Parallel construction of independent spanning trees and an application in diagnosis on Möbius cubes

A fully parallelized scheme of constructing independent spanning trees on Möbius cubes

“…Indeed, all ISTs constructed in here are isomorphic and have height n + 1, which is the same as the height of those in [7,10]. To the best of our knowledge, for class of networks without node-symmetry, the present paper is the first to employ the fully parallelized approach for constructing ISTs.…”

“…Also, this conjecture has been confirmed for several restricted classes of graphs, e.g., graphs related to planarity [18,19,25,26], graphs defined by Cartesian product [6,27,29,30,34,40,44], variations of hypercubes [5,[8][9][10]24,32,33,38,49], special Cayley graphs [22,23,28,39,42,43], and chordal ring [21,41]. In particular, [5,[7][8][9][10][32][33][34]40,49] are published after 2012.…”

“…First of all, we would like to give a brief introduction on what the previous algorithm in [7] is doing. Then, we show the comparison of execution time and measured speedup between the previous algorithm and the new one.…”

“…Cheng et al [7] provided the following algorithm, called PMCIST (see Fig. 4), to construct n ISTs rooted at an arbitrary node x on M Q n based on the descending circular permutation n − 1, n − 2, .…”

A fully parallelized scheme of constructing independent spanning trees on Möbius cubes

“…Indeed, all ISTs constructed in here are isomorphic and have height n + 1, which is the same as the height of those in [7,10]. To the best of our knowledge, for class of networks without node-symmetry, the present paper is the first to employ the fully parallelized approach for constructing ISTs.…”

“…Also, this conjecture has been confirmed for several restricted classes of graphs, e.g., graphs related to planarity [18,19,25,26], graphs defined by Cartesian product [6,27,29,30,34,40,44], variations of hypercubes [5,[8][9][10]24,32,33,38,49], special Cayley graphs [22,23,28,39,42,43], and chordal ring [21,41]. In particular, [5,[7][8][9][10][32][33][34]40,49] are published after 2012.…”

“…First of all, we would like to give a brief introduction on what the previous algorithm in [7] is doing. Then, we show the comparison of execution time and measured speedup between the previous algorithm and the new one.…”

“…Cheng et al [7] provided the following algorithm, called PMCIST (see Fig. 4), to construct n ISTs rooted at an arbitrary node x on M Q n based on the descending circular permutation n − 1, n − 2, .…”

At/kdiagnosis algorithm on hypercube‐like networks

“…So it is attractive to propose a t / k diagnosis algorithm for larger k ∈[ n +2,2 n −1]. Besides, for some special hypercube‐like network such as crossed cube and some special diagnosis strategy such as the fault diagnosis strategy based on independent spanning tree, to design (or re‐design) an algorithm with the time complexity less than the one in this paper is possible and interesting, and is naturally our research work in the future.…”

A fully parallelized scheme of constructing independent spanning trees on Möbius cubes

“…Indeed, all ISTs constructed in here are isomorphic and have height n + 1, which is the same as the height of those in [7,10]. To the best of our knowledge, for class of networks without node-symmetry, the present paper is the first to employ the fully parallelized approach for constructing ISTs.…”

“…Also, this conjecture has been confirmed for several restricted classes of graphs, e.g., graphs related to planarity [18,19,25,26], graphs defined by Cartesian product [6,27,29,30,34,40,44], variations of hypercubes [5,[8][9][10]24,32,33,38,49], special Cayley graphs [22,23,28,39,42,43], and chordal ring [21,41]. In particular, [5,[7][8][9][10][32][33][34]40,49] are published after 2012.…”

“…First of all, we would like to give a brief introduction on what the previous algorithm in [7] is doing. Then, we show the comparison of execution time and measured speedup between the previous algorithm and the new one.…”

“…Cheng et al [7] provided the following algorithm, called PMCIST (see Fig. 4), to construct n ISTs rooted at an arbitrary node x on M Q n based on the descending circular permutation n − 1, n − 2, .…”

At/kdiagnosis algorithm on hypercube‐like networks

“…So it is attractive to propose a t / k diagnosis algorithm for larger k ∈[ n +2,2 n −1]. Besides, for some special hypercube‐like network such as crossed cube and some special diagnosis strategy such as the fault diagnosis strategy based on independent spanning tree, to design (or re‐design) an algorithm with the time complexity less than the one in this paper is possible and interesting, and is naturally our research work in the future.…”

A fully parallelized scheme of constructing independent spanning trees on Möbius cubes

“…Indeed, all ISTs constructed in here are isomorphic and have height n + 1, which is the same as the height of those in [7,10]. To the best of our knowledge, for class of networks without node-symmetry, the present paper is the first to employ the fully parallelized approach for constructing ISTs.…”

“…Also, this conjecture has been confirmed for several restricted classes of graphs, e.g., graphs related to planarity [18,19,25,26], graphs defined by Cartesian product [6,27,29,30,34,40,44], variations of hypercubes [5,[8][9][10]24,32,33,38,49], special Cayley graphs [22,23,28,39,42,43], and chordal ring [21,41]. In particular, [5,[7][8][9][10][32][33][34]40,49] are published after 2012.…”

“…First of all, we would like to give a brief introduction on what the previous algorithm in [7] is doing. Then, we show the comparison of execution time and measured speedup between the previous algorithm and the new one.…”

“…Cheng et al [7] provided the following algorithm, called PMCIST (see Fig. 4), to construct n ISTs rooted at an arbitrary node x on M Q n based on the descending circular permutation n − 1, n − 2, .…”

At/kdiagnosis algorithm on hypercube‐like networks

“…So it is attractive to propose a t / k diagnosis algorithm for larger k ∈[ n +2,2 n −1]. Besides, for some special hypercube‐like network such as crossed cube and some special diagnosis strategy such as the fault diagnosis strategy based on independent spanning tree, to design (or re‐design) an algorithm with the time complexity less than the one in this paper is possible and interesting, and is naturally our research work in the future.…”