Many-to-many disjoint paths in hypercubes with faulty vertices

Li, Xiangjun; Liu, Bin; Ma, Meijie; Xu, Jie

doi:10.1016/j.dam.2016.09.013

Cited by 16 publications

(16 citation statements)

References 25 publications

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“…Li et al proposed an algorithm that solves the problem in an ndimensional hypercube with f faulty nodes. The algorithm can construct k disjoint paths between two node sets that include at least (2 n − 2f ) nodes if f ≤ 2n − 2f − 2 and each non-faulty node has at least two non-faulty neighbor nodes [31]. Chen proposed an algorithm that solves the problem in an n-dimensional hypercube with f n faulty nodes and f e faulty edges.…”

Section: Related Workmentioning

confidence: 99%

Pairwise Disjoint Paths Routing in Tori

2020

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The pairwise disjoint paths problem is to construct c disjoint paths s i d i (1 ≤ i ≤ c) between given pairs of nodes (s i , d i) (1 ≤ i ≤ c) in a graph whose connectivity is no less than 2c. It is a major problem for interconnection networks, together with the node-to-node disjoint paths problem, the node-to-set disjoint paths problem, and the set-to-set disjoint paths problem. In this paper, we propose an algorithm that solves this problem for a torus. In a previous work, Bossard and Kaneko have developed an algorithm that constructs disjoint paths between c pairs of nodes in a k-ary n-dimensional torus, where c ≤ n. The time complexity of their algorithm is O(c 4 n + kcn), and the maximum path length is k/2 n + 2k(c − 1). However, the algorithm proposed in this paper achieves a time complexity of O(c 3 n + kcn), and its maximum path length is k/2 n + (3k/2 − 2)(c − 1), which are both improvements over the previous algorithm. We also conducted an evaluation experiment to show that the average path lengths are proportional to k if n is fixed, and to n if k is fixed. The theoretical maximum path lengths were not attained by the paths constructed by our algorithm, and the average execution time was proportional to k if n is fixed, and to n 2 if k is fixed. Additional experimental results show that, compared to the previous algorithm by Bossard and Kaneko, our algorithm achieves a better performance with respect to the maximum path lengths, but both algorithms achieve a similar level of performance with respect to the average maximum path lengths. Also, it is shown that the average execution time of our algorithm is about O(n 2), which is better than the average execution time of the previous algorithm O(n 3) in the experimental framework.

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Section: Related Workmentioning

confidence: 99%

Pairwise Disjoint Paths Routing in Tori

2020

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“…Obviously, Q 3 1 is a cycle of length 3, and Q 3 2 is a 3 × 3 wraparound mesh. We can partition Q 3 n along j-dimension, 0 ≤ j ≤ n − 1, by deleting all the j-dimension edges, into three disjoint subcubes, Q 3 [1] and Q [2], if there are no ambiguities). Q[j] is isomorphic to Q 3 n− 1 for 0 ≤ j ≤ 2.…”

Section: E K-ary N-cubementioning

confidence: 99%

“…(1) If |F| ≤ 2n − 2, then the graph Q k n − F is hamiltonian. (2) If |F| ≤ 2n − 3, then the graph Q k n − F is hamiltonianconnected.…”

Section: E K-ary N-cubementioning

confidence: 99%

“…en, the theorem holds. us, we suppose s i t j ∉ E(Q 3 n ) for all 1 ≤ i, j ≤ m. We decompose Q 3 n into three isomorphic subgraphs [2]). By Lemma 1, there is a Hamilton path M � (x 2 , y 2 ) in Q [2].…”

Section: Many-to-many Disjoint Pathsmentioning

confidence: 99%

“…Shih and Kao [1] constructed the oneto-one m-disjoint path cover of k-ary n-cube for 1 ≤ m ≤ 2n, n ≥ 2 and k ≥ 3. Li et al [2] studied many-to-many k-disjoint paths in hypercubes with f ≤ 2n − 2k − 2, 1 ≤ k ≤ n − 1 and each fault-free vertex has at least two fault-free neighbors. Chen in [3] considered many-to-many k disjoint (S, T)-paths in the hypercube with f v + f e ≤ n − k − 1, 1 ≤ k ≤ n − 1 and any two sets S and T of k fault-free vertices in different parts.…”

Section: Introductionmentioning

confidence: 99%

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Vertex-Disjoint Paths in a 3-Ary n-Cube with Faulty Vertices

Wang

2020

Mathematical Problems in Engineering

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The construction of vertex-disjoint paths (disjoint paths) is an important research topic in various kinds of interconnection networks, which can improve the transmission rate and reliability. The k-ary n-cube is a family of popular networks. In this paper, we determine that there are m2≤m≤n disjoint paths in 3-ary n-cube covering Qn3−F from S to T (many-to-many) with F≤2n−2m and from s to T (one-to-many) with F≤2n−m−1 where s is in a fault-free cycle of length three.

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