Regularities in strings arise in various areas of science, including coding and automata theory, formal language theory, combinatorics, molecular biology and many others. A common notion to describe regularity in a string T is a cover, which is a string C for which every letter of T lies within some occurrence of C. The alignment of the cover repetitions in the given text is called a tiling. In many applications finding exact repetitions is not sufficient, due to the presence of errors. In this paper, we use a new approach for handling errors in coverable phenomena and define the approximate cover problem (ACP), in which we are given a text that is a sequence of some cover repetitions with possible mismatch errors, and we seek a string that covers the text with the minimum number of errors. We first show that the ACP is NP-hard, by studying the cover-size relaxation of the ACP, in which the requested size of the approximate cover is also given with the input string. We show this relaxation is already NP-hard. We also study another two relaxations of the ACP, which we call the partial-tiling relaxation of the ACP and the full-tiling relaxation of the ACP, in which a tiling of the requested cover is also given with the input string. A given full tiling retains all the occurrences of the cover before the errors, while in a partial tiling there can be additional occurrences of the cover that are not marked by the tiling. We show that the partial-tiling relaxation has a polynomial time complexity and give experimental evidence that the full-tiling also has polynomial time complexity. The study of these relaxations, besides shedding another light on the complexity of the ACP, also involves a deep understanding of the properties of covers, yielding some key lemmas and observations that may be helpful for a future study of regularities in the presence of errors.
|Title of host publication||28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017|
|Editors||Jakub Radoszewski, Juha Karkkainen, Jakub Radoszewski, Wojciech Rytter|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|State||Published - 1 Jul 2017|
|Event||28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017 - Warsaw, Poland|
Duration: 4 Jul 2017 → 6 Jul 2017
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017|
|Period||4/07/17 → 6/07/17|
Bibliographical notePublisher Copyright:
© Amihood Amir, Avivit Levy, Ronit Lubin, and Ely Porat.
- Approximate cover