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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.
Approximating the Traffic Grooming Problem in Tree and Star Networks ⋆ (Extended Abstract)
"... Abstract. 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 ..."
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Abstract. 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.
de rechercheOptimal Solution of the Maximum All Request Path Grooming Problem
"... Abstract: We give an optimal solution to the Maximum All Request Path Grooming (MARPG) problem motivated by a traffic grooming application. The MARPG problem consists in finding the maximum number of connections which can be established in a path of size N, where each arc has a capacity or bandwidth ..."
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Abstract: We give an optimal solution to the Maximum All Request Path Grooming (MARPG) problem motivated by a traffic grooming application. The MARPG problem consists in finding the maximum number of connections which can be established in a path of size N, where each arc has a capacity or bandwidth C (grooming factor). We present a greedy algorithm to solve the problem and an explicit formula for the maximum number of requests that can be groomed. In particular, if C = s(s + 1)/2 and N> s(s − 1), an optimal solution is obtained by taking all the requests of smallest length, that is of length 1 to s. However this is not true in general since anomalies can exist. We give a complete analysis and the exact number of such anomalies. Keywords: Grooming, requests, path, capacity, coloration of interval graphs This work has been partially funded by European projects ist fet Aeolus and COST 293 Graal, and has been done within the crc Corso with France Telecom. Unité de recherche INRIA Sophia Antipolis
1 Grooming
, 2010
"... 1.1 Remark Traffic grooming in networks refers to group low rate traffic into higher speed streams (containers) so as to minimize the equipment cost [11, 7, 13, 12, 8, 9]. There are many variants according to the type of network considered, the constraints used and the parameters one wants to optimi ..."
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1.1 Remark Traffic grooming in networks refers to group low rate traffic into higher speed streams (containers) so as to minimize the equipment cost [11, 7, 13, 12, 8, 9]. There are many variants according to the type of network considered, the constraints used and the parameters one wants to optimize which give rise to a lot of interesting design problems (graph decompositions). To fix ideas, suppose that we have an optical network represented by a directed graph G (in many cases a symmetric one) on n vertices, for example a unidirectional ring ⃗ Cn or a bidirectional ring C ∗ n. We are given also a traffic matrix, that is a family of connection requests represented by a multidigraph I (the number of arcs from i to j corresponding to the number of requests from i to j). An interesting case is when there is exactly one request from i to j; then I = K ∗ n. Satisfying a request from i to j consists in finding a route (dipath) in G and assigning it a wavelength. The grooming factor, g, means that a request uses only 1/g of the bandwidth available on a wavelength along its route. Said otherwise, for each arc e of G and for each wavelength w, there are at most g dipaths with wavelength w which contain the arc
The Maximum All Request Path Grooming (MARPG)
, 2009
"... Abstract — We give an optimal solution to the Maximum All Request Path Grooming (MARPG) problem motivated by a traffic grooming application. The MARPG problem consists in finding the maximum number of connections which can be established in a path of size N, where each arc has a capacity or bandwidt ..."
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Abstract — We give an optimal solution to the Maximum All Request Path Grooming (MARPG) problem motivated by a traffic grooming application. The MARPG problem consists in finding the maximum number of connections which can be established in a path of size N, where each arc has a capacity or bandwidth C (grooming factor). We present a greedy algorithm to solve the problem and an explicit formula for the maximum number of requests that can be groomed. In particular, if C = s(s+1)/2 and N> s(s−1), an optimal solution is obtained by taking all the requests of smallest length, that is of length 1 to s. However this is not true in general since anomalies can exist. We give a complete analysis and the exact number of such anomalies.
Directed acyclic graphs with the unique dipath property
, 2013
"... Let P be a family of dipaths of a DAG (Directed Acyclic Graph) G. The load of an arc is the number of dipaths containing this arc. Let π(G, P) be the maximum of the load of all the arcs and let w(G, P) be the minimum number of wavelengths (colors) needed to color the family of dipaths P in such a wa ..."
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Let P be a family of dipaths of a DAG (Directed Acyclic Graph) G. The load of an arc is the number of dipaths containing this arc. Let π(G, P) be the maximum of the load of all the arcs and let w(G, P) be the minimum number of wavelengths (colors) needed to color the family of dipaths P in such a way that two dipaths with the same wavelength are arcdisjoint. There exist DAGs such that the ratio between w(G, P) and π(G, P) cannot be bounded. An internal cycle is an oriented cycle such that all the vertices have at least one predecessor and one successor in G (said otherwise every cycle contains neither a source nor a sink of G). We prove that, for any family of dipaths P, w(G, P) = π(G, P) if and only if G is without internal cycle. We also consider a new class of DAGs, called UPPDAGs, for which there is at most one dipath from a vertex to another. For these UPPDAGs we show that the load is equal to the maximum size of a clique of the conflict graph. We prove that the ratio between w(G, P) and π(G, P) cannot be bounded (a result conjectured in an other article). For that we introduce “good labelings ” of the conflict graph associated to G and P, namely labelings of the edges such that for any ordered pair of vertices (x, y) there do not exist two paths from x to y with increasing labels. Keywords: DAG (Directed acyclic graphs); load; wavelengths; dipaths; good labelings; conflict graphs; intersection graphs; chromatic number. 1