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Periodicity Property Testing on Strings with Wildcards

Authors Carl Barton , Panagiotis Charalampopoulos , Taha El Ghazi , Jonas Ellert , Oded Lachish , Tatiana Starikovskaya

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Subject Classification

ACM Subject Classification
  • Theory of computation → Probabilistic computation
  • Mathematics of computing → Combinatorics on words
Keywords
  • periodicity
  • property testing
  • wildcards

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Abstract

In this work, we study periodicity in strings with wildcards. A string T with at most k wildcards is called strongly (p,k)-periodic if the wildcards in T can be replaced with alphabet symbols to obtain a string with period p, and weakly (p,k)-periodic if T[i] matches T[i+p] for all i. Intuitively, both generalize to (≤ g, k)-periodicity, which is the property of being (p,k)-periodic for some p ∈ [1..g].
An ε-tester for a property 𝒫 is an algorithm that distinguishes between strings that satisfy 𝒫 and strings where one needs to change at least an ε-fraction of the symbols to obtain a string that satisfies 𝒫. We study one-sided error testers, where strings satisfying 𝒫 must always be accepted, while strings that are ε-far must be rejected with probability at least 2/3. The complexity of a tester is the worst-case number of symbols of an input of length n it must read to make the decision.
We design the following testers for p,g ≤ n/2:
1) An ε-tester for strong (p,k)-periodicity with complexity Õ_ε(1) . 
2) An ε-tester for strong (≤ g,k)-periodicity with complexity Õ_ε(√g). 
3) An ε-tester for weak (p,k)-periodicity with complexity Õ_ε(min(k, n /(k+p))). 
4) An ε-tester for weak (≤ g,k)-periodicity with complexity Õ_ε(min(k+ √{gk}, n/√k)). 
Additionally, we show a lower bound on the complexity of ε-testers for weak (≤ g,k)-periodicity, implying that our tester for weak (≤ g,k)-periodicity is optimal up to a multiplicative (ε^{-1} ln(gk))^O(1) factor for a wide range of g and k. Finally, our tester for strong (≤ g,k)-periodicity generalizes the one of [Lachish and Newman; Algorithmica 2011] for strings without wildcards, matching (up to polylogarithmic factors) the unconditional lower bound of ̃Ω(√g) in said work for constant ε.

Cite As Get BibTex

Carl Barton, Panagiotis Charalampopoulos, Taha El Ghazi, Jonas Ellert, Oded Lachish, and Tatiana Starikovskaya. Periodicity Property Testing on Strings with Wildcards. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 28:1-28:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CPM.2026.28

Author Details

Carl Barton
  • Birkbeck, University of London, UK
Panagiotis Charalampopoulos
  • King’s College London, UK
Taha El Ghazi
  • DIENS, École normale supérieure de Paris, PSL Research University, France
Jonas Ellert
  • CWI, Amsterdam, The Netherlands
Oded Lachish
  • Birkbeck, University of London, UK
Tatiana Starikovskaya
  • DIENS, École normale supérieure de Paris, PSL Research University, France

Funding

Parts of the presented ideas were conceived during a research visit funded by a Royal Society International Exchanges Award. Taha El Ghazi, Jonas Ellert, and Tatiana Starikovskaya were partially funded by grant ANR20-CE48-0001 from the French National Research Agency. Jonas Ellert was partially supported by an ERCIM Alain Bensoussan fellowship.

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