Papers
Topics
Authors
Recent
Search
2000 character limit reached

String 2-Covers with No Length Restrictions

Published 19 May 2024 in cs.DS | (2405.11475v1)

Abstract: A $\lambda$-cover of a string $S$ is a set of strings ${C_i}_1\lambda$ such that every index in $S$ is contained in an occurrence of at least one string $C_i$. The existence of a $1$-cover defines a well-known class of quasi-periodic strings. Quasi-periodicity can be decided in linear time, and all $1$-covers of a string can be reported in linear time plus the size of the output. Since in general it is NP-complete to decide whether a string has a $\lambda$-cover, the natural next step is the development of efficient algorithms for $2$-covers. Radoszewski and Straszy\'nski [ESA 2020] analysed the particular case where the strings in a $2$-cover must be of the same length. They provided an algorithm that reports all such $2$-covers of $S$ in time near-linear in $|S|$ and in the size of the output. In this work, we consider $2$-covers in full generality. Since every length-$n$ string has $\Omega(n2)$ trivial $2$-covers (every prefix and suffix of total length at least $n$ constitute such a $2$-cover), we state the reporting problem as follows: given a string $S$ and a number $m$, report all $2$-covers ${C_1,C_2}$ of $S$ with length $|C_1|+|C_2|$ upper bounded by $m$. We present an $\tilde{O}(n + Output)$ time algorithm solving this problem, with Output being the size of the output. This algorithm admits a simpler modification that finds a $2$-cover of minimum length. We also provide an $\tilde{O}(n)$ time construction of a $2$-cover oracle which, given two substrings $C_1,C_2$ of $S$, reports in poly-logarithmic time whether ${C_1,C_2}$ is a $2$-cover of $S$.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (30)
  1. Computing covers using prefix tables. Discret. Appl. Math., 212:2–9, 2016. URL: https://doi.org/10.1016/j.dam.2015.05.019, doi:10.1016/J.DAM.2015.05.019.
  2. Can we recover the cover? Algorithmica, 81(7):2857–2875, 2019. URL: https://doi.org/10.1007/s00453-019-00559-8, doi:10.1007/S00453-019-00559-8.
  3. Approximate cover of strings. Theor. Comput. Sci., 793:59–69, 2019. URL: https://doi.org/10.1016/j.tcs.2019.05.020, doi:10.1016/J.TCS.2019.05.020.
  4. Efficient detection of quasiperiodicities in strings. Theoretical Computer Science, 119(2):247–265, 1993. URL: https://www.sciencedirect.com/science/article/pii/030439759390159Q, doi:10.1016/0304-3975(93)90159-Q.
  5. The "runs" theorem. SIAM J. Comput., 46(5):1501–1514, 2017. doi:10.1137/15M1011032.
  6. Indexing weighted sequences: Neat and efficient. Inf. Comput., 270, 2020. URL: https://doi.org/10.1016/j.ic.2019.104462, doi:10.1016/J.IC.2019.104462.
  7. The subtree max gap problem with application to parallel string covering. Inf. Comput., 123(1):127–137, 1995. URL: https://doi.org/10.1006/inco.1995.1162, doi:10.1006/INCO.1995.1162.
  8. Bernard Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM J. Comput., 17(3):427–462, 1988. doi:10.1137/0217026.
  9. The complexity of the minimum k-cover problem. Journal of Automata, Languages and Combinatorics, 10(5-6):641–653, 2005.
  10. Squares, cubes, and time-space efficient string searching. Algorithmica, 13(5):405–425, 1995.
  11. Experimental evaluation of algorithms for computing quasiperiods. Theoretical Computer Science, 854:17–29, 2021.
  12. Making data structures persistent. J. Comput. Syst. Sci., 38(1):86–124, 1989. doi:10.1016/0022-0000(89)90034-2.
  13. Optimal square detection over general alphabets. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 5220–5242. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch189, doi:10.1137/1.9781611977554.CH189.
  14. Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc., 16:109–114, 1965.
  15. Data structures and algorithms for approximate string matching. Journal of Complexity, 4(1):33–72, 1988. doi:10.1016/0885-064X(88)90008-8.
  16. Quasi-periodicity in streams. In Nadia Pisanti and Solon P. Pissis, editors, 30th Annual Symposium on Combinatorial Pattern Matching, CPM 2019, June 18-20, 2019, Pisa, Italy, volume 128 of LIPIcs, pages 22:1–22:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.CPM.2019.22, doi:10.4230/LIPICS.CPM.2019.22.
  17. Computing the λ𝜆\lambdaitalic_λ-covers of a string. Information Sciences, 177(19):3957–3967, 2007.
  18. Fast pattern matching in strings. SIAM J. Comput., 6(2):323–350, 1977.
  19. Fast algorithm for partial covers in words. Algorithmica, 73(1):217–233, 2015. URL: https://doi.org/10.1007/s00453-014-9915-3, doi:10.1007/S00453-014-9915-3.
  20. Optimal data structure for internal pattern matching queries in a text and applications. CoRR, abs/1311.6235, 2013. URL: http://arxiv.org/abs/1311.6235, arXiv:1311.6235.
  21. Fast string matching with k differences. Journal of Computer and System Sciences, 37(1):63–78, 1988. doi:10.1016/0022-0000(88)90045-1.
  22. Suffix tree-based linear algorithms for multiple prefixes, single suffix counting and listing problems. CoRR, abs/2203.16908, 2022. URL: https://doi.org/10.48550/arXiv.2203.16908, arXiv:2203.16908, doi:10.48550/ARXIV.2203.16908.
  23. An optimal algorithm to compute all the covers of a string. Information Processing Letters, 50(5):239–246, 1994. URL: https://www.sciencedirect.com/science/article/pii/002001909400045X, doi:10.1016/0020-0190(94)00045-X.
  24. A correction to “an optimal algorithm to compute all the covers of a string”. Information Processing Letters, 54(2):101–103, 1995. URL: https://www.sciencedirect.com/science/article/pii/002001909400235Q, doi:10.1016/0020-0190(94)00235-Q.
  25. An output-sensitive algorithm for the minimization of 2-dimensional string covers. In T. V. Gopal and Junzo Watada, editors, Theory and Applications of Models of Computation - 15th Annual Conference, TAMC 2019, Kitakyushu, Japan, April 13-16, 2019, Proceedings, volume 11436 of Lecture Notes in Computer Science, pages 536–549. Springer, 2019. doi:10.1007/978-3-030-14812-6\_33.
  26. Efficient computation of 2-covers of a string. In 28th Annual European Symposium on Algorithms (ESA 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020.
  27. Counting palindromes in substrings. In Gabriele Fici, Marinella Sciortino, and Rossano Venturini, editors, String Processing and Information Retrieval - 24th International Symposium, SPIRE 2017, Proceedings, volume 10508 of Lecture Notes in Computer Science, pages 290–303. Springer, 2017.
  28. Smyth. Computing the cover array in linear time. Algorithmica, 32:95–106, 2002.
  29. Dan E. Willard. New data structures for orthogonal range queries. SIAM J. Comput., 14(1):232–253, 1985. doi:10.1137/0214019.
  30. Algorithms for computing the λ𝜆\lambdaitalic_λ-regularities in strings. Fundamenta Informaticae, 84(1):33–49, 2008.

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Authors (3)

  1. Itai Boneh 
  2. Shay Golan 
  3. Arseny Shur 

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 1 tweet with 0 likes about this paper.

Sign Up for Free