We survey the theoretical results obtained for wavelength routing in all–optical networks, present some new results and propose several open problems. In all–optical networks the vast bandwidth available is utilized through wavelength division multiplexing: a single physical optical link can carry several logical signals, provided that they are transmitted on different wavelengths. The information, once transmitted as light, reaches its destination without being converted to electronic form in between, thus reaching high data transmission rates. We consider both networks with arbitrary topologies and particular networks of practical interest.

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. We address the issue of efficiently assigning wavelengths to communication requests in wavelength division multiplexing (WDM) optical networks. We consider directed tree and tree of rings topologies. These are topologies of concrete practical relevance for which undirected underlying graph models have been studied by Raghavan and Upfal [6]. Directed models were first studied by Mihail et al [4]. For trees, we give a polynomial time routing algorithm that, for requests of maximum load L per directed fiber link, uses at most 7=4L wavelengths. This improves the bound of Mihail et al.. As a corollary we also give an algorithm for trees of rings that uses 7=2L wavelengths. 1 Introduction Optics is a major technology that drives very high speed networking to the future. A single optical wavelength supports rates of gigabits-per-second (which in turn support multiple channels of voice, data, and video [3] [5]). Multiple laser beams that are propagated over the same fiber on distinct optica...

This paper studies the problem of All-to-All Communication for optical networks. In such networks the vast bandwidth available is utilized through wavelength division multiplexing (WDM): a single physical optical link can carry several logical signals, provided that they are transmitted on different wavelengths. In this paper we consider all-optical (or singlehop) networks, where the information, once transmitted as light, reaches its destination without being converted to electronic form in between, thus reaching high data transmission rates. In this model, we give optimal all-to-all protocols, using minimum numbers of wavelengths, for particular networks of practical interest, namely the d-dimensional square tori with even side, the corresponding meshes and the Cartesian sums of complete graphs.

Bandwidth is a very valuable resource in wavelength division multiplexed optical networks. The problem of finding an optimal assignment of wavelengths to requests is of fundamental importance in bandwidth utilization. We present a polynomialtime algorithm for this problem on fixed constant-size topologies. We combine this algorithm with ideas from Raghavan and Upfal [15] to obtain an optimal assignment of wavelengths on constant degree undirected trees. Mihail, Kaklamanis, and Rao [14] posed the following open question: what is the complexity of this problem on directed trees? We show that it is NP-complete both on binary and constant depth directed trees. Keywords: Algorithms, Combinatorial Problems, Computational Complexity, Interconnection Networks. 1 Introduction Motivation. Developments in fiber-optic networking technology using Wavelength Division Multiplexing (WDM) have finally reached the point where it 1 Supported by ONR Young Investigator Award N00014-93-1-0590. This work ...

Abstract. This paper addresses multicast routing in multi-hop optical networks employing wavelength-division multiplexing (WDM). We consider a model in which multicast communication requests are made and released dynamically over time. A multicast connection is realized by constructing a multicast tree which distributes the message from the source node to all destination nodes such that the wavelengths used on each link and the receivers and transmitters used at each node are not used by existing circuits. We show that although the routing and wavelength assignment in this model is NP-complete, the wavelength assignment problem can be solved in linear time. 1

) Luisa Gargano Dipartimento di Informatica ed Applicazioni Universit`a di Salerno 84081 Baronissi (SA), Italy. Abstract Let T be a symmetric directed tree, i.e., a tree with each edge viewed as two opposite directed links. We consider the problem of routing arbitrary sets of connection requests in T . In all-optical communication tree networks with WDM (wavelength-division multiplexing) this is equivalent to color (assign wavelengths to) a given set of directed paths so that no two directed paths of the same color use the same link of T . Let W be the number of available wavelengths. The load, that is, the maximum number of directed paths passing through a link of T cannot exceed W . If there is no wavelength conversion available then each request (directed path) is restricted to a single wavelength and it is known that the minimum number of colors needed to color any set of directed paths in a tree is lower bounded away from the load L of the paths on the tree; moreover, no algori...

We study the problem of assigning a minimum number of colors to directed paths (dipaths) of a tree, so that any two dipaths that share a directed edge of the tree are not assigned the same color. The problem has applications to wavelength routing in WDM all-optical tree networks, an important engineering problem. Dipaths represent communication requests, while colors correspond to wavelengths that must be assigned to requests so that multiple users can communicate simultaneously through the same optical fiber. Recent work on wavelength routing in trees has studied a special class of algorithms which are called greedy. Although these algorithms are simple and implementable in a distributed setting, it has been proved that there are cases where a bandwidth utilization of 100% is not possible. Thus, in this work, we relax the constraints of the original engineering problem and use devices called wavelength converters that are able to convert the wavelength a...

. In this paper we consider the problem of allocating optical bandwidth on a tree-shaped network when wavelength conversion is allowed. We consider both converters with full conversion capabilities and converters with limited capabilities. We give upper and lower bounds on the number of full wavelength converters necessary to fully utilize the optical bandwidth available. For converters of limited capabilities we show that, for general trees, converters of degree dL=3e + 1 are sufficient to route a set of requests of load L using at most 3=2L wavelengths. For the special case of binary trees, we show that converters of degree O( p L) are sufficient to route a set of requests of load L using exactly L wavelengths. The study of limited capabilities converters is based on graph construction problems that might be of their own interest. 1 Introduction 1.1 Background Optical fiber is rapidly becoming the standard transmission medium for networks, and can provide the required data rate,...

: This paper surveys theoretical results for wavelength--routing in all-- optical networks and presents several open problems. We focus our attention on graph-theoretical problems and proof techniques. 1 INTRODUCTION Optical networks are emerging as key technology in communication networks and are expected to dominate many applications, such as video conferencing, scientific visualisation, real-time medical imaging, high--speed super-computing and distributed computing [18, 19, 33, 36]. The books of Green [18] and McAulay [30] offer a comprehensive overview of the physical theory and applications of this emerging technology. In WDM (Wavelength Division Multiplexing) optical networks, the bandwidth available in optical fiber is utilised by partitioning it into several channels, each at a different wavelength. Each wavelength can carry a separate stream of data. In general, a WDM network consists of routing nodes interconnected by point--to--point unidirectional optic fiber links. Each...