1 . Figure 1 depicts the first three alternating group networks AN 3 , AN 4 , and AN 5 .…”

“…is the distance between u and v. Table 1 The structure of the proof of Theorem 1 in terms of parts (A/B), cases (1)(2)(3)(4)(5)(6), and subcases (a/b). The label given in the leftmost column corresponds to the (sub)section in Appendix A or B where the corresponding proof can be found Proof By the vertex symmetry of AN n , it suffices to show the result for one vertex labeled with an arbitrary even permutation p = C 1 C 2 .…”

The vulnerability of the diameter of the enhanced hypercubes

“…These parameters were studied by several authors for some Cartesian product graphs [7,25,26] and for the hypercube and its variants [3,5,8,10,18,19,20].…”