Skip to main content
SpringerLink
Log in

Embedding hierarchical folded cubes into linear arrays and complete binary trees with minimum wirelength

  • Published:
The Journal of Supercomputing Aims and scope Submit manuscript

Abstract

Graph embedding maps a guest graph into a host graph, thus enabling structural simulation, processor allocation, and algorithm porting. It is used to design the physical layout of Network-on-Chip (NoC) and to study the simulation capabilities of a parallel architecture. Wirelength is one of the indicators to measure the quality of graph embedding. Minimum wirelength in NoC design means a smaller wiring area and less wiring cost. In parallel computing, it means shorter communication time and delay. In this paper, the guest graph is the hierarchical folded cube with good communication and fault tolerance capabilities. The host graphs are the linear array and the complete binary tree, both of which are widely used in graph embeddings. We solve the embedding problems in linear time for hierarchical folded cubes into linear arrays and complete binary trees with minimum wirelength, respectively.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Data availability

All of the material is owned by the authors and/or no permissions are required.

References

  1. Keshavarz-Kohjerdi F (2022) Embedding linear arrays of the maximum length in o-shaped meshes. J Supercomput 78(1):884–918. https://doi.org/10.1007/s11227-021-03895-1

    Article  Google Scholar 

  2. Rajan RS, Kalinowski T, Klavzar S, Mokhtar H, Rajalaxmi TM (2021) Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes. J Supercomput 77(4):4135–4150. https://doi.org/10.1007/s11227-020-03420-w

    Article  Google Scholar 

  3. Wang XJ, Shi F, Zhang H (2019) KLSAT: An Application Mapping Algorithm Based on Kernighan-Lin Partition and Simulated Annealing for a Specific WK-Recursive NoC Architecture. In: Proceedings of the 16th International Conference on Network and Parallel Computing, Hohhot, China, 23–24 August 2019. https://doi.org/10.1007/978-3-030-30709-7_3

  4. Quadras J, Solomon SS (2015) Embedding of the folded hypercubes into tori. Math Comput Sci 9(2):177–183. https://doi.org/10.1007/s11786-015-0223-3

    Article  MathSciNet  MATH  Google Scholar 

  5. Lai Y-L, Williams K (1999) A survey of solved problems and applications on bandwidth, edge sum, and profile of graphs. J Graph Theoy 31(2):75–94

    Article  MATH  Google Scholar 

  6. Bezrukov SL, Chavez JD, Harper LH, Rottger M, Schroeder U-P (1998) Embedding of Hypercubes into Grids. In: Proceedings of the 23rd International Symposium on Mathematical Foundations of Computer Science, Brno, Czech Republic, 24–28 August 1998. https://doi.org/10.1007/BFb0055820

  7. Bezrukov SL, Chavez JD, Harper LH, Rottger M, Schroeder U-P (2000) The congestion of \(n\)-cube layout on a rectangular grid. Discret Math 213(1–3):13–19. https://doi.org/10.1016/S0012-365X(99)00162-4

    Article  MathSciNet  MATH  Google Scholar 

  8. Rajasingh I, William A, Quadras J, Manuel PD (2004) Embedding of cycles into arbitrary trees. Networks 44(3):173–178. https://doi.org/10.1002/net.20027

    Article  MathSciNet  MATH  Google Scholar 

  9. Klugerman M, Russell A, Sundaram R (1998) On embedding complete graphs into hypercubes. Discret Math 186(1–3):289–293. https://doi.org/10.1016/S0012-365X(97)00239-2

    Article  MathSciNet  MATH  Google Scholar 

  10. Manuel PD, Rajasingh I, Rajan B, Mercy H (2009) Exact wirelength of hypercubes on a grid. Discret Appl Math 157(7):1486–1495. https://doi.org/10.1016/j.dam.2008.09.013

    Article  MathSciNet  MATH  Google Scholar 

  11. Arockiaraj M, Abraham J, Quadras J, Shalini AJ (2017) Linear layout of locally twisted cubes. Int J Comput Math 94(1):56–65. https://doi.org/10.1080/00207160.2015.1088943

    Article  MathSciNet  MATH  Google Scholar 

  12. Fan W, Fan J, Lin C-K, Wang Y, Han Y, Wang R (2019) Optimally embedding \(3\)-ary \(n\)-cubes into grids. J Comput Sci Technol 34(2):372–387. https://doi.org/10.1007/s11390-019-1893-0

    Article  MathSciNet  Google Scholar 

  13. Fan W, Fan J, Lin C-K, Wang G, Cheng B, Wang R (2019) An efficient algorithm for embedding exchanged hypercubes into grids. J Supercomput 75(2):783–807. https://doi.org/10.1007/s11227-018-2612-2

    Article  Google Scholar 

  14. Xia J, Wang Y, Fan J, Fan W, Han Y (2020) Embedding Augmented Cubes into Grid Networks for Minimum Wirelength. In: Proceedings of the 20th International Conference on Algorithms and Architectures for Parallel, New York, USA, 2–4 October 2020. https://doi.org/10.1007/978-3-030-60239-0_4

  15. Manuel PD (2011) Minimum average congestion of enhanced and augmented hypercubes into complete binary trees. Discret Appl Math 159(5):360–366. https://doi.org/10.1016/j.dam.2010.12.001

    Article  MathSciNet  MATH  Google Scholar 

  16. Rajasingh I, Manuel PD, Rajan B, Arockiaraj M (2012) Wirelength of hypercubes into certain trees. Discret Appl Math 160(18):2778–2786. https://doi.org/10.1016/j.dam.2011.12.007

    Article  MathSciNet  MATH  Google Scholar 

  17. Arockiaraj M, Quadras J, Rajasingh I, Shalini AJ (2014) Embedding of hypercubes into sibling trees. Discret Appl Math 169(31):9–14. https://doi.org/10.1016/j.dam.2014.01.002

    Article  MathSciNet  MATH  Google Scholar 

  18. Shantrinal AA, Rajan RS, Babu AR, Anil S, Ahmed MA (2020) Embedding complete multipartite graphs into certain trees. J Comb Math Comb Comput 112:273–286. https://doi.org/10.48550/arXiv.1910.10643

    Article  MathSciNet  MATH  Google Scholar 

  19. Arockiaraj M, Delaila J, Abraham J (2021) Optimal wirelength of balanced complete multipartite graphs onto cartesian product of path, cycle and trees. Fundam Inform 178(3):187–202. https://doi.org/10.3233/FI-2021-2003

    Article  MathSciNet  MATH  Google Scholar 

  20. Abd-El-Barr MIH, Al-Somani TF (2011) Topological properties of hierarchical interconnection networks: a review and comparison. J Electr Comput Eng 2011(189434):1–12. https://doi.org/10.1155/2011/189434

    Article  Google Scholar 

  21. Shi Y, Hou Z, Song J (2000) Hierarchical Interconnection Networks with Folded Hypercubes as Basic Cluster. In: Proceedings of the 4th International Conference on High Performance Computing in the Asia-pacific Region, Beijing, China, 14–17 May 2000. https://doi.org/10.1109/HPC.2000.846533

  22. Sun X, Dong Q, Zhou S, Lv M, Lian G, Liu J (2019) Fault tolerance analysis of hierarchical folded cube. Theor Comput Sci 790:117–130

    Article  MathSciNet  MATH  Google Scholar 

  23. Sun X, Fan J, Cheng B, Liu Z, Yu J (2021) Component conditional fault tolerance of hierarchical folded cubic networks. Theor Comput Sci 883:44–58. https://doi.org/10.1016/j.tcs.2021.06.001

    Article  MathSciNet  MATH  Google Scholar 

  24. Bezrukov SL, Das SK, Elsasser R (2000) An edge-isoperimetric problem for powers of the petersen graph. Ann Comb 4:153–169. https://doi.org/10.1007/s000260050003

    Article  MathSciNet  MATH  Google Scholar 

  25. Arockiaraj M, Quadras J, Rajasingh I, Shalini AJ (2015) Embedding hypercubes and folded hypercubes onto cartesian product of certain trees. Discret Optim 17:1–13. https://doi.org/10.1016/j.disopt.2015.03.001

    Article  MathSciNet  MATH  Google Scholar 

  26. Rajasingh I, Arockiaraj M (2011) Linear wirelength of folded hypercubes. Math Comput Sci 5(1):101–111. https://doi.org/10.1007/s11786-011-0085-2

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhang J, Yang X, Yu C, He L, Yang L-X (2013) Implementing duplex crossed cube communication patterns on optical linear arrays. Optik 124(24):6496–6500. https://doi.org/10.1016/j.ijleo.2013.07.001

    Article  Google Scholar 

  28. Glantz R, Meyerhenke H, Noe A (2015) Algorithms for Mapping Parallel processes onto Grid and Torus Architectures. In: Proceedings of the 23rd Euromicro International Conference on Parallel, Distributed, and Network-Based Processing, Turku, Finland, 236–243 March 2015. https://doi.org/10.1109/PDP.2015.21

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. U1905211), A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), Natural Science Foundation of China under grant (No. 62102196), Natural Science Foundation of Jiangsu Province (No. BK20200753), Jiangsu Postdoctoral Science Foundation Funded Project (No. 2021K096A), the Future Network Scientific Research Fund Project (No. FNSRFP-2021-YB-60), the Natural Science Fund for Colleges and Universities in Jiangsu Province(No. 21KJB520026), and the Fundamental Research Funds for the Central Universities of Jilin University (No. 93K172020K25).

Funding

No funding.

Author information

Author notes

  1. All the authors have contributed equally to this study.

Authors and Affiliations

  1. School of Computer Science and Technology, Soochow University, Suzhou, 215006, China

    Ruyan Guo, Yan Wang & Jianxi Fan

  2. School of Computer, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China

    Weibei Fan

Authors
  1. Ruyan Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianxi Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Weibei Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

RG wrote the main manuscript text. YW, JF, and WF reviewed and revised the manuscript.

Corresponding author

Correspondence to Yan Wang.

Ethics declarations

Conflict of interest

I declare that the authors have no competing interests as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, R., Wang, Y., Fan, J. et al. Embedding hierarchical folded cubes into linear arrays and complete binary trees with minimum wirelength. J Supercomput 79, 11300–11327 (2023). https://doi.org/10.1007/s11227-023-05095-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11227-023-05095-5

Keywords

Search

Navigation