Skip to main content
Springer Nature Link
Log in

Fault-tolerant basis of generalized fat trees and perfect binary tree derived architectures

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

Abstract

The ability to uniquely identify all nodes in a network based on network distances has proven to be highly beneficial despite the computational challenges involved in discovering minimal resolving sets within an interconnection network. A subset R of vertices of a graph G is referred to as a resolving set of the graph if each node can be uniquely identified by its distance code with respect to R, with its minimal cardinality defining the metric dimension of G. Similarly, a resolving set \(F \subseteq V\) is designated as a fault-tolerant resolving set if \(F {\setminus } \{s\}\) serves as a resolving set for each \(s \in F\). The minimum cardinality of F represents the fault-tolerant metric dimension of G. Although determining the precise metric dimension of a given graph remains challenging, various effective techniques and meaningful constraints have been developed for different graph families. However, no notable technique has been developed to find fault-tolerant metric dimension of a given graph. Recently, Prabhu et al. have shown that each twin vertex of G belongs to every fault-tolerant resolving set of G. Consequently, the fault-tolerant metric dimension is equal to the order of the graph G if and only if each vertex of G is a twin vertex, a characterization proved in Appl Math Comput 420:126897, 2022, corrects a wrong characterization in the literature. It is also interesting to note from the above literature correction that the twin vertices are necessary to form the fault-tolerant resolving set, but determining whether they are sufficient is challenging. Evidence of this context is also discussed in this paper through the amalgamation of perfect binary trees. This article focuses on determining the exact value of the fault-tolerant metric dimension of generalized fat trees. For the amalgamation of perfect binary trees, both the metric dimension and fault-tolerant metric dimension were precisely found.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Similar content being viewed by others

Data availability

No data associated with the manuscript.

References

  1. Beerliova Z, Eberhard F, Erlebach T, Hall A, Hoffman M, Mihalák M (2006) Network discovery and verification. IEEE J Sel Areas Commun 24(12):2168–2181

    Google Scholar 

  2. Khuller S, Ragavachari B, Rosenfield A (1996) Landmarks in graphs. Discret Appl Math 70(3):217–229

    MathSciNet  Google Scholar 

  3. Harary F, Melter RA (1976) On the metric dimension of a graph. Ars Combin 2:191–195

    MathSciNet  Google Scholar 

  4. Slater PJ (1975) Leaves of trees. Congr Numer 14:549–559

    MathSciNet  Google Scholar 

  5. Slater PJ (1988) Dominating and reference sets in a graph. J Math Phys Sci 22(4):445–455

    MathSciNet  Google Scholar 

  6. Chartrand G, Zhang P (2003) The theory and applications of resolvability in graphs, a survey. Congr Numer 160:47–68

    MathSciNet  Google Scholar 

  7. Javaid I, Salman M, Chaudhry MA, Shokat S (2009) Fault-tolerance in resolvability. Utilitas Math 80:263–275

    MathSciNet  Google Scholar 

  8. Salman M, Javaid I, Chaudhry MA (2018) Minimum fault-tolerant, local and strong metric dimension of graphs. Ars Combin 138:333–353

    MathSciNet  Google Scholar 

  9. Wang WH, Palaniswami M, Low SH (2003) Optimal flow control and routing in multi-path networks. Perform Eval 52:119–132

    Google Scholar 

  10. Chartrand G, Eroh L, Johnson MA, Oellermann O (2000) Resolvability in graphs and the metric dimension of a graph. Discret Appl Math 105:99–113

    MathSciNet  Google Scholar 

  11. Johnson M (1993) Structure-activity maps for visualizing the graph variables arising in drug design. J Biopharm Stat 3(2):203–236

    MathSciNet  Google Scholar 

  12. Manuel P, Abd-El-Barr MI, Rajasingh I, Rajan B (2008) An efficient representation of Benes networks and its applications. J Discrete Alg 6(1):11–19

    MathSciNet  Google Scholar 

  13. Díaz J, Pottonen O, Serna M, Van Leeuwen EJ (2012) On the complexity of metric dimension. European symposium on algorithms. Springer, Berlin, pp 419–430

    Google Scholar 

  14. Epstein L, Levin A, Woeginger GJ (2015) The (weighted) metric dimension of graphs: hard and easy cases. Algorithmica 72:1130–1171

    MathSciNet  Google Scholar 

  15. Rajan B, Rajasingh I, Cynthia JA, Manuel P (2014) Metric dimension of directed graphs. Int J Comput Math 91(7):1397–1406

    MathSciNet  Google Scholar 

  16. Foucaud F, Mertzios GB, Naserasr R, Parreau A, Valicov P (2017) Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity. Algorithmica 78:914–944

    MathSciNet  Google Scholar 

  17. Liu K, Abu-Ghazaleh N (2006) Virtual coordinates with backtracking for void traversal in geographic routing. Lect Notes Comput Sci 4104:46–59

    Google Scholar 

  18. Söderberg S, Shapiro HS (1963) A combinatory detection problem. Am Math Mon 70(10):1066–1070

    MathSciNet  Google Scholar 

  19. Sebő A, Tannier E (2004) On metric generators of graphs. Math Oper Res 29(2):383–393

    MathSciNet  Google Scholar 

  20. Ahmad S, Chaudhry MA, Javaid I, Salman M (2013) On the metric dimension of generalized Petersen graphs. Quaest Math 36(3):421–435

    MathSciNet  Google Scholar 

  21. Rajan B, Rajasingh I, Monica MC, Manuel P (2008) Metric dimension of enhanced hypercube networks. J Comb Math Comb Comput 67:5–15

    MathSciNet  Google Scholar 

  22. Manuel P, Rajan B, Rajasingh I, Monica MC (2008) On minimum metric dimension of honeycomb networks. J Discrete Alg 6(1):20–27

    MathSciNet  Google Scholar 

  23. Rajan B, Rajasingh I, Gopal PV, Monica MC (2014) Minimum metric dimension of Illiac networks. Ars Combin 117:95–103

    MathSciNet  Google Scholar 

  24. Behtoei A, Davoodi A, Jannesari M, Omoomi B (2017) A characterization of some graphs with metric dimension two. Discrete Math Alg Appl 9(2):1750027

    MathSciNet  Google Scholar 

  25. Yero IG, Estrada-Moreno A, Rodríguez-Velázquez JA (2017) Computing the \(k\)-metric dimension of graphs. Appl Math Comput 300:60–69

    MathSciNet  Google Scholar 

  26. Prabhu S, Deepa S, Arulperumjothi M, Susilowati L, Liu JB (2022) Resolving-power domination number of probabilistic neural networks. J Intell Fuzzy Syst 43(5):6253–6263

    Google Scholar 

  27. Umilasari R, Susilowati L, Slamin, Prabhu S (2022) On the dominant local metric dimension of corona product graphs. IAENG Int J Appl Math 52(4):1–7

    Google Scholar 

  28. Alfarisi R, Susilowati L, Dafik D, Prabhu S (2023) Local multiset dimension of amalgamation graphs. F1000Research 12:95

    Google Scholar 

  29. Cynthia VJA, Ramya M, Prabhu S (2023) Local metric dimension of certain classes of circulant networks. J Adv Comput Intell Intell Informat 27(4):554–560

    Google Scholar 

  30. Prabhu S, Sagaya Rani Jeba D, Arulperumjothi M, Klavžar S (2023) Metric dimension of irregular convex triangular networks. AKCE Int J Graphs Combin. https://doi.org/10.1080/09728600.2023.2280799

    Article  Google Scholar 

  31. Hernando C, Mora M, Slater PJ, Wood DR (2008) Fault-tolerant metric dimension of graphs. In: Convexity in discrete structures in Ramanujan mathematical society lecture notes, vol 5, pp 81–85

  32. Simić A, Bogdanović M, Maksimović Z, Milošević J (2018) Fault-tolerant metric dimension problem: a new integer linear programming formulation and exact formula for grid graphs. Kragujevac J Math 42(4):495–503

    MathSciNet  Google Scholar 

  33. Basak M, Saha L, Das GK, Tiwary K (2020) Fault-tolerant metric dimension of circulant graphs \(C_n(1, 2, 3)\). Theoret Comput Sci 817:66–79

    MathSciNet  Google Scholar 

  34. Saha L, Basak M, Tiwary K (2022) All metric bases and fault-tolerant metric dimension for square of grid. Opuscula Math 42(1):93–111

    MathSciNet  Google Scholar 

  35. Koam ANA, Ahmad A, Abdelhag ME, Azeem M (2021) Metric and fault-tolerant metric dimension of hollow coronoid. IEEE Access 2021:81527–81534

    Google Scholar 

  36. Wang H, Azeem M, Nadeem MF, Ur-Rehman A, Aslam A (2021) On fault-tolerant resolving sets of some families of ladder networks. Complexity 2021:9939559

    Google Scholar 

  37. Raza H, Hayat S, Pan XF (2018) On the fault-tolerant metric dimension of convex polytopes. Appl Math Comput 339:172–185

    MathSciNet  Google Scholar 

  38. Prabhu S, Manimozhi V, Arulperumjothi M, Klavžar S (2022) Twin vertices in fault-tolerant metric sets and fault-tolerant metric dimension of multistage interconnection networks. Appl Math Comput 420:126897

    MathSciNet  Google Scholar 

  39. Hayat S, Khan A, Malik MYH, Imran M, Siddiqui MK (2020) Fault-tolerant metric dimension of interconnection networks. IEEE Access 8:145435–145445

    Google Scholar 

  40. Arulperumjothi M, Klavžar S, Prabhu S (2023) Redefining fractal cubic networks and determining their metric dimension and fault-tolerant metric dimension. Appl Math Comput 452:128037

    MathSciNet  Google Scholar 

  41. Lüdtke D, Tutsch D (2009) The modeling power of CINSim: performance evaluation of interconnection networks. Comput Netw 53:1274–1288

    Google Scholar 

  42. Saad Y, Schultz MH (1988) Topological properties of hypercubes. IEEE Trans Comput 37(7):867–872

    Google Scholar 

  43. Chan TF, Saad Y (1986) Multigrid algorithms on the hypercube multiprocessor. IEEE Trans Comput 35(11):969–977

    Google Scholar 

  44. Hwang K, Ghosh J (1987) Hypernet: a communication-efficient architecture for constructing massively parallel computers. IEEE Trans Comput 36(12):1450–1466

    Google Scholar 

  45. Xu J (2013) Topological structures and analysis of interconnection networks, vol 7. Springer, Berlin

    Google Scholar 

  46. Ohring SR, Ibel M, Das SK, Kumar MJ (1995) On generalized fat trees. In: Proceedings of 9th International Parallel Processing Symposium, pp 37–44

  47. Leiserson CE (1985) Fat-trees: universal networks for hardware-efficient supercomputing. IEEE Trans Comput 34(10):892–901

    Google Scholar 

  48. Frank S, Burkhardt H, Rothnie J (1993) The KSR 1: bridging the gap between shared memory and MPPs. Digest of Papers. Compcon Spring, pp 285–294

  49. Bay P, Bilardi G (1990) Deterministic on-line routing on area-universal networks. In: Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science, pp 297–306

  50. Greenberg RI, Leiserson CE (1989) Randomized routing on fat trees. JAI Press 5:345–374

    Google Scholar 

  51. Prabhu S, Flora T, Arulperumjothi M (2018) On independent resolving number of TiO\(_2 [m, n]\) nanotubes. J Intell Fuzzy Syst 35(6):6421–6425

    Google Scholar 

  52. Rajan B, William A, Prabhu S (2015) On certain resolving parameters of tree derived architectures. J Comb Math Comb Comput 92:233–242

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Rajalakshmi Engineering College, Thandalam, Chennai, 602105, India

    S. Prabhu

  2. Department of Mathematics, Panimalar Engineering College, Chennai, 600123, India

    V. Manimozhi

  3. The Czech Academy of Sciences, Institute of Computer Science, Pod Vodárenskou věží 2, 182 07, Prague, Czech Republic

    Akbar Davoodi

  4. Department of Applied Mathematics and Statistics, Technical University of Cartagena, Cartagena, Spain

    Juan Luis García Guirao

Authors
  1. S. Prabhu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. V. Manimozhi
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Akbar Davoodi
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Juan Luis García Guirao
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to S. Prabhu.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of 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

Prabhu, S., Manimozhi, V., Davoodi, A. et al. Fault-tolerant basis of generalized fat trees and perfect binary tree derived architectures. J Supercomput 80, 15783–15798 (2024). https://doi.org/10.1007/s11227-024-06053-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11227-024-06053-5

Keywords

Mathematics Subject Classification