Fault‐tolerant routings in chordal ring networks

Barrière, Lali; Fàbrega, Josep M.; Simó, Ester; Zaragozá, Marisa

doi:10.1002/1097-0037(200010)36:3<180::aid-net5>3.3.co;2-i

Cited by 2 publications

(6 citation statements)

References 21 publications

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“…(d) The spanning tree T 0 obtained by Algorithm 1 is not unique. For example, instead of A 2,0 = H(d)( [1], [2]) and A 2,1 = H(d)( [1], [3d + 3]) as in (c) above, we may choose A 2,0 = H(d)( [1], [2]) and A 2,1 = H(d)( [1], [3d 2 + 1]) at distance two level.…”

Section: Gossiping and Routingmentioning

confidence: 99%

“…In this article, we will study these problems for TL n d , and prove that it admits "perfect" gossiping and routing schemes with interesting features. By a result of Morillo (see [1,Section 2] and Remark 2), up to isomorphism TL n d is the only undirected triple-loop network of order n d and diameter d. Thus, without loss of generality we can focus on TL n d as far as optimal undirected triple-loop networks are concerned.…”

Section: Introductionmentioning

confidence: 99%

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Gossiping and routing in undirected triple-loop networks

Thomson

Zhou

2009

Networks

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Given integers n ≥ 7 and a, b, c with 1 ≤ a, b, c ≤ n − 1 such that a, n − a, b, n − b, c, n − c are pairwise distinct, the (undirected) triple-loop network TL n (a, b, c) is the degree-six graph with vertices 0, 1, 2, . . . , n − 1 such that each vertex x is adjacent to x ± a, x ± b, and x ± c, where the operation is modulo n. It is known that the maximum order of a connected triple-loop network of the form TL n (a, b, n − (a + b)) with given diameter d ≥ 2 is n d = 3d 2 + 3d + 1, which is achieved by2 −1). In this article, we study the routing and gossiping problems for such optimal tripleloop networks under the store-and-forward, all-port, and full-duplex model, and prove that they admit "perfect" gossiping and routing schemes which exhibit many interesting features. Using a group-theoretic approach we develop for TL n d a method for systematically producing such optimal gossiping and routing schemes. Moreover, we determine the minimum gossip time, the edge-and arc-forwarding indices, and the minimal edge-and arcforwarding indices of TL n d , and prove that our routing schemes are optimal with respect to these four indices simultaneously. As a key step towards these results, we prove that TL n d is a Frobenius graph on a Frobenius group with Frobenius kernel Z n d , and that TL n d is arc-transitive with respect to this Frobenius group. In addition, we show that TL n d admits complete rotations.

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Section: Gossiping and Routingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gossiping and routing in undirected triple-loop networks

Thomson

Zhou

2009

Networks

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“…Associated labelings are also used in [9] to determine the broadcast time of some double and triple step graphs; see also [2,3].…”

Section: Proposition 1 Let G Be a Gcr Of Diameter D And Let T Be A mentioning

confidence: 99%

“…The standard way to treat generalized chordal rings involves the use of plane tessellations [2,13]. We associate with each CR n (a, b, c) a labeling of the cells of the standard triangular grid ( Fig.…”

Section: Introductionmentioning

confidence: 99%

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Broadcasting in generalized chordal rings

Comellas

Hell

2003

Networks

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Broadcasting is an information dissemination process in which a message originating at one node of a communication network (modeled as an undirected graph) is sent to all other nodes by means of calls involving two nodes at a time, with each node participating in at most one call at any time. We are interested in efficient broadcasting in a class of cubic graphs known as generalized chordal rings. These graphs have been found useful for having a small diameter D, among graphs with a given number of vertices and maximum degree. We show that the minimum broadcast time in any generalized chordal ring is D, D ؉ 1, or D ؉ 2. For the generalized chordal rings of diameter D which have the greatest (or greatestknown) number of nodes, we then evaluate exactly the minimum broadcast time. It turns out to be D ؉ 1 when D is even and D ؉ 2 when D is odd. For these purposes, we review the construction of these extremal generalized chodal rings. We also review the optimal broadcast schemes for infinite triangular grids, which we use to prove our bounds. Finally, we ask for the maximum number of nodes that can be informed by a broadcast in time t in any generalized chordal ring. We answer this completely for even t and almost completely for odd t. We use a geometric approach, based on plane tessellations.

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Fault‐tolerant routings in chordal ring networks

Cited by 2 publications

References 21 publications

Gossiping and routing in undirected triple-loop networks

Gossiping and routing in undirected triple-loop networks

Broadcasting in generalized chordal rings

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