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.

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...

This paper addresses the natural relaxation of the path coloring problem, in which one needs to color directed paths on a symmetric directed graph with a minimum number of colors, in such a way that paths using the same arc of the graph have different colors. This classic combinatorial problem finds applications in the minimization of the number of wavelengths in wavelength division multiplexing (WDM) all-optical networks.

. A bidirected tree is the directed graph obtained from an undirected tree by replacing each undirected edge by two directed edges with opposite directions. Given a set of directed paths in a bidirected tree, the goal of the maximum edge-disjoint paths problem is to select a maximumcardinality subset of the paths such that the selected paths are edge-disjoint. This problem can be solved optimally in polynomial time for bidirected trees of constant degree, but is MAXSNP-hard for bidirected trees of arbitrary degree. For every fixed " ? 0, a polynomial-time (5=3+ ")-approximation algorithm is presented. Key words. approximation algorithms, edge-disjoint paths, bidirected trees AMS subject classifications. 68Q25, 68R10 1. Introduction. Research on disjoint paths problems in graphs has a long history [12]. In recent years, edge-disjoint paths problems have been brought into the focus of attention by advances in the field of communication networks. Many modern network architectures estab...

A tree of rings is an undirected graph obtained from the union of rings, which intersect two by two in at most one node, such that any two nodes are connected by exactly two edge-disjoint paths. In this paper, we consider symmetric directed trees of rings with weighted nodes. A routing for a weighted digraph is a collection of directed paths (dipaths), such that for each ordered pair of nodes (x 1 ; x 2 ) with respective weights w 1 and w 2 , there are w 1 w 2 dipaths (possibly not distinct) from x 1 to x 2 . Motivated by the Wavelength Division Multiplexing (WDM) technology in all-optical networks, we study the problem of nding a routing which can be colored by the fewest number of colors so that dipaths of the same color are arc-disjoint. We prove that this minimum number of colors (wavelengths) is equal to the maximum number of dipaths that share one arc (load), minimized over all routings. The problem can be efficiently solved (dipaths found and colored) using cut properties.

We study the wavelength problem and arc (edge) congestion problem for communicating permutation instances on a ring. We prove the best possible upper bounds on the number of wavelengths and arc (edge) congestion in both directed and undirected cases. Keywords: all-optical network, congestion, permutation, wavelength 1

We consider the problem of routing in networks employing all-optical routing technology. In such networks, information between nodes of the network is transmitted as light on fiber-optic lines without being converted to electronic form in between. We consider switched optical networks that use the wavelength-division multiplexing (or WDM) approach. A WDM network consists of nodes connected by point-to-point fiber-optic links, each of which can support a fixed number of wavelengths. The switches are capable of redirecting incoming streams based on wavelengths, without changing the wavelengths. Different messages may use the same link concurrently if they are assigned distinct wavelengths. However, messages assigned the same wavelength must be assigned edge-disjoint paths. Given a communication instance in a network, the optical routing problem is the assignment of routes to communication requests of the instance, as well as wavelengths to routes so that the number of wavelen...

We consider all-to-all routing problem in an optical ring network that uses the wavelength-division multiplexing (WDM). Since one-hop, all-to-all optical routing in a WDM optical ring of n nodes needs d 1 2 b n 2 4 ce wavelengths (see [3]), which can be too large even for moderate values of n, we consider in this paper j-hop implementations of all-to-all routing in a WDM optical ring, j 2. From among the possible routings we focus our attention on uniform routings, in which each node of the ring uses the same communication pattern and the communication load is distributed evenly among the nodes. We show that there exists a uniform 2-hop implementation of all-to-all routing that needs at most n 4 ( 3 p n+ 3) wavelengths. This value is within multiplicative constants of a lower bound. We then give a uniform 3-hop, 4-hop implementation of all-to-all routing that needs at most n 2 ( 7 q n 16 +3), n 2 ( 15 q n 2 +6) wavelengths, respectively. 1