Skip to main content
Log in

Embedding linear arrays of the maximum length in O-shaped meshes

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

Abstract

Embedding an interconnection network into another network is one of the important problems in parallel processing. In this paper, we study embedding of linear arrays (paths) of maximum length in O-shaped meshes (O-shaped grid graphs). This is equal to finding a longest path in an O-shaped mesh (grid graph). An O-shaped mesh is a 2D mesh that a smaller 2D mesh is removed from it. The removed nodes can be considered as faulty processor. We give a linear-time parallel algorithm for this problem. To show the algorithm finds an optimal path, first we prove some upper bounds on the length of the longest paths, then we show that how our algorithm meets these upper bounds.

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

Access this article

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
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34

Similar content being viewed by others

References

  1. Asgharian-Sardroud A, Bagheri A (2016) An approximation algorithm for the longest path problem in solid grid graphs. Optim Methods Softw 31(3):47–493

    Article  MathSciNet  Google Scholar 

  2. Abuelrub EM (1993) Interconnection networks embeddings and efficient parallel computations, Louisiana State University and Agricultural & Mechanical College, PhD Thesis

  3. Björklund A, Husfeldt T (2003) Finding a path of superlogarithmic length, SIAM. J Comput 32:1395–1402

    MathSciNet  MATH  Google Scholar 

  4. Chen SD, Shen H, Topor R (2002) An efficient algorithm for constructing Hamiltonian paths in meshes. Parallel Comput 28:1293–1305

    Article  MathSciNet  Google Scholar 

  5. Chang RY, Hsu CH, Peng S (2012) The longest path problem on permutation graphs, The 29th Workshop on Combinatorial Mathematics and Computation Theory, pp. 294–297

  6. Cohen Y, Stern R, Felner A (2020) Solving the Longest Simple Path Problem with Heuristic Search, Proceedings of the Thirtieth International Conference on Automated Planning and Scheduling (ICAPS 2020), pp. 75–79

  7. Fieger K, Balyo T, Schulz C, Schreiber D (2019) Finding optimal longest paths by dynamic programming in parallel, In Proceedings of the Twelfth International Symposium on Combinatorial Search (SoCS 2019), Napa, California, AAAI Press, pp. 61–69

  8. Ghosh E, Narayanaswamy NS, Rangan CP (2011) A polynomial time algorithm for longest paths in biconvex graphs, International Workshop on Algorithms and Computation WALCOM 2011: WALCOM: Algorithms and Computation, pp. 191–201

  9. Guo YL, Ho CW, Ko MT (2013) The longest path problem on distance-hereditary graphs. Adv Intell Syst Appl 1:69–77

    Google Scholar 

  10. Gutin G (1993) Finding a longest path in a complete multipartite digraph. SIAM J. Discrete Math. 6:270–273

    Article  MathSciNet  Google Scholar 

  11. Ioannidou K, Mertzios GB, Nikolopoulos SD (2009) The longest path problem is polynomial on interval graphs, International Symposium on Mathematical Foundations of Computer Science MFCS 2009: Mathematical Foundations of Computer Science, pp. 403–414

  12. Itai A, Papadimitriou CH, Szwarcfiter JL (1982) Hamiltonian paths in grid graphs, SIAM. J Comput 11:676–686

    MathSciNet  MATH  Google Scholar 

  13. Karger D, Montwani R, Ramkumar GDS (1997) On approximating the longest path in a graph. Algorithmica 18:82–98

    Article  MathSciNet  Google Scholar 

  14. Keshavarz-Kohjerdi F, Bagheri A, Asgharian-Sardroud A (2012) A linear-time algorithm for the longest path problem in rectangular grid graphs. Discrete Appl Math 160:210–217

    Article  MathSciNet  Google Scholar 

  15. Keshavarz-Kohjerdi F, Bagheri A (2013) An efficient parallel algorithm for the longest path problem in rectangular grid graphs. J Supercomput 65:723–741

    Article  Google Scholar 

  16. Keshavarz-Kohjerdi F, Bagheri A (2016) Hamiltonian paths in \(L\)-shaped grid graphs. Theor Comput Sci 621:37–56

    Article  MathSciNet  Google Scholar 

  17. Keshavarz-Kohjerdi F, Bagheri A (2018) Longest \((s, t)\)-paths in \(L\)-shaped grid graphs. Optim Methods Softw 34:797–826

    Article  Google Scholar 

  18. Keshavarz-Kohjerdi F, Bagheri A (2017) A linear-time algorithm for finding Hamiltonian \((s, t)\)-paths in even-sized rectangular grid graphs with a rectangular hole. Theor Comput Sci 690:26–58

    Article  MathSciNet  Google Scholar 

  19. Keshavarz-Kohjerdi F, Bagheri A (2017) A linear-time algorithm for finding Hamiltonian \((s, t)\)-paths in odd-sized rectangular grid graphs with a rectangular hole. J Supercomput 73:3821–3860

    Article  Google Scholar 

  20. Keshavarz-Kohjerdi F, Bagheri A (2020) Linear-time algorithms for finding Hamiltonian and longest \((s, t)\)-paths in \(C\)-shaped grid graphs. Discrete Optim. https://doi.org/10.1016/j.disopt.2019.100554

  21. Mertzios GB, Corneil DG (2013) The longest path problem is polynomial on cocomparability graphs. Algorithmica 65:177–205

    Article  MathSciNet  Google Scholar 

  22. Rahman MS, Kaykobad M (2005) On Hamiltonian cycles and Hamiltonian paths. Inf Process Lett 94:37–41

    Article  MathSciNet  Google Scholar 

  23. Uehara R, Uno Y (2007) On computing longest paths in small graph classes. Int J Found Comput Sci 18:911–930

    Article  MathSciNet  Google Scholar 

  24. Xu JM, Ma M (2009) Survey on path and cycle embedding in some networks. Front Math China 4:217–252

    Article  MathSciNet  Google Scholar 

  25. Zhang Z, Li H (2007) Algorithms for long paths in graphs. Theor Comput Sci 377:25–34

    Article  MathSciNet  Google Scholar 

  26. Zhang WQ, Liu YJ (2011) Approximating the longest paths in grid graphs. Theor Comput Sci 412:5340–5350

    Article  MathSciNet  Google Scholar 

  27. Du ZZ, Xu JM (2011) A note on cycle embedding in hypercubes with faulty vertices. Inf Process Lett 111(12):557–560

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Fatemeh Keshavarz-Kohjerdi.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11227-021-03895-1

Keywords

Navigation