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Traffic Grooming on the Path
"... In a WDM network, routing a request consists in assigning it a route in the physical network and a wavelength. If each request uses at most 1/C of the bandwidth of the wavelength, we will say that the grooming factor is C. That means that on a given edge of the network we can groom (group) at most C ..."
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In a WDM network, routing a request consists in assigning it a route in the physical network and a wavelength. If each request uses at most 1/C of the bandwidth of the wavelength, we will say that the grooming factor is C. That means that on a given edge of the network we can groom (group) at most C requests on the same wavelength. With this constraint the objective can be either to minimize the number of wavelengths (related to the transmission cost) or minimize the number of Add Drop Multiplexers (shortly ADM) used in the network (related to the cost of the nodes). We consider here the case where the network is a path on N nodes, PN. Thus the routing is unique. For a given grooming factor C minimizing the number of wavelengths is an easy problem, well known and related to the load problem. But minimizing the number of ADM’s is NPcomplete for a general set of requests and no results are known. Here we show how to model the problem as a graph partition problem and using tools of design theory we completely solve the case where C = 2 and where we have a static uniform alltoall traffic (one request for each pair of vertices).
Traffic grooming in unidirectional wdm rings with boundeddegree request graph
 In 34th International Workshop on GraphTheoretic Concepts in Computer Science (WG 2008
, 2008
"... Abstract. Traffic grooming is a major issue in optical networks. It refers to grouping low rate signals into higher speed streams, in order to reduce the equipment cost. In SONET WDM networks, this cost is mostly given by the number of electronic terminations, namely AddDrop Multiplexers (ADMs for ..."
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Abstract. Traffic grooming is a major issue in optical networks. It refers to grouping low rate signals into higher speed streams, in order to reduce the equipment cost. In SONET WDM networks, this cost is mostly given by the number of electronic terminations, namely AddDrop Multiplexers (ADMs for short). We consider the unidirectional ring topology with a generic grooming factor C, and in this case, in graphtheoretical terms, the traffic grooming problem consists in partitioning the edges of a request graph into subgraphs with at most C edges, while minimizing the total number of vertices of the decomposition. We consider the case when the request graph has bounded degree Δ, and our aim is to design a network (namely, place the ADMs at each node) being able to support any request graph with maximum degree at most Δ. The existing theoretical models in the literature are much more rigid, and do not allow such adaptability. We formalize the problem, and we solve the cases Δ = 2 (for all values of C) andΔ =3(exceptthecase C = 4). We also provide lower and upper bounds for the general case.
Traffic grooming in bidirectional WDM ring networks
 In IEEELEOS ICTON / COST 293 GRAAL
, 2006
"... We study the minimization of ADMs (AddDrop Multiplexers) in Optical WDM Networks with Bidirectional Ring topology considering symmetric shortest path routing and alltoall unitary requests. We insist on the statement of the problem, which had not been clearly stated before in the bidirectional cas ..."
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We study the minimization of ADMs (AddDrop Multiplexers) in Optical WDM Networks with Bidirectional Ring topology considering symmetric shortest path routing and alltoall unitary requests. We insist on the statement of the problem, which had not been clearly stated before in the bidirectional case. Optimal solutions had not been found up to date. In particular, we study the case C = 2 and C = 3 (giving either optimal constructions or nearoptimal solutions) and the case C = k(k+1)/2 (giving optimal decompositions for specific congruence classes of N). We state a general Lower Bound for all the values of C and N, and we improve this Lower Bound for C=2 and C=3 (when N=4t+3). We also include some comments about the simulation of the problem using Linear Programming.
Approximations for Alltoall Uniform Traffic Grooming on Unidirectional Ring
, 2008
"... Traffic grooming in a WDM network consists of assigning to each request (lightpath) a wavelength with the constraint that a given wavelength can carry at most C requests or equivalently a request uses 1/C of the bandwidth. C is known as the grooming ratio. A request (lightpath) needs two SONET addd ..."
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Traffic grooming in a WDM network consists of assigning to each request (lightpath) a wavelength with the constraint that a given wavelength can carry at most C requests or equivalently a request uses 1/C of the bandwidth. C is known as the grooming ratio. A request (lightpath) needs two SONET adddrop multiplexers (ADMs) at each end node; using grooming, different requests can share the same ADM. The so called traffic grooming problem consists of minimizing the total number of ADMs to be used (in order to reduce the overall cost of the network). Here we consider the traffic grooming problem in WDM unidirectional rings which has been recently shown to be APXhard and for which no constant approximations are known. We furthermore suppose an all to all uniform unitary traffic. This problem has been optimally solved for specific values of the grooming ratio, namely C = 2, 3, 4, 5, 6. In this paper we present various simple constructions for the grooming problem providing approximation of the total number of ADMs with a small constant ratio. For that we use the fact that the problem corresponds to a partition of the edges of the complete graph into subgraphs, where each subgraph has at most C edges and where the total number of vertices has to be minimized.
Grooming for twoperiod optical networks
 Networks
"... Minimizing the number of adddrop multiplexers (ADMs) in a unidirectional SONET ring can be formulated as a graph decomposition problem. When traffic requirements are uniform and alltoall, groomings that minimize the number of ADMs (equivalently, the drop cost) have been characterized for grooming ..."
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Minimizing the number of adddrop multiplexers (ADMs) in a unidirectional SONET ring can be formulated as a graph decomposition problem. When traffic requirements are uniform and alltoall, groomings that minimize the number of ADMs (equivalently, the drop cost) have been characterized for grooming ratio at most six. However, when two different traffic requirements are supported, these solutions do not ensure optimality. In twoperiod optical networks, n vertices are required to support a grooming ratio of C in the first time period, while in the second time period a grooming ratio of C ′, C ′ < C, is required for v ≤ n vertices. This allows the twoperiod grooming problem to be expressed as an optimization problem on graph decompositions of Kn that embed graph decompositions of Kv for v ≤ n. Using this formulation, optimal twoperiod groomings are found for small grooming ratios using techniques from the theory of graphs and designs.
Approximating the Traffic Grooming Problem in Tree and Star Networks (Extended Abstract)
, 2006
"... We consider the problem of grooming paths in alloptical networks with tree topology so as to minimize the switching cost, measured by the total number of used ADMs. We first present efficient approximation algorithms with approximation factor of 2ln(δ · g)+o(ln(δ · g)) for any fixed node degree b ..."
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We consider the problem of grooming paths in alloptical networks with tree topology so as to minimize the switching cost, measured by the total number of used ADMs. We first present efficient approximation algorithms with approximation factor of 2ln(δ · g)+o(ln(δ · g)) for any fixed node degree bound δ and grooming factor g, and2lng + o(ln g) in unbounded degree directed trees, respectively. In the attempt of extending our results to general undirected trees we completely characterize the complexity of the problem in star networks by providing polynomial time optimal algorithms for g ≤ 2 and proving the intractability of the problem for any fixed g>2. While for general topologies the problem was known to be NPhard g not constant, the complexity for fixed values of g was still an open question.
Graph partitioning and traffic grooming with bounded degree request graph
 GraphTheoretic Concepts in Computer Science, volume 5911 of Lecture Notes in Computer Science
, 2010
"... We study a graph partitioning problem which arises from traffic grooming in optical networks. We wish to minimize the equipment cost in a SONET WDM ring network by minimizing the number of AddDrop Multiplexers (ADMs) used. We consider the version introduced by Muñoz and Sau [12] where the ring is ..."
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Cited by 2 (0 self)
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We study a graph partitioning problem which arises from traffic grooming in optical networks. We wish to minimize the equipment cost in a SONET WDM ring network by minimizing the number of AddDrop Multiplexers (ADMs) used. We consider the version introduced by Muñoz and Sau [12] where the ring is unidirectional with a grooming factor C, and we must design the network (namely, place the ADMs at the nodes) so that it can support any request graph with maximum degree at most ∆. This problem is essentially equivalent to finding the least integer M(C,∆) such that the edges of any graph with maximum degree at most ∆ can be partitioned into subgraphs with at most C edges and each vertex appears in at most M(C,∆) subgraphs [12]. The cases where ∆ = 2 and ∆ = 3, C 6 = 4 were solved by Muñoz and Sau [12]. In this article we establish the value of M(C,∆) for many more cases, leaving open only the case where ∆ ≥ 5 is odd, ∆ (mod 2C) is between 3 and C − 1, C ≥ 4, and the request graph does not contain a perfect matching. In particular, we answer a conjecture of [12].
Traffic Grooming in Unidirectional WavelengthDivision Multiplexed Rings with Grooming Ratio C = 6
, 2005
"... SONET/WDM networks using wavelength adddrop multiplexing can be constructed using certain graph decompositions used to form a grooming, consisting of unions of primitive rings. The cost of such a decomposition is the sum, over all graphs in the decomposition, of the number of vertices of nonzero de ..."
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SONET/WDM networks using wavelength adddrop multiplexing can be constructed using certain graph decompositions used to form a grooming, consisting of unions of primitive rings. The cost of such a decomposition is the sum, over all graphs in the decomposition, of the number of vertices of nonzero degree in the graph. The existence of such decompositions with minimum cost, when every pair of sites employs no more than 1 6 of the wavelength capacity, is determined with a finite number of possible exceptions. Indeed, when the number N of sites satisfies N ≡ 1 (mod 3), the determination is complete, and when N ≡ 2 (mod 3), the only value left undetermined is N = 17. When N ≡ 0 (mod 3), a finite number of values of N remain, the largest being N = 2580. The techniques developed rely heavily on tools from combinatorial design theory.
DROP COST AND WAVELENGTH OPTIMAL TWOPERIOD GROOMING WITH RATIO 4
"... Abstract. We study grooming for twoperiod optical networks, a variation of the traffic grooming problem for WDM ring networks introduced by Colbourn, Quattrocchi, and Syrotiuk. In the twoperiod grooming problem, during the first period of time there is alltoall uniform traffic among n nodes, eac ..."
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Abstract. We study grooming for twoperiod optical networks, a variation of the traffic grooming problem for WDM ring networks introduced by Colbourn, Quattrocchi, and Syrotiuk. In the twoperiod grooming problem, during the first period of time there is alltoall uniform traffic among n nodes, each request using 1/C of the bandwidth; and during the second period, there is alltoall uniform traffic only among a subset V of v nodes, each request now being allowed to use 1/C ′ of the bandwidth, where C ′ < C. We determine the minimum drop cost (minimum number of ADMs) for any n, v and C = 4 and C ′ ∈ {1, 2, 3}. To do this, we use tools of graph decompositions. Indeed the twoperiod grooming problem corresponds to minimizing the total number of vertices in a partition of the edges of the complete graph Kn into subgraphs, where each subgraph has at most C edges and where furthermore it contains at most C ′ edges of the complete graph on v specified vertices. Subject to the condition that the twoperiod grooming has the least drop cost, the minimum number of wavelengths required is also determined in each case. Key words. theory. traffic grooming, SONET ADM, optical networks, graph decomposition, design
Lower bounds for twoperiod grooming via linear programming duality
 Networks
"... In a problem arising in grooming for twoperiod optical networks, it is required to decompose the complete graph on n vertices into subgraphs each containing at most C edges, so that the induced subgraphs on a specified set of v ≤ n vertices each contain at most C ′ < C edges. The cost of the gro ..."
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In a problem arising in grooming for twoperiod optical networks, it is required to decompose the complete graph on n vertices into subgraphs each containing at most C edges, so that the induced subgraphs on a specified set of v ≤ n vertices each contain at most C ′ < C edges. The cost of the grooming is the sum, over all subgraphs, of the number of vertices of nonzero degree in the subgraph. The optimum grooming is the one of lowest cost. An integer linear programming formulation is used to determine precise lower bounds on this minimum cost for all choices of n and v when 1 ≤ C ′ < C ≤ 3. In most cases, this approach determines not only the bound but also the specific structure of any grooming that could realize the bound.