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.

This paper proposes optical wavelength division multiplexed (WDM) networks with limited wavelength conversion that can efficiently support lightpaths (connections) between nodes. Each lightpath follows a route in the network and must be assigned a channel along each link in its route. The load max of a set of lightpath requests is the maximum over all links of the number of lightpaths that use the link. At least max wavelengths will be needed to assign channels to the lightpaths. If the network has full wavelength conversion capabilities then max wavelengths are sufficient to perform the channel assignment. We propose ring networks with fixed wavelength conversion capability within the nodes that can support all lightpath request sets with load max at most W \Gamma 1, where W is the number of wavelengths in each link. We also propose ring networks with selective pairwise wavelength conversion capability within the nodes that can support all lightpath request sets with l...

This work deals with wavelength assignment for lightpaths. We restrict our attention to the near term, in which WDM networks will be in the form of rings and higher level networks will be SONET/SDH self-healing rings. This view changes the goal of wavelength assignment (WLA) vs. previous work on the subject in a number of aspects. First, a pair of SONET add/drop multiplexers (ADMs) terminates each lightpath. These ADMs also terminate adjacent lightpaths to form rings, implying that the WLA has to support this type of sharing. Second, following [GRS98], we argue that the first-order optimization goal should be to minimize the overall network cost which is dominated by the number of required ADMs and not the number of wavelengths. We show that these two minimization problems are intrinsically different, and there exist cases where the two minima cannot be simultaneously achieved. We derive a simple lower bound to the number of ADMs. Depending on the given lightpaths, we show that this lo...

this paper, we assume that all call requests are available at time 0 and do not have to be completed by a certain deadline. There is no precedence relation among the calls. We consider the off-line version of the problem, where call durations are known in advance. In section 2 and 3 we restrict our attention to the case that all edges have unit capacity, all calls require unit bandwidth, and all call durations are the same (unit duration). In section 4 we present some results for the more general setting of arbitrary call durations and arbitrary bandwidth requirements.

We present lower bounds on the competitive ratio of randomized algorithms for a wide class of on-line graph optimization problems and we apply such results to on-line virtual circuit and optical routing problems. Lund and Yannakakis [LY93a] give inapproximability results for the problem of finding the largest vertex induced subgraph satisfying any non-trivial, hereditary, property . E.g., independent set, planar, acyclic, bipartite, etc. We consider the on-line version of this family of problems, where some graph G is fixed and some subgraph H is presented on-line, vertex by vertex. The on-line algorithm must choose a subset of the vertices of H , choosing or rejecting a vertex when it is presented, whose vertex induced subgraph satisfies property . Furthermore, we study the on-line version of graph coloring whose off-line version has also been shown to be inapproximable [LY93b], on-line max edge-disjoint paths and on-line path coloring problems. Irrespective of the time complexity, w...

This paper studies the problems of broadcasting and gossiping in optical networks. In such 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. In this paper we consider both single--hop and multihop optical networks. In single--hop networks the information, once transmitted as light, reaches its destination without being converted to electronic form in between, thus reaching high speed communication. In multi hop networks a packet may have to be routed through a few intermediate nodes before reaching its final destination. In both models, we give efficient broadcasting and gossiping algorithms, in terms of time and number of wavelengths. We consider both networks with arbitrary topologies and particular networks of practical interest. Several of our algorithms exhibit optimal performances. 1 Introduction Motivations. Op...

We present improved bounds for efficient bandwidth allocation in a WDM optical network whose topology is that of a directed tree of fiber-optic links. The problem of bandwidth allocation is modeled as a coloring problem, where each path in a set of communication requests must be assigned a color (representing a wavelength) in such a way that no two paths using the same link in the same direction are assigned the same color. Letting L be the largest number of paths using any directed link, we show that for an arbitrary set of paths, 7L=4 colors are sufficient to route all paths. This improves an upper bound of 15L=8 due to Mihail, Kaklamanis and Rao [8]. In addition, we show that a family of problem instances given by Mihail, Kaklamanis and Rao [8] to establish a worst-case lower bound of 3L=2 for the problem can in most cases be colored with only 5L=4 colors (technically, d5L=4e). Finally, we show that in all cases 5L=4 colors are in fact necessary for this family of instances, yield...

Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specifics of the computational techniques employed. We consider eight variant vertexcoloring problems here. This article begins with a gentle introduction to the problem of computing a sparse Jacobian, followed by an overview of the historical development of the research area. Then we present a unifying framework for the graph models of the variant matrixestimation problems. The framework is based upon the viewpoint that a partition of a matrixinto structurally orthogonal groups of columns corresponds to distance-2 coloring an appropriate graph representation. The unified framework helps integrate earlier work and leads to fresh insights; enables the design of more efficient algorithms for many problems; leads to new algorithms for others; and eases the task of building graph models for new problems. We report computational results on two of the coloring problems to support our claims. Most of the methods for these problems treat a column or a row of a matrixas an atomic entity, and partition the columns or rows (or both). A brief review of methods that do not fit these criteria is provided. We also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here.

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The paper deals with on-line routing in WDM (wavelength division multiplexing) optical networks. A sequence of requests arrives over time, each is a pair of nodes to be connected by a path. The problem is to assign a wavelength and a path to each pair, so that no two paths sharing a link are assigned the same wavelength. The goal is to minimize the number of wavelengths used to establish all connections. Raghavan and Upfal [RU94] considered the off-line version of the problem, which was further studied in [AR95, KP96, MKR95, Ra96]. For a line topology, the problem is the well-studied interval graph coloring problem. On-line algorithms for this problem have been analyzed in [KT81, Ki88]. We consider trees, trees of rings, and meshes topologies, previously studied in the off-line case. We give on-line algorithms with competitive ratio O(log n) for all these topologies. We give a matching \Omega\Gammaing n) lower bound for meshes. We also prove that any algorithm for trees canno...