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.

Modern high-performance communication networks pose a number of challenging problems concerning the efficient allocation of resources to connection requests. In all-optical networks with wavelength-division multiplexing, connection requests must be assigned paths and colors (wavelengths) such that intersecting paths receive different colors, and the goal is to minimize the number of colors used. This path coloring problem is proved NP-hard for undirected and bidirected ring networks. Path coloring in undirected tree networks is shown to be equivalent to edge coloring of multigraphs, which implies a polynomial-time optimal algorithm for trees of constant degree as well as NP-hardness and an approximation algorithm with absolute approximation ratio 4:3 and asymptotic approximation ratio 1:1 for trees of arbitrary degree. For bidirected trees, path coloring is shown to be NP-hard even in the binary case. A polynomial-time optimal algorithm is given for path coloring in undirected or bidir...

. In optical networks with wavelength division multiplexing (WDM), multiple connections can share a link if they are transmitted on different wavelengths. We study the problem of satisfying a maximum number of connection requests in a directed tree network if only a limited number W of wavelengths are available. In optical networks without wavelength converters this is the maximum path coloring (MaxPC) problem, in networks with full wavelength conversion this is the maximum path packing (MaxPP) problem. MaxPC and MaxPP are shown to be polynomial-time solvable to optimality if the tree has height one or if both W and the degree of the tree are bounded by a constant. If either W or the degree of the tree is not bounded by a constant, MaxPC and MaxPP are proved NP-hard. Polynomial-time approximation algorithms with performance ratio 5=3 + " for arbitrarily small " are presented for the case W = 1, in which MaxPC and MaxPP are equivalent. For arbitrary W , a 2-approximation for MaxPP in arbitrary trees, a 1:58-approximation for MaxPC in trees of bounded degree, and a 2:22-approximation for MaxPC in arbitrary trees are obtained. 1

. Let T be a symmetric directed tree, i.e., an undirected tree with each edge viewed as two opposite arcs. We prove that the minimum number of colours needed to colour the set of all directed paths in T , so that no two paths of the same colour use the same arc of T , is equal to the maximum number of paths passing through an arc of T . This result is applied to solve the all-to-all communication problem in wavelength-- division--multiplexing (WDM) routing in all--optical networks, that is, we give an efficient algorithm to optimally assign wavelengths to the all the paths of a tree network. It is known that the problem of colouring a general subset of all possible paths in a symmetric directed tree is an NPhard problem. We study conditions for a given set S of paths be coloured efficiently with the minimum possible number of colours/wavelengths. 1 Introduction Let T be a tree and x; y two vertices of T . The dipath P (x; y) in T is the undirected path joining x to y, in which each ed...

. Motivated by the problem of WDM routing in all--optical networks, we study the following NP--hard problem. We are given a directed binary tree T and a set R of directed paths on T . We wish to assign colors to paths in R, in such a way that no two paths that share a directed arc of T are assigned the same color and that the total number of colors used is minimized. Our results are expressed in terms of the depth of the tree and the maximum load l of R, i.e., the maximum number of paths that go through a directed arc of T . So far, only deterministic greedy algorithms have been presented for the problem. The best known algorithm colors any set R of maximum load l using at most 5l=3 colors. Alternatively, we say that this algorithm has performance ratio 5=3. It is also known that no deterministic greedy algorithm can achieve a performance ratio better than 5=3. In this paper we define the class of greedy algorithms that use randomization. We study their limitations and pr...

. Given a communication network and a set of call requests, the goal is to find a minimum makespan schedule for the calls such that the sum of the bandwidth requirements of simultaneously active calls using the same link does not exceed the capacity of that link. In this paper the call-scheduling problem is studied for star and tree networks. Lower and upper bounds on the worst-case performance of List-Scheduling (LS) and variants of it are obtained for call-scheduling with arbitrary bandwidth requirements and either unit call durations or arbitrary call durations. LS does not require advance knowledge of call durations and, hence, is an on-line algorithm. It has performance ratio (competitive ratio) at most 5 in star networks. A variant of LS for calls with unit durations is shown to have performance ratio at most 2 2 3 . In tree networks with n nodes, a variant of LS for calls with unit durations has performance ratio at most 6, and a variant for calls with arbitrary d...

) 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...

. 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...

In this paper, we address the issue of efficiently allocating wavelengths to communication requests in wavelength division multiplexing (WDM) optical networks. We present a simple greedy algorithm and study its performance on directed binary fiber trees. We prove that our algorithm guarrantees a 5=3-approximation for wavelengths assignment in binary tree shaped WDM optical networks. We also prove that no greedy algorithm can go below the ratio of 5=3, even for the case of leaf--to--leaf communication. 1 Introduction Optics is emerging as a key technology in state-of-the-art communication networks. A single optical wavelength supports rates of gigabits-per-second (which in turn support multiple channels of voice, data, and video [5,10]. Multiple laser beams that are propagated over the same fiber on distinct optical wavelengths can increase this capacity much further; this is achieved through WDM (Wavelength Division Multiplexing). The model we use for the underlying fiber network is a...