A Fast Parallel Algorithm for Constructing Independent Spanning Trees on Parity Cubes

Chang, Yu-Huei; Yang, Jinn‐Shyong; Chang, Jou–Ming; Wang, Yue-Li

doi:10.1016/j.amc.2015.06.081

Cited by 12 publications

(10 citation statements)

References 19 publications

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“…We set the edge (marked with a blue X) from used to unused and use another edge (marked with a blue arrow) to traverse c. Hence, the triangle shape is retained. For example, in Figures 8,9, and 10, the frame nodes of T 5, 3 1 , T 5,3 2 , and T 5,3 3 are indicated by ellipses with blue borders.…”

Section: A Buildframe Functionmentioning

confidence: 99%

“…For example, S 5,3 has four trees and three germs: 541 in T 5, 3 1 , 542 in T 5,3 2 , and 543 in T This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/.…”

Section: Recursive Algorithmmentioning

confidence: 99%

“…The constructions of ISTs on several interconnection networks have been solved, including crossed cubes [7], Möbius cubes [8], twisted cubes [25], locally twisted cubes [13], [22], hypercubes [27], parity cubes [24], folded hypercubes [29], and bubble-sort networks [19], [20], alternating group networks [16]. In those studies, some parallel algorithms have been proposed on twisted cubes [5], locally twisted cubes [4], parity cubes [3], and hypercubes [28], [26]. To the best of our knowledge, the method of constructing ISTs in (n,k)star graphs is still unknown.…”

Section: Introductionmentioning

confidence: 99%

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Two Algorithms for Constructing Independent Spanning Trees in (n,k)-Star Graphs

2020

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In a graph G, two spanning trees T 1 and T 2 are rooted at the same vertex r. If, for every v ∈ V (G), the paths from v to the root r in T 1 and T 2 are internally vertex-disjoint, they are called independent spanning trees (ISTs). ISTs can be applied in numerous fields, such as fault-tolerant broadcasting and secure message distribution. The (n,k)-star graphs S n,k constitute a generalized version of the star network. The method of constructing ISTs in (n,k)-star graphs remains unknown. In this paper, we propose one recursive algorithm and one parallel algorithm for constructing ISTs on (n,k)-star graphs. The main ideas of the recursive algorithm are to use induction to change small trees into large trees, to use a modified breadth-first search (MBFS) traversal to create a frame for an IST, and to use a breadth-first search (BFS) traversal to connect the rest of nodes. The main ideas of the parallel algorithm are to create frames through MBFS traversals in parallel, and to use some specific rules to connect the rest of nodes in parallel. We also present validation proofs for both algorithms, and analyze the time complexities of both algorithms. The time complexity of the recursive algorithm in S n,k is O(n × n! (n−k)!), where n! (n−k)! is the number of nodes of S n,k. The time complexity of the parallel algorithm can be reduced to O(n! (n−k)!) if the system has n − 1 processors computing in parallel. Both algorithms are correct with the stated time complexity values; the parallel algorithm is more efficient than the recursive algorithm.

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Section: A Buildframe Functionmentioning

confidence: 99%

Section: Recursive Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Two Algorithms for Constructing Independent Spanning Trees in (n,k)-Star Graphs

2020

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“…, S k have optimal height. Observe that if equality in (4) or equality in (5) for bipartite graphs holds for all vertices (up to the root vertex) in LDPs, then LDPs have optimal height. See examples on Figure 4(b) and on Figure 3.…”

Section: Level-disjoint Partitions Of Optimal Heightmentioning

confidence: 99%

“…Furthermore, it is useful for security reasons to avoid two messages meeting at a vertex at the same time, if untrustworthy vertex is able to decode secret information from having access to more messages at the same time. Similarly, an algorithm for secure message distribution with even more restricted condition when different messages can arrive in the same vertex only via distinct paths is proposed in [5]. The condition (b) can be useful for example in streaming context when every message needs to be sent immediately without delay.…”

Section: Introductionmentioning

confidence: 99%

Modelling simultaneous broadcasting by level-disjoint partitions

Gregor

Škrekovski

Vukašinović

2018

Applied Mathematics and Computation

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Simultaneous broadcasting of multiple messages from the same source vertex in synchronous networks is considered under restrictions that each vertex receives at most one message in a unit time step, every received message can be sent out only in the next time step, no message is sent to already informed vertex. The number of outgoing messages in unrestricted, messages have unit length, and we assume full-duplex mode. In [9] we developed a concept of level-disjoint partitions to study simultaneous broadcasting under this model. In this work we consider the optimal number of level-disjoint partitions. We also provide a necessary condition in terms of eccentricity and girth on existence of k v-rooted level-disjoint partitions of optimal height. In particular, we provide a structural characterization of graphs admitting two level-disjoint partitions with the same root.

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