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Embedding complete binary trees into parity cubes

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Abstract

The complete binary tree as an important network structure has long been investigated for parallel and distributed computing, which has many nice properties and used to be embedded into other interconnection architectures. The parity cube is an important variant of the hypercube. It has many attractive features superior to those of the hypercube. In this paper, we prove that the complete binary tree with \(2^n-1\) vertices can be embedded with dilation 1, congestion 1, load 1 into the \(n\)-dimensional parity cube \(PQ_n\) and expansion tending to 1. Furthermore, we provide an \(O(NlogN)\) algorithm to construct the complete binary tree with \(2^n-1\) vertices in \(PQ_n\), where \(N\) denotes the number of vertices in \(PQ_n\) and \(n\ge 1\).

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Acknowledgments

We would like to express our sincerest appreciation to Prof. Guoliang Chen for his constructive suggestions. This work is supported by National Natural Science Foundation of China (No. 61170021 and No. 61303205), and Application Foundation Research of Suzhou of China (No. SYG201240).

Author information

Authors and Affiliations

  1. School of Computer Science and Technology, Soochow University, Suzhou, China

    Zhao Liu & Jianxi Fan

  2. Provincial Key Laboratory for Computer Information Processing Technology, Soochow University, Suzhou, China

    Zhao Liu

  3. Department of Computer Science, City University of Hong Kong, Kowloon, Hong Kong

    Xiaohua Jia

Authors
  1. Zhao Liu
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  2. Jianxi Fan
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  3. Xiaohua Jia
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Corresponding author

Correspondence to Jianxi Fan.

Appendix

Appendix

Here, the main C functions of algorithm PQCBT are as follows:

figure c
figure d

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Liu, Z., Fan, J. & Jia, X. Embedding complete binary trees into parity cubes. J Supercomput 71, 1–27 (2015). https://doi.org/10.1007/s11227-014-1274-y

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