Quasi-Periodicity Under Mismatch Errors
Tracing regularities plays a key role in data analysis for various areas of science, including coding and automata theory, formal language theory, combinatorics, molecular biology and many others. Part of the scientific process is understanding and explaining these regularities. A common notion to describe regularity in a string T is a cover or quasi-period, which is a string C for which every letter of T lies within some occurrence of C. In many applications finding exact repetitions is not sufficient, due to the presence of errors. In this paper we initiate the study of quasi-periodicity persistence under mismatch errors, and our goal is to characterize situations where a given quasi-periodic string remains quasi-periodic even after substitution errors have been introduced to the string. Our study results in proving necessary conditions as well as a theorem stating sufficient conditions for quasi-periodicity persistence. As an application, we are able to close the gap in understanding the complexity of Approximate Cover Problem (ACP) relaxations studied by [Amir 2017a, Amir 2017b] and solve an open question.
Periodicity
Quasi-Periodicity
Cover
Approximate Cover
Mathematics of computing~Combinatorics on words
Theory of computation~Pattern matching
4:1-4:15
Regular Paper
Amihood
Amir
Amihood Amir
Bar-Ilan University and Johns Hopkins University, Ramat-Gan, Israel
Avivit
Levy
Avivit Levy
Shenkar College, Ramat-Gan, Israel
Ely
Porat
Ely Porat
Bar-Ilan University, Ramat-Gan, Israel
10.4230/LIPIcs.CPM.2018.4
A. Amir, E. Eisenberg, and A. Levy. Approximate periodicity. In Proc. ISAAC 2010, LNCS 6506, pages 25-36. Springer, 2010.
A. Amir, E. Eisenberg, A. Levy, E. Porat, and N. Shapira. Cycle detection and correction. ACM Transactions on Algorithms, 9(1)(13), 2012.
A. Amir, C. S. Iliopoulos, and J. Radoszewski. Two strings at hamming distance 1 cannot be both quasiperiodic. Information Processing Letters, 128:54-57, 2017.
A. Amir, A. Levy, M. Lewenstein, R. Lubin, and B. Porat. Can we recover the cover? In Proc. CPM, 2017.
A. Amir, A. Levy, R. Lubin, and E. Porat. Approximate cover of strings. In Proc. CPM, 2017.
P. Antoniou, M. Crochemore, C. S. Iliopoulos, I. Jayasekera, and G. M. Landau. Conservative string covering of indeterminate strings. In Proc. Stringology, pages 108-115, 2008.
A. Aposolico and A. Ehrenfeucht. Efficient detection of quasiperiodicities in strings. Theoret. Comput. Sci., 119:247-265, 1993.
A. Apostolico and D. Breslauer. Of periods, quasiperiods, repetitions and covers. In Proc. Structures in Logic and Computer Science, LNCS 1261, pages 236-248, 1997.
A. Apostolico, M. Farach, and C. S. Iliopoulos. Optimal superprimitivity testing for strings. Information Processing Letters, 39:17-20, 1991.
D. Breslauer. An on-line string superprimitivity test. Information Processing Letters, 44:345-347, 1992.
D. Breslauer. Testing string superprimitivity in parallel. Information Processing Letters, 49(5):235-241, 1994.
M. Christodoulakis, C. S. Iliopoulos, K. Park, and J. S. Sim. Approximate seeds of strings. Journal of Automata, Languages and Combinatorics, 10:609-626, 2005.
T. Crawford, C. S. Iliopoulos, and R. Raman. String matching techniques for musical similarity and melodic recognition. Comput. Musicol., 11:73-100, 1998.
M. Crochemore, C. S. Iliopoulos, S. P. Pissis, and G. Tischler. Cover array string reconstruction. In Proc. CPM, pages 251-259, 2010.
M. Crochemore, C. S. Iliopoulos, and H. Yu. Algorithms for computing evolutionary chains in molecular and musical sequences. In Proc. 9th Austral. Workshop on Combinatorial Algorithms, pages 172-185, 1998.
T. Flouri, C. S. Iliopoulos, T. Kociumaka, S. P. Pissis, S. J. Puglisi, W. F. Smyth, and W. Tyczynski. Enhanced string covering. Theor. Comput. Sci., 506:102-114, 2013.
O. Guth and B. Melichar. Using Finite Automata Approach for Searching Approximate Seeds of Strings, pages 347-360. Springer, 2010.
C. S. Iliopoulos and L. Mouchard. Quasiperiodicity and string covering. Theor. Comput. Sci., 218(1):205-216, 1999.
C. S. Iliopoulos and W. F. Smyth. An on-line algorithm of computing a minimum set of k-covers of a string. In Proc. 9th Australasian Workshop on Combinatorial Algorithms (AWOCA), pages 97-106, 1998.
C. S. Iliopoulus, D. W. G. Moore, and K. Park. Covering a string. Algorithmica, 16(3):288-297, 1996.
T. Kociumaka, S. P. Pissis, J. Radoszewski, W. Rytter, and T. Walen. Fast algorithm for partial covers in words. In Proc. CPM, pages 177-188, 2013.
R. M. Kolpakov and G. Kucherov. Finding approximate repetitions under hamming distance. Theor. Comput. Sci., 303:135-156, 2003.
G. M. Landau and J. P. Schmidt. An algorithm for approximate tandem repeats. In Proc. 4th Symp. Combinatorial Pattern Matching, LNCS 648, pages 120-133, 1993.
G. M. Landau, J. P. Schmidt, and D. Sokol. An algorithm for approximate tandem repeats. J. of Computational Biology, 8(1):1-18, 2001.
Y. Li and W. F. Smyth. Computing the cover array in linear time. Algorithmica, 32(1):95-106, 2002.
M. Lothaire. Combinatorics on words. Addison-Wesley, 1983.
D. Moore and W. F. Smyth. An optimal algorithm to compute all the covers of a string. Information Processing Letters, 50(5):239-246, 1994.
D. Moore and W. F. Smyth. A correction to: An optimal algorithm to compute all the covers of a string. Information Processing Letters, 54:101-103, 1995.
B. Melichar O. Guth and M. Balik. All Approximate Covers and Their Distance using Finite Automata, pages 21-26. CEUR-WS, 2009.
J. S. Sim, C. S. Iliopoulos, K. Park, and W. F. Smyth. Approximate periods of strings. Theor. Comput. Sci., 262:557-568, 2001.
W. F. Smyth. Repetitive perhaps, but certainly not boring. Theor. Comput. Sci., 249(2):343-355, 2000.
H. Zhang, Q. Guo, and C. S. Iliopoulos. Algorithms for computing the lambda-regularities in strings. Fundam. Inform., 84(1):33-49, 2008.
H. Zhang, Q. Guo, and C. S. Iliopoulos. Varieties of regularities in weighted sequences. In Proc. AAIM, LNCS 6142, pages 271-280, 2010.
Amihood Amir, Avivit Levy, and Ely Porat
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