Finding the Anticover of a String

Authors Mai Alzamel , Alessio Conte , Shuhei Denzumi, Roberto Grossi, Costas S. Iliopoulos, Kazuhiro Kurita , Kunihiro Wasa

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Author Details

Mai Alzamel
  • Department of Informatics, King’s College London, UK
  • Department of Computer Science, King Saud University, KSA
Alessio Conte
  • Dipartimento di Informatica, Università di Pisa, Italy
Shuhei Denzumi
  • Graduate School of Information Science and Technology, The University of Tokyo, Japan
Roberto Grossi
  • Dipartimento di Informatica, Università di Pisa, Italy
Costas S. Iliopoulos
  • Department of Informatics, King’s College London, UK
Kazuhiro Kurita
  • IST, Hokkaido University, Sapporo, Japan
Kunihiro Wasa
  • National Institute of Informatics, Tokyo, Japan

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Mai Alzamel, Alessio Conte, Shuhei Denzumi, Roberto Grossi, Costas S. Iliopoulos, Kazuhiro Kurita, and Kunihiro Wasa. Finding the Anticover of a String. In 31st Annual Symposium on Combinatorial Pattern Matching (CPM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 161, pp. 2:1-2:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


A k-anticover of a string x is a set of pairwise distinct factors of x of equal length k, such that every symbol of x is contained into an occurrence of at least one of those factors. The existence of a k-anticover can be seen as a notion of non-redundancy, which has application in computational biology, where they are associated with various non-regulatory mechanisms. In this paper we address the complexity of the problem of finding a k-anticover of a string x if it exists, showing that the decision problem is NP-complete on general strings for k ≥ 3. We also show that the problem admits a polynomial-time solution for k=2. For unbounded k, we provide an exact exponential algorithm to find a k-anticover of a string of length n (or determine that none exists), which runs in O*(min {3^{(n-k)/3)}, ((k(k+1))/2)^{n/(k+1)) time using polynomial space.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
  • Anticover
  • String algorithms
  • Stringology
  • NP-complete


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