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>3.0.co;2-r

Cited by 12 publications

(2 citation statements)

References 23 publications

(28 reference statements)

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“…A parametric description of the form (n; S) completely specifies the circulant of order n and dimension k. Since circulants belong to the family of Cayley graphs, any undirected circulant graph C n (S) is vertex-transitive and |S|-regular. Circulant graphs are also known as starpolygon graphs [4], cyclic graphs [7], distributed loop networks [3], chordal rings [1], multiple fixed step graphs [10], point-symmetric graphs [17], in Russian as Diophantine structures [15].…”

Section: Circulant Graphsmentioning

confidence: 99%

Diameter of generalized Petersen graphs

Loudiki¹,

Kchikech²,

Essaky³

2021

Preprint

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Due to their broad application to different fields of theory and practice, generalized Petersen graphs GP G(n, s) have been extensively investigated. Despite the regularity of generalized Petersen graphs, determining an exact formula for the diameter is still a difficult problem. In their paper, Beenker and Van Lint have proved that if the circulant graph C n (1, s) has diameter d, then GP G(n, s) has diameter at least d + 1 and at most d + 2. In this paper, we provide necessary and sufficient conditions so that the diameter of GP G(n, s) is equal to d+ 1, and sufficient conditions so that the diameter of GP G(n, s) is equal to d + 2. Afterwards, we give exact values for the diameter of GP G(n, s) for almost all cases of n and s. Furthermore, we show that there exists an algorithm computing the diameter of generalized Petersen graphs with running time O(logn).Keywords Diameter • Generalized Petersen graphs • Circulant graphs, • Edge-contraction s gcd(n,s) , the paths P 1 c (i), P 2 c (i), P 1,t c (i), P 2,t c (i), P 3,t c (i), P 4,t c (i) are pairwise nonequivalent and walk through outer edges before entering to inner edges.

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Section: Circulant Graphsmentioning

confidence: 99%

Diameter of generalized Petersen graphs

Loudiki¹,

Kchikech²,

Essaky³

2021

Preprint

View full text Add to dashboard Cite

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“…There are few works in the literature devoted to the search for the shortest paths in three-dimensional circulants. There are solutions [ 23 , 24 ] for particular circulant families of order

, where

– diameter, and work [ 22 ] presents a simple analytical method for finding the shortest path in circulants of maximum order for a given diameter. There is neither universal routing algorithm for three-dimensional circulants nor for the subfamily of ring circulants.…”

Section: Study Areamentioning

confidence: 99%

Routing in triple loop circulants: A case of networks-on-chip

Romanov

Starykh

2020

Heliyon

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In this paper we propose and analyze various approaches to organizing routing in a triple loop circulant topologies as applied to networks-on-chip: static routing based on universal graph search algorithms, such as Dijkstra's algorithm and a possible implementation using Table routing; algorithms created analytically based on an engineering approach with taking into account the structural features of triple loop circulant graphs (Advanced clockwise, Direction selection); an algorithm created on the basis of a mathematical analysis of graph structure and solving the problem of enumerating coefficients at generators (Coefficients finding algorithm). Efficiency, maximum graph paths, occupied memory resources, and calculation time of the algorithms developed are estimated. Comparison of various variants of the algorithms is made and recommendations on their application for the development of networks-on-chip with triple loop circulant topologies are given. It is shown that Advanced clockwise and Direction selection algorithms guarantee that the packet reaches the destination node, but often in more steps than the shortest path. Nevertheless, they themselves are simpler and require less hardware resources than other algorithms. In turn, Coefficients finding algorithm has great computational complexity, but is optimal and, in comparison with Dijkstra's algorithm, is much simpler for RTL implementation which reduces network-on-chip routers resources cost.

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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 12 publications

References 23 publications

Diameter of generalized Petersen graphs

Diameter of generalized Petersen graphs

Routing in triple loop circulants: A case of networks-on-chip

Broadcasting in generalized chordal rings

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