Forwarding and optical indices of 4-regular circulant networks

Abstract

An all-to-all routing in a graph $G$ is a set of oriented paths of $G$, with exactly one path for each ordered pair of vertices. The load of an edge under an all-to-all routing $R$ is the number of times it is used (in either direction) by paths of $R$, and the maximum load of an edge is denoted by $\pi(G,R)$. The edge-forwarding index $\pi(G)$ is the minimum of $\pi(G,R)$ over all possible all-to-all routings $R$, and the arc-forwarding index $\overrightarrow{\pi}(G)$ is defined similarly by taking direction into consideration, where an arc is an ordered pair of adjacent vertices. Denote by $w(G,R)$ the minimum number of colours required to colour the paths of $R$ such that any two paths having an edge in common receive distinct colours. The optical index $w(G)$ is defined to be the minimum of $w(G,R)$ over all possible $R$, and the directed optical index $\overrightarrow{w}(G)$ is defined similarly by requiring that any two paths having an arc in common receive distinct colours. In this paper we obtain lower and upper bounds on these four invariants for $4$-regular circulant graphs with connection set $\{\pm 1,\pm s\}$, $1<s<n/2$. We give approximation algorithms with performance ratio a small constant for the corresponding forwarding index and routing and wavelength assignment problems for some families of $4$-regular circulant graphs.Comment: 19 pages, no figure in Journal of Discrete Algorithms 201

Forwarding and Optical Indices in an All-Optical BCube Networks

“…In the part of path allocation, we aim to find a set of dipaths that minimizes the maximum link load; in the part of wavelength assignment, we aim to minimize the usage of wavelengths. Specifically, the minimum of the maximum link load over all possible routings is referred to as the forwarding index when link load is measured by the number of paths passing through it [11,12]. We refer to the minimum number of wavelengths, used to support simultaneous host-to-host communication, as the optical index.…”

“…We refer to the minimum number of wavelengths, used to support simultaneous host-to-host communication, as the optical index. It has been shown that the optical index is naturally lower bounded by the forwarding index [11,12].…”

Recursive cubes of rings as models for interconnection networks

“…Various factors need to be considered in order to achieve high performance and low construction costs of an interconnection network. Among them, vertex-transitivity, small and fixed node degree, small diameter, recursive construction, existence of efficient routing algorithms are some of the desirable properties [10,13,18,21]. For example, networks with smaller diameters will lead to shorter data transmission delay.…”

A Few Families of Cayley Graphs and Their Efficiency as Communication Networks

“…We give approximation algorithms for the corresponding problems of the forwarding indices and optical indices with a small constant performance ratio. Our results on the family of circulant graphs of degree 4 are published in [3].The family of recursive cubes of rings has received a lot of attention for communication networks [10], but many aspects of them have remained unknown. We study this family of graphs by redefining each of them as a Cayley graph on the semidirect product of an elementary abelian group by a cyclic group in order to facilitate the study of them by using algebraic tools.…”

“…We give approximation algorithms for the corresponding problems of the forwarding indices and optical indices with a small constant performance ratio. Our results on the family of circulant graphs of degree 4 are published in [3].…”