A note on cycle covering

Bermond, Jean‐Claude; Coudert, David; Chacon, L.; Tilerot, François

doi:10.1145/378580.378716

Cited by 3 publications

(3 citation statements)

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“…For parts (3), (4), (5) and (6), we only give in Figure 4 the basic cycles and the routings corresponding to C x,y as the routings corresponding to C x,y are just the transformation of C x,y under α. and C x,y = α(C x,y ).…”

Section: Small Cycle Covering and Its Routingmentioning

confidence: 99%

“…(More precisely one can associate two wavelengths to each cycle of the covering: one for the normal traffic and another as a spare one.) This problem was asked by France Telecom R & D (see [3] for more details). Similar problems were also considered by several authors [6,7,8,9].…”

Section: Introductionmentioning

confidence: 99%

“…In [3] and [4], the case is studied when the physical graph G is C n , a cycle of length n, and the logical graph I is K n , which corresponds to the instance of communication called total exchange or all-to-all, where each vertex wants to communicate with all the others simultaneously. In this case, the DR constraint implies that the paths associated with the routing of a request cycle form the C n and they give a load 1 to each edge.…”

Section: Introductionmentioning

confidence: 99%

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Vertex disjoint routings of cycles over tori

Bermond

2007

Networks

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We study the problem of designing a survivable WDM network based on covering the communication requests with subnetworks that are protected independently from each other. We consider here the case when the physical network is T (n), a torus of size n by n, the subnetworks are cycles and the communication scheme is all-to-all or total exchange (where all pairs of vertices communicate). We will represent the communication requests by a logical graph: a complete graph for the scheme of all-toall. This problem can be modeled as follows: find a cycle partition or covering of the request edges of K n 2 , such that for each cycle in the partition, its request edges can be routed in the physical network T (n) by a set of vertex disjoint paths (equivalently, the routings with the request cycle form an elementary cycle in T (n)). Let the load of an edge of the WDM network be the number of paths associated with the requests using the edge. The cost of the network depends on the total load (the cost of transmission) and the maximum load (the cost of equipment). To minimize these costs, we will search for an optimal (or quasi optimal) routing satisfying the following two conditions: (a) 1 each request edge is routed by a shortest path over T (n), and (b) the load of each physical edge resulting from the routing of all cycles of S is uniform or quasi uniform.In this paper, we find a covering or partition of the request edges of K n 2 into cycles with an associated optimal or quasi optimal routing such that either (1) the number of cycles of the covering is minimum, or (2) the cycles have size 3 or 4.

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Section: Small Cycle Covering and Its Routingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Vertex disjoint routings of cycles over tori

Bermond

2007

Networks

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On DRC‐covering of K_n by cycles

Bermond

Coudert

2003

J of Combinatorial Designs

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International audienceThis paper considers the cycle covering of complete graphs motivated by the design of survivable WDM networks, where the requests are routed on INF-networks which are protected independently from each other. The problem can be stated as follows: for a given graph G, find a cycle covering of the edge set of Kn, where V(Kn) = V(G), such that each cycle in the covering satisfies the disjoint routing constraint (DRC), relatively to G, which can be stated as follows: to each edge of Kn we associate in G a path and all the paths associated to the edges of a cycle of the covering must be vertex disjoint. Here we consider the case where G = Cn, a ring of size n and we want to minimize the number of cycles in the covering. We give optimal solutions for the problem as well as for variations of the problem, namely, its directed version and the case when the cycle length is fixed to 4

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