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A stochastic link-fault-tolerant routing algorithm in folded hypercubes

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Abstract

A folded hypercube is obtained by adding complementary links to a hypercube. The diameter of the folded hypercube is almost half of that of the hypercube, while its degree is larger than the degree of the hypercube by only one. In this paper, we propose a stochastic link-fault-tolerant routing algorithm in a folded hypercube by introducing a limited global information called routing probabilities. For an n-dimensional folded hypercube, we have proved that the routing probabilities for all distances can be calculated in \(O(n^2\log n)\) time at each node and the message can be forwarded to its neighbor node at each node in O(n) time. We also conducted a computer experiment, the results of which show that our algorithm achieves a better performance than the best algorithm for a hypercube.

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Abbreviations

n :

Dimension of a folded hypercube

\(\hbox {FQ}_n\) :

n-Dimensional folded hypercube

\(\varvec{a}\), \(\varvec{b}\), ...:

Nodes

\(\varvec{a}^\mathrm{c}\), \(\varvec{a}^{(i)}\) (\(1\le i\le n\)):

For a node \(\varvec{a}=(a_1,a_2,\ldots ,a_n)\), \(\varvec{a}^\mathrm{c}=(\bar{a}_1,\bar{a}_2,\ldots ,\bar{a}_n)\) and \(\varvec{a}^{(i)}=(a_1,a_2,\ldots ,a_{i-1},\bar{a}_i,a_{i+1},\ldots ,a_n)\) where \(\bar{a}_i=1-a_i\)

\((\varvec{a},\varvec{b})\) :

Link between two adjacent nodes \(\varvec{a}\) and \(\varvec{b}\)

\((\varvec{a},\varvec{a}^\mathrm{c})\) :

Complementary link

\(H(\varvec{a},\varvec{b})\) :

Hamming distance between two nodes \(\varvec{a}\) and \(\varvec{b}\)

\(d(\varvec{a},\varvec{b})\) :

Distance between two nodes \(\varvec{a}\) and \(\varvec{b}\)

\(N(\varvec{a})\) :

Set of neighbor nodes of \(\varvec{a}\)

\(C(\varvec{a})\) :

Singleton set of the node \(\varvec{a}^\mathrm{c}\)

\(N_0(\varvec{a},\varvec{b})\) :

Subset of \(N(\varvec{a}){\setminus } C(\varvec{a})\) such that Hamming distance between each node in \(N_0(\varvec{a},\varvec{b})\) and \(\varvec{b}\) is less than that between \(\varvec{a}\) and \(\varvec{b}\)

\(N_1(\varvec{a},\varvec{b})\) :

Subset of \(N(\varvec{a}){\setminus } C(\varvec{a})\) such that Hamming distance between each node in \(N_1(\varvec{a},\varvec{b})\) and \(\varvec{b}\) is more than that between \(\varvec{a}\) and \(\varvec{b}\)

\(Pre(\varvec{a},\varvec{b})\) :

Subset of \(N(\varvec{a})\), the elements of which are on the shortest paths from \(\varvec{a}\) to \(\varvec{b}\)

\(Spr(\varvec{a},\varvec{b})\) :

Subset of \(N(\varvec{a})\), the elements of which are on the detour paths from \(\varvec{a}\) to \(\varvec{b}\)

\(\gamma (\varvec{a},\varvec{b})\) :

Function that expresses the status of a link \((\varvec{a},\varvec{b})\). If the link is faulty, \(\gamma (\varvec{a},\varvec{b})=0\). Otherwise, \(\gamma (\varvec{a},\varvec{b})=1\)

\(P_h(\varvec{a})\) :

Routing probability of a node \(\varvec{a}\) with respect to the Hamming distance h

\(n\atopwithdelims ()h\) :

Number of combinations of h elements from a set of n elements

\(p^\mathrm{c}\) :

\(\gamma (\varvec{a},\varvec{a}^\mathrm{c})P_{n-h}(\varvec{a}^\mathrm{c})\)

\(p_1<p_2<\cdots <p_n\) :

Obtained by sorting \(\gamma (\varvec{a},\varvec{a}^{(i)}) P_{h-1}(\varvec{a}^{(i)})\) (\(1\le i\le n\)) in ascending order

\(q_1<q_2<\cdots <q_n\) :

Obtained by sorting \(\gamma (\varvec{a},\varvec{a}^{(i)}) P_{h+1}(\varvec{a}^{(i)})\) (\(1\le i\le n\)) in ascending order

\(\hat{p}_i\) (\(1\le i\le n\)):

\(\hat{p}_i=\gamma (\varvec{a},\varvec{a}^{(j)})P_{h-1}(\varvec{a}^{(j)})\) if \(q_i=\gamma (\varvec{a},\varvec{a}^{(j)})P_{h+1}(\varvec{a}^{(j)})\)

\(\hat{q}_i\) (\(1\le i\le n\)):

\(\hat{q}_i=\gamma (\varvec{a},\varvec{a}^{(j)})P_{h+1}(\varvec{a}^{(j)})\) if \(p_i=\gamma (\varvec{a},\varvec{a}^{(j)})P_{h-1}(\varvec{a}^{(j)})\)

\(\delta _\mathrm{p}(i)\) (\(1\le i\le n\)):

\(\delta _\mathrm{p}(i)=0\), if \(\exists \hat{q}_j\) (\(i+1\le j\le n\)) such that \(\hat{q}_j>\max \{p^\mathrm{c},p_i\}\). Otherwise, \(\delta _\mathrm{p}(i)=1\)

\(\delta _\mathrm{q}(i)\) (\(1\le i\le n\)):

\(\delta _\mathrm{q}(i)=0\), if \(\exists \hat{p}_j\) (\(i+1\le j\le n\)) such that \(\hat{p}_j>\max \{p^\mathrm{c},q_i\}\). Otherwise, \(\delta _\mathrm{q}(i)=1\)

x(i) (\(1\le i\le n\)):

\(x(i)=|\{\hat{q}_j\mid 1\le j\le i-1,\hat{q}_j>p_i\}|\)

y(i) (\(1\le i\le n\)):

\(y(i)=|\{\hat{p}_j\mid 1\le j\le i-1,\hat{p}_j>q_i\}|\)

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Acknowledgements

We appreciate Dr. Bipin Indurkhya for proofreading this paper. We also appreciate all of the reviewers for their insightful comments and suggestions, which contributed to improvement of this paper. This study was partly supported by JSPS Grant-in-Aid for Scientific Research (C), Grant No. 17K00093.

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  1. Tokyo University of Agriculture and Technology, Tokyo, Japan

    Bui Thi Thuan, Lam Boi Ngoc & Keiichi Kaneko

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Correspondence to Keiichi Kaneko.

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Thuan, B.T., Ngoc, L.B. & Kaneko, K. A stochastic link-fault-tolerant routing algorithm in folded hypercubes. J Supercomput 74, 5539–5557 (2018). https://doi.org/10.1007/s11227-018-2456-9

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