Methods and problems of wavelength-routing in all-optical networks

We give a survey of recent theoretical results obtained for wavelength-routing in all-optical networks. The survey is based on the previous survey in [Beauquier, B., Bermond, J-C., Gargano, L., Hell, P., Perennes, S., Vaccaro, U.: Graph problems arising from wavelength-routing in all-optical networks. In: Proc. of the 2nd Workshop on Optics and Computer Science, part of IPPS'97, 1997]. We focus our survey on the current research directions and on the used methods. We also state several open problems connected with this line of research, and give an overview of several related research directions

Forwarding and optical indices of 4-regular circulant networks

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

Virtual network embedding in the cycle

AbstractWe consider a problem motivated by the design of Asynchronous transfer mode (ATM) networks. Given a physical network and an all-to-all traffic, the problem consists in designing a virtual network with a given diameter, which can be embedded in the physical one with a minimum congestion (the congestion is the maximum load of a physical link). Here we solve the problem when the physical network is a ring. We give an almost optimal solution for diameter 2 and bounds for large diameters

Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes

Interconnection networks provide an effective mechanism for exchanging data between processors in a parallel computing system. One of the most efficient interconnection networks is the hypercube due to its structural regularity, potential for parallel computation of various algorithms, and the high degree of fault tolerance. Thus it becomes the first choice of topological structure of parallel processing and computing systems. In this paper, lower bounds for the dilation, wirelength, and edge congestion of an embedding of a graph into a hypercube are proved. Two of these bounds are expressed in terms of the bisection width. Applying these results, the dilation and wirelength of embedding of certain complete multipartite graphs, folded hypercubes, wheels, and specific Cartesian products are computed