Document Open Access Logo

Generalized Fibonacci Cubes Based on Swap and Mismatch Distance

Authors Marcella Anselmo , Giuseppa Castiglione , Manuela Flores , Dora Giammarresi , Maria Madonia , Sabrina Mantaci

PDF HTML 5

Files

Thumbnail PDF
PDF
OASIcs.Grossi.5.pdf
  • Filesize: 0.91 MB
  • 14 pages
HTML5 Logo HTML (experimental)
OASIcs.Grossi.5.html

Document Identifiers

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Formal languages and automata theory
Keywords
  • Swap and mismatch distance
  • Isometric words
  • Hypercube

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

The hypercube of dimension n is the graph with 2ⁿ vertices associated to all binary words of length n and edges connecting pairs of vertices with Hamming distance equal to 1. Here, an edit distance based on swaps and mismatches is considered and referred to as tilde-distance. Accordingly, the tilde-hypercube is defined, with edges linking words having tilde-distance equal to 1. The focus is on the subgraphs of the tilde-hypercube obtained by removing all vertices having a given word as factor. If the word is 11, then the subgraph is called tilde-Fibonacci cube; in the case of a generic word, it is called generalized tilde-Fibonacci cube. The paper surveys recent results on the definition and characterization of those words that define generalized tilde-Fibonacci cubes that are isometric subgraphs of the tilde-hypercube. Finally, a special attention is given to the study of the tilde-Fibonacci cubes.

Cite As Get BibTex

Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci. Generalized Fibonacci Cubes Based on Swap and Mismatch Distance. In From Strings to Graphs, and Back Again: A Festschrift for Roberto Grossi's 60th Birthday. Open Access Series in Informatics (OASIcs), Volume 132, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/OASIcs.Grossi.5

Author Details

Marcella Anselmo
  • Dipartimento di Informatica, Università di Salerno, Italy
Giuseppa Castiglione
  • Dipartimento di Matematica e Informatica, Università di Palermo, Italy
Manuela Flores
  • Dipartimento di Bioscienze e Territorio, Università del Molise, Campobasso, Italy
Dora Giammarresi
  • Dipartimento di Matematica, Università Roma "Tor Vergata", Italy
Maria Madonia
  • Dipartimento di Matematica e Informatica, Università di Catania, Italy
Sabrina Mantaci
  • Dipartimento di Matematica e Informatica, Università di Palermo, Italy

Funding

Partially supported by INdAM-GNCS Project 2025 - CUP E53C24001950001, FARB Project ORSA240597 of University of Salerno, TEAMS Project and PNRR MUR Project PE0000013-FAIR University of Catania, FFR fund University of Palermo, MUR Excellence Department Project MatMod@TOV, CUP E83C23000330006, awarded to the Department of Mathematics, University of Rome Tor Vergata, “ACoMPA – Algorithmic and Combinatorial Methods for Pangenome Analysis” (CUP B73C24001050001) funded by the NextGeneration EU programme PNRR ECS00000017 Tuscany Health Ecosystem (Spoke 6).

References

  1. Amihood Amir, Estrella Eisenberg, and Ely Porat. Swap and mismatch edit distance. Algorithmica, 45(1):109-120, 2006. URL: https://doi.org/10.1007/978-3-540-30140-0_4.
  2. Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci. Hypercubes and isometric words based on swap and mismatch distance. In Descriptional Complexity of Formal Systems. DCFS23, volume 13918 of Lect. Notes Comput. Sci., pages 21-35. Springer Nature, 2023. URL: https://doi.org/10.1007/978-3-031-34326-1_2.
  3. Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci. Isometric words based on swap and mismatch distance. In Developments in Language Theory. DLT23, volume 13911 of Lect. Notes Comput. Sci., pages 23-35. Springer Nature, 2023. URL: https://doi.org/10.1007/978-3-031-33264-7_3.
  4. Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci. Isometric sets of words and generalizations of the Fibonacci cubes. In Computability in Europe. CIE24, volume LNCS 14773 of Lect. Notes Comput. Sci., pages 1-14. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-64309-5_35.
  5. Marcella Anselmo, Giuseppa Castiglione, Manuela Flores, Dora Giammarresi, Maria Madonia, and Sabrina Mantaci. Characterization of isometric words based on swap and mismatch distance. International Journal of Foundations of Computer Science, 36(03):221-245, 2025. URL: https://doi.org/10.1142/S0129054125430051.
  6. Marcella Anselmo, Manuela Flores, and Maria Madonia. Fun slot machines and transformations of words avoiding factors. In Fun with Algorithms, volume 226 of LIPIcs, pages 4:1-4:15, 2022. URL: https://doi.org/10.4230/LIPIcs.FUN.2022.4.
  7. Marcella Anselmo, Manuela Flores, and Maria Madonia. On k-ary n-cubes and isometric words. Theor. Comput. Sci., 938:50-64, 2022. URL: https://doi.org/10.1016/j.tcs.2022.10.007.
  8. Marcella Anselmo, Manuela Flores, and Maria Madonia. Density of Ham- and Lee- non-isometric k-ary words. In ICTCS'23 Italian Conference on Theor. Comput. Sci., volume 3587 of CEUR Workshop Proceedings, pages 116-128, 2023. URL: https://ceur-ws.org/Vol-3587/3914.pdf.
  9. Marcella Anselmo, Manuela Flores, and Maria Madonia. Computing the index of non-isometric k-ary words with Hamming and Lee distance. Computability, IOS press, 13(3-4):199-222, 2024. URL: https://doi.org/10.3233/COM-230441.
  10. Marcella Anselmo, Manuela Flores, and Maria Madonia. Density of k-ary words with 0, 1, 2 - error overlaps. Theor. Comput. Sci., 1025:114958, 2025. URL: https://doi.org/10.1016/j.tcs.2024.114958.
  11. Jernej Azarija, Sandi Klavžar, Jaehun Lee, Jay Pantone, and Yoomi Rho. On isomorphism classes of generalized Fibonacci cubes. Eur. J. Comb., 51:372-379, 2016. URL: https://doi.org/10.1016/j.ejc.2015.05.011.
  12. Marie-Pierre Béal and Maxime Crochemore. Checking whether a word is Hamming-isometric in linear time. Theor. Comput. Sci., 933:55-59, 2022. URL: https://doi.org/10.1016/j.tcs.2022.08.032.
  13. Laxmi N. Bhuyan and Dharma P. Agrawal. Generalized hypercube and hyperbus structures for a computer network. IEEE Transactions on Computers, C-33(4):323-333, 1984. URL: https://doi.org/10.1109/TC.1984.1676437.
  14. Fred Buckley. Self-centered graphs. Ann. New York Acad. Sci, 576:71-78, 1989. URL: https://doi.org/10.1111/j.1749-6632.1989.tb16384.x.
  15. Sergio Cabello, David Eppstein, and Sandi Klavzar. The Fibonacci dimension of a graph. Electron. J. Comb., 18(1), 2011. URL: https://doi.org/10.37236/542.
  16. Giuseppa Castiglione, Manuela Flores, and Dora Giammarresi. Isometric words and edit distance: Main notions and new variations. In Cellular Automata and Discrete Complex Systems. AUTOMATA 2023, volume 14152 of Lect. Notes Comput. Sci., pages 6-13. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-42250-8_1.
  17. W.J. Dally. Performance analysis of k-ary n-cube interconnection networks. IEEE Transactions on Computers, 39(6):775-785, 1990. URL: https://doi.org/10.1109/12.53599.
  18. Ahmed El-Amawy and Shaharam Latifi. Properties and performance of folded hypercubes. IEEE Transactions on Parallel and Distributed Systems, 2(1):31-42, 1991. URL: https://doi.org/10.1109/71.80187.
  19. Simone Faro and Arianna Pavone. An efficient skip-search approach to swap matching. Comput. J., 61(9):1351-1360, 2018. URL: https://doi.org/10.1093/comjnl/bxx123.
  20. Frank Harary, John P. Hayes, and Horng J. Wu. A survey of the theory of hypercube graphs. Comput. Math. Appl., 15(4):277-289, 1988. URL: https://doi.org/10.1016/0898-1221(88)90213-1.
  21. Wen-Jing Hsu. Fibonacci cubes-a new interconnection topology. IEEE Transactions on Parallel and Distributed Systems, 4(1):3-12, 1993. URL: https://doi.org/10.1109/71.205649.
  22. Aleksandar Ilić, Sandi Klavžar, and Yoomi Rho. Generalized Fibonacci cubes. Discrete Math., 312(1):2-11, 2012. URL: https://doi.org/10.1016/j.disc.2011.02.015.
  23. Aleksandar Ilić, Sandi Klavžar, and Yoomi Rho. The index of a binary word. Theor. Comput. Sci., 452:100-106, 2012. URL: https://doi.org/10.1016/j.tcs.2012.05.025.
  24. Sandi Klavžar. Structure of Fibonacci cubes: A survey. J. Comb. Optim., 25(4):505-522, 2013. URL: https://doi.org/10.1007/s10878-011-9433-z.
  25. Sandi Klavžar and Sergey V. Shpectorov. Asymptotic number of isometric generalized Fibonacci cubes. Eur. J. Comb., 33(2):220-226, 2012. URL: https://doi.org/10.1016/j.ejc.2011.10.001.
  26. Sandi Klavžar. On median nature and enumerative properties of Fibonacci-like cubes. Discrete Mathematics, 299(1):145-153, 2005. URL: https://doi.org/10.1016/j.disc.2004.02.023.
  27. Sandi Klavžar and Petra Žigert. Fibonacci cubes are the resonance graphs of fibonaccenes. The Fibonacci Quarterly, 43(3):269-276, 2005. URL: https://doi.org/10.1080/00150517.2005.12428368.
  28. Emanuele Munarini and Norma Zagaglia Salvi. Structural and enumerative properties of the Fibonacci cubes. Discrete Math., 255(1-3):317-324, 2002. URL: https://doi.org/10.1016/S0012-365X(01)00407-1.
  29. Gonzalo Navarro. A guided tour to approximate string matching. ACM Comput. Surv., 33(1):31-88, 2001. URL: https://doi.org/10.1145/375360.375365.
  30. P. Tolstrup Nielsen. A note on bifix-free sequences (corresp.). IEEE Transactions on Information Theory, 19(5):704-706, 1973. URL: https://doi.org/10.1109/TIT.1973.1055065.
  31. N.J.A. Sloane. On-line encyclopedia of integer sequences. URL: http://oeis.org/.
  32. Nian-Feng Tzeng and Sizheng Wei. Enhanced hypercubes. IEEE Trans. Computers, 40(3):284-294, 1991. URL: https://doi.org/10.1109/12.76405.
  33. Robert A. Wagner. On the complexity of the extended string-to-string correction problem. In Symposium on Theory of Computing, pages 218-223, 1975. URL: https://doi.org/10.1145/800116.803771.
  34. Robert A. Wagner and Michael J. Fischer. The string-to-string correction problem. Journal of the ACM, 21(1):168-173, 1974. URL: https://doi.org/10.1145/321796.321811.
  35. Jianxin Wei. The structures of bad words. Eur. J. Comb., 59:204-214, 2017. URL: https://doi.org/10.1016/j.ejc.2016.05.003.
  36. Jianxin Wei. Proof of a conjecture on 2-isometric words. Theor. Comput. Sci., 855:68-73, 2021. URL: https://doi.org/10.1016/j.tcs.2020.11.026.
  37. Jianxin Wei, Yujun Yang, and Guangfu Wang. Circular embeddability of isometric words. Discret. Math., 343(10):112024, 2020. URL: https://doi.org/10.1016/j.disc.2020.112024.
  38. Jianxin Wei, Yujun Yang, and Xuena Zhu. A characterization of non-isometric binary words. Eur. J. Comb., 78:121-133, 2019. URL: https://doi.org/10.1016/j.ejc.2019.02.001.
  39. Jianxin Wei and Heping Zhang. Proofs of two conjectures on generalized Fibonacci cubes. Eur. J. Comb., 51:419-432, 2016. URL: https://doi.org/10.1016/j.ejc.2015.07.018.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact