Fault-tolerant Hamiltonian laceability of balanced hypercubes

Paired many-to-many two-disjoint path cover of balanced hypercubes with faulty edges

“…Lemma 9 [32]. Let F be a set of faulty edges of BH n with |F | ≤ 2n − 2 for n ≥ 2 and let x and y be two vertices in different partite sets of BH n .…”

“…Thus tasks running on a faulty processor can be shifted to its backup processor [26]. With such novel properties above, different aspects of the balanced hypercube are studied extensively, including Hamiltonian embedding issues [5,10,15,17,27,29,32], connectivity issues [18,31], matching preclusion and extendability [16,19], and symmetric issues [33,34] and some other topics [11,30]. In this paper, we will consider the problem of paired 2-DPC of the balanced hypercube with faulty edges.…”

A parallel algorithm for motion estimation in video coding using the bilinear transformation

“…Specifically, we propose a parallel algorithm for this computation on the well-known parallel model of the hypercube network (Leighton 1992 ). This network as well as its numerous variations have been intensively studied in the literature (Hsieh and Lee 2010 ; Shih et al 2008 ; Fu 2008 ; Lai 2012 ; Kuo et al 2013 ; Zhou et al 2015 ). The rich interconnection structure of this network favours the design of “elegant” parallel algorithms for a number of problems (Grama et al 2002 ) which can be used in other parallel models (Sundar et al 2013 ), as well.…”

Fault-Tolerant Cycle Embedding in Balanced Hypercubes with Faulty Vertices and Faulty Edges

“…12 ([4]) Let F v be the set of faulty vertices in BH n with |F v | ≤ n − 1. Then every fault-free edge of BH n − F v lies on a fault-free cycle of every even length from 4 to 2 2n − 2|F v | inclusive, where n ≥ 1.Hao et al[8] and Zhou et al[26] obtained the following two results, respectively.Lemma 2.13 ([8]) Let F e be a faulty edge set of the balanced hypercube BH n with |F e | ≤ 2n − 2, n ≥ 1, then there exists a fault-free Hamiltonian path between any two adjacent vertices x and y in BH n . Lemma 2.14 ([26]) BH n is (2n − 2)-edge-fault Hamiltonian laceable for n ≥ 2.…”

“…faulty edges) in the n-dimensional balanced hypercube BH n . As examples, Zhou et al [26] derived that BH n is (2n − 2)-edge-fault Hamiltonian laceable for n ≥ 2, i.e., if at most 2n − 2 faulty edges occurs, the resulting graph remains Hamiltonian laceable. Cheng et al in [4] proved that if |F v | ≤ n − 1 faulty vertices occurs, every fault-free edge of BH n − F v lies on a faultfree cycle of every even length from 4 to 2 2n − 2|F v |.…”