Quasi-Periodicity in Streams

In this work, we show two streaming algorithms for computing the length of the shortest cover of a string of length n. We start by showing a two-pass algorithm that uses O(log2 n) space and then show a one-pass streaming algorithm that uses O( √ n logn) space. Both algorithms run in near-linear time. The algorithms are randomized and compute the answer incorrectly with probability inverse-polynomial in n. We also show that there is no sublinear-space streaming algorithm for computing the length of the shortest seed of a string. 2012 ACM Subject Classification Theory of computation → Pattern matching

[1]  Wojciech Rytter,et al.  Efficient seed computation revisited , 2013, Theor. Comput. Sci..

[2]  Zvi Galil,et al.  Real-Time Streaming String-Matching , 2014, TALG.

[3]  Mohammad Sohel Rahman,et al.  Computing covers using prefix tables , 2014, Discret. Appl. Math..

[4]  Ely Porat,et al.  Approximate Cover of Strings , 2017, CPM.

[5]  Ely Porat,et al.  Exact and Approximate Pattern Matching in the Streaming Model , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[6]  Funda Ergün,et al.  Periodicity in Data Streams with Wildcards , 2018, CSR.

[7]  Costas S. Iliopoulos,et al.  Algorithms for Computing the λ-regularities in Strings , 2008 .

[8]  Ondřej Guth.  On approximate enhanced covers under Hamming distance , 2020, Discret. Appl. Math..

[9]  William F. Smyth,et al.  A Correction to "An Optimal Algorithm to Compute all the Covers of a String" , 1995, Inf. Process. Lett..

[10]  William F. Smyth,et al.  String covering with optimal covers , 2018, J. Discrete Algorithms.

[11]  Ziv Bar-Yossef,et al.  Information theory methods in communication complexity , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[12]  Funda Ergün,et al.  Streaming Periodicity with Mismatches , 2017, APPROX-RANDOM.

[13]  Andrzej Ehrenfeucht,et al.  Efficient Detection of Quasiperiodicities in Strings , 1993, Theor. Comput. Sci..

[14]  William F. Smyth,et al.  Frequency Covers for Strings , 2018, Fundam. Informaticae.

[15]  H. Wilf,et al.  Uniqueness theorems for periodic functions , 1965 .

[16]  Wojciech Rytter,et al.  Efficient Algorithms for Shortest Partial Seeds in Words , 2014, CPM.

[17]  Richard M. Karp,et al.  Efficient Randomized Pattern-Matching Algorithms , 1987, IBM J. Res. Dev..

[18]  Moshe Lewenstein,et al.  Can We Recover the Cover? , 2017, CPM.

[19]  Dany Breslauer,et al.  An On-Line String Superprimitivity Test , 1992, Inf. Process. Lett..

[20]  Yin Li,et al.  Computing the Cover Array in Linear Time , 2001, Algorithmica.

[21]  Costas S. Iliopoulos,et al.  New complexity results for the k-covers problem , 2011, Inf. Sci..

[22]  Costas S. Iliopoulos,et al.  Optimal Superprimitivity Testing for Strings , 1991, Inf. Process. Lett..

[23]  Wojciech Rytter,et al.  Fast Algorithm for Partial Covers in Words , 2014, Algorithmica.

[24]  Costas S. Iliopoulos,et al.  Computing the λ-Seeds of a String , 2006 .

[25]  Richard Cole,et al.  The Complexity of the Minimum k-Cover Problem , 2005, J. Autom. Lang. Comb..

[26]  Costas S. Iliopoulos,et al.  Enhanced string covering , 2013, Theor. Comput. Sci..

[27]  Maxime Crochemore,et al.  Quasiperiodicities in Fibonacci strings , 2012, Ars Comb..

[28]  Robert Krauthgamer,et al.  The Sketching Complexity of Pattern Matching , 2004, APPROX-RANDOM.

[29]  Jeong Seop Sim,et al.  Approximate Seeds of Strings , 2003, Stringology.

[30]  Maxime Crochemore,et al.  On left and right seeds of a string , 2012, J. Discrete Algorithms.

[31]  Costas S. Iliopoulos,et al.  Two strings at Hamming distance 1 cannot be both quasiperiodic , 2017, Inf. Process. Lett..

[32]  Mohammad Sohel Rahman,et al.  Enhanced Covers of Regular and Indeterminate Strings Using Prefix Tables , 2015, J. Autom. Lang. Comb..

[33]  Ely Porat,et al.  Quasi-Periodicity Under Mismatch Errors , 2018, CPM.

[34]  Funda Ergün,et al.  Periodicity in Streams , 2010, APPROX-RANDOM.

[35]  William F. Smyth,et al.  Computing the covers of a string in linear time , 1994, SODA '94.