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41
Constrained Bipartite Edge Coloring with Applications to Wavelength Routing
, 1997
"... . Motivated by the problem of efficient routing in alloptical networks, we study a constrained version of the bipartite edge coloring problem. We show that if the edges adjacent to a pair of opposite vertices of an Lregular bipartite graph are already colored with ffL different colors, then the re ..."
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Cited by 33 (15 self)
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. Motivated by the problem of efficient routing in alloptical networks, we study a constrained version of the bipartite edge coloring problem. We show that if the edges adjacent to a pair of opposite vertices of an Lregular bipartite graph are already colored with ffL different colors, then the rest of the edges can be colored using at most (1+ff=2)L colors. We also show that this bound is tight by constructing instances in which (1 + ff=2)L colors are indeed necessary. We also obtain tight bounds on the number of colors that each pair of opposite vertices can see. Using the above results, we obtain a polynomial time greedy algorithm that assigns proper wavelengths to a set of requests of maximum load L per directed fiber link on a directed fiber tree using at most 5=3L wavelengths. This improves previous results of [9, 7, 6, 10]. We also obtain that no greedy algorithm can in general use less than 5=3L wavelengths for a set of requests of load L in a directed fiber tree, and thus t...
OnLine Routing in AllOptical Networks
 IN PROCEEDINGS OF THE 24TH INTERNATIONAL COLLOQIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING, LNCS 1256
, 1997
"... The paper deals with online 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 assigne ..."
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Cited by 31 (7 self)
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The paper deals with online 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 offline version of the problem, which was further studied in [AR95, KP96, MKR95, Ra96]. For a line topology, the problem is the wellstudied interval graph coloring problem. Online algorithms for this problem have been analyzed in [KT81, Ki88]. We consider trees, trees of rings, and meshes topologies, previously studied in the offline case. We give online 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...
Wavelength assignment and generalized interval graph coloring
 In Proceedings of the 14th Annual ACMSIAM Symposium on Discrete Algorithms
, 2003
"... Abstract In this paper we study wavelength assignment on an optical linesystem without wavelength conversion. Consider a set of undirected demands along the line. Each demand is carried on a wavelength and any two overlapping demands require distinct wavelengths. Suppose _ wavelengths are available ..."
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Cited by 26 (4 self)
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Abstract In this paper we study wavelength assignment on an optical linesystem without wavelength conversion. Consider a set of undirected demands along the line. Each demand is carried on a wavelength and any two overlapping demands require distinct wavelengths. Suppose _ wavelengths are available in the system. We define `(e), the load on link e, to be the smallest integer such that `(e) _ is at least the number of demands passing through e. Hence, `(e) is the minimum number of fibers required on e in order to support all demands. We present a polynomialtime wavelength assignment algorithm that guarantees each wavelength appears at most `(e) times on each link e. (This generalizes the wellknown fact that interval graphs are perfect.) In the presence of MOADMs (mesh optical add/drop multiplexers), devices that multiplex distinct wavelengths from different fibers into a new fiber, we only need to deploy `(e) fibers per link. On the other hand, if each demand has to stay on a single fiber, as is the case without MOADMs, we show that some links may require more than `(e) fibers. In fact, we show that it is NPcomplete to decide if a set of demands can be carried on a given set of fibers, or if there exists a set of fibers with a given total length that can carry all the demands.
The Complexity of Path Coloring and Call Scheduling
 Theoretical Computer Science
, 2000
"... Modern highperformance communication networks pose a number of challenging problems concerning the efficient allocation of resources to connection requests. In alloptical networks with wavelengthdivision multiplexing, connection requests must be assigned paths and colors (wavelengths) such that i ..."
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Cited by 22 (6 self)
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Modern highperformance communication networks pose a number of challenging problems concerning the efficient allocation of resources to connection requests. In alloptical networks with wavelengthdivision 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 NPhard 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 polynomialtime optimal algorithm for trees of constant degree as well as NPhardness 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 NPhard even in the binary case. A polynomialtime optimal algorithm is given for path coloring in undirected or bidir...
The Maximum EdgeDisjoint Paths Problem In Bidirected Trees
 SIAM Journal on Discrete Mathematics
, 1998
"... . 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 edgedisjoint paths problem is to select a maximumcardinality subse ..."
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Cited by 17 (3 self)
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. 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 edgedisjoint paths problem is to select a maximumcardinality subset of the paths such that the selected paths are edgedisjoint. This problem can be solved optimally in polynomial time for bidirected trees of constant degree, but is MAXSNPhard for bidirected trees of arbitrary degree. For every fixed " ? 0, a polynomialtime (5=3+ ")approximation algorithm is presented. Key words. approximation algorithms, edgedisjoint 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, edgedisjoint paths problems have been brought into the focus of attention by advances in the field of communication networks. Many modern network architectures estab...
Maximizing the Number of Connections in Optical Tree Networks
 In Proceedings of the 9th Annual International Symposium on Algorithms and Computation (1998), LNCS 1533
, 1998
"... . 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 wavelength ..."
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Cited by 15 (4 self)
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. 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 polynomialtime 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 NPhard. Polynomialtime 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 2approximation for MaxPP in arbitrary trees, a 1:58approximation for MaxPC in trees of bounded degree, and a 2:22approximation for MaxPC in arbitrary trees are obtained. 1
Randomized Path Coloring on Binary Trees
 3rd International Workshop on Approximation ALgorihms for Combinatorial Optimization Problems (APPROXâ€™00), Vol.1913
, 2000
"... . Motivated by the problem of WDM routing in alloptical networks, we study the following NPhard 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 assig ..."
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Cited by 12 (4 self)
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. Motivated by the problem of WDM routing in alloptical networks, we study the following NPhard 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...
Sparse and Limited Wavelength Conversion in AllOptical Tree Networks
, 2000
"... 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 alloptical tree networks, an important engine ..."
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Cited by 11 (0 self)
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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 alloptical 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...