International Symposium on String Processing and Information Retrieval

SPIRE 2015: String Processing and Information Retrieval pp 246-257

# Computing the Longest Unbordered Substring

• Paweł Gawrychowski
• Gregory Kucherov
• Benjamin Sach
• Tatiana Starikovskaya
Conference paper

DOI: 10.1007/978-3-319-23826-5_24

Volume 9309 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Gawrychowski P., Kucherov G., Sach B., Starikovskaya T. (2015) Computing the Longest Unbordered Substring. In: Iliopoulos C., Puglisi S., Yilmaz E. (eds) String Processing and Information Retrieval. SPIRE 2015. Lecture Notes in Computer Science, vol 9309. Springer, Cham

## Abstract

A substring of a string is unbordered if its only border is the empty string. The study of unbordered substrings goes back to the paper of Ehrenfeucht and Silberger [Discr. Math 26 (1979)]. The main focus of their and subsequent papers was to elucidate the relationship between the longest unbordered substring and the minimal period of strings. In this paper, we consider the algorithmic problem of computing the longest unbordered substring of a string. The problem was introduced recently by G. Kucherov et al. [CPM (2015)], where the authors showed that the average-case running time of the simple, border-array based algorithm can be bounded by $$\mathcal {O}(\max \{n, n^2/\sigma ^4\})$$ for $$\sigma$$ being the size of the alphabet. (The worst-case running time remained $$\mathcal {O}(n^2)$$.) Here we propose two algorithms, both presenting substantial theoretical improvements to the result of [11]. The first algorithm has $$\mathcal {O}(n \log n)$$ average-case running time and $$\mathcal {O}(n^2)$$ worst-case running time, and the second algorithm has $$\mathcal {O}(n^{1.5})$$ worst-case running time.

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© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Paweł Gawrychowski
• 1
• Gregory Kucherov
• 2
• Benjamin Sach
• 3
• Tatiana Starikovskaya
• 3
1. 1.University of WarsawWarsawPoland
2. 2.Laboratoire d’Informatique Gaspard MongeUniversité Paris-Est and CNRSChamps-sur-MarneFrance
3. 3.University of BristolBristolEngland