The divide-and-swap cube: a new hypercube variant with small network cost

Abstract

The hypercube is one of the most popular interconnection networks. Its network cost is \(O(n^2)\). In this paper, we propose a new hypercube variant, the divide-and-swap cube \(\textit{DSC}(n)\,(n=2^d,\,d\ge 1)\), which reduces the network cost to \(O(n \log n)\) while maintaining the same number of nodes and the same asymptotic performances for fundamental algorithms such as the broadcasting. The new network has nice hierarchical properties. We first show that the diameter of \(\textit{DSC}(n)\) is lower than or equal to \(\frac{5n}{4}-1\). However, unlike the hypercube of dimension n whose degree is n, the node degree of the network is \(\log n + 1\), resulting in a network cost of \(O(n \log n)\). We then examine the one-to-all and all-to-all broadcasting times of \(\textit{DSC}(n)\), based on the single-link-available and multiple-link-available models. We also present an upper bound on the bisection width of the \(\textit{DSC}(n)\) and show that \(\textit{DSC}(n)\) is Hamiltonian. Finally, we introduce the folded divide-and-swap cube, \(\textit{FDSC}(n)\), a variant of the \(\textit{DSC}(n)\) and study its many properties including its hierarchical structure, routing algorithm, broadcasting algorithms, bisection width, and its Hamiltonicity. All the broadcasting algorithms presented in this paper are asymptotically optimal.

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Acknowledgements

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2017R1D1A3B03032173). We are grateful to the anonymous referees for their helpful comments and suggestions.

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Correspondence to Hyeong-Ok Lee.

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Kim, J., Kim, D., Qiu, K. et al. The divide-and-swap cube: a new hypercube variant with small network cost. J Supercomput 75, 3621–3639 (2019). https://doi.org/10.1007/s11227-018-2712-z

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Keywords

  • Interconnection network
  • Divide-and-swap cube
  • Folded divide-and-swap cube
  • Network cost
  • Diameter
  • Routing
  • Broadcasting
  • Bisection width
  • Hamiltonian cycle