License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ISAAC.2016.34
URN: urn:nbn:de:0030-drops-68041
URL: https://drops.dagstuhl.de/opus/volltexte/2016/6804/
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### Space-Time Trade-Offs for the Shortest Unique Substring Problem

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### Abstract

Given a string X[1, n] and a position k in [1, n], the Shortest Unique Substring of X covering k, denoted by S_k, is a substring X[i, j] of X which satisfies the following conditions: (i) i leq k leq j, (ii) i is the only position where there is an occurrence of X[i, j], and (iii) j - i is minimized. The best-known algorithm [Hon et al., ISAAC 2015] can find S k for all k in [1, n] in time O(n) using the string X and additional 2n words of working space. Let tau be a given parameter. We present the following new results. For any given k in [1, n], we can compute S_k via a deterministic algorithm in O(n tau^2 log n tau) time using X and additional O(n/tau) words of working space. For every k in [1, n], we can compute S_k via a deterministic algorithm in O(n tau^2 log n/tau) time using X and additional O(n/tau) words and 4n + o(n) bits of working space. For both problems above, we present an O(n tau log^{c+1} n)-time randomized algorithm that uses n/ log c n words in addition to that mentioned above, where c geq 0 is an arbitrary constant. In this case, the reported string is unique and covers k, but with probability at most n^{-O(1)} , may not be the shortest. As a consequence of our techniques, we also obtain similar space-and-time tradeoffs for a related problem of finding Maximal Unique Matches of two strings [Delcher et al., Nucleic Acids Res. 1999].

### BibTeX - Entry

```@InProceedings{ganguly_et_al:LIPIcs:2016:6804,
author =	{Arnab Ganguly and Wing-Kai Hon and Rahul Shah and Sharma V. Thankachan},
title =	{{Space-Time Trade-Offs for the Shortest Unique Substring Problem}},
booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
pages =	{34:1--34:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-026-2},
ISSN =	{1868-8969},
year =	{2016},
volume =	{64},
editor =	{Seok-Hee Hong},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address =	{Dagstuhl, Germany},
URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6804},
URN =		{urn:nbn:de:0030-drops-68041},
doi =		{10.4230/LIPIcs.ISAAC.2016.34},
annote =	{Keywords: Suffix Tree, Sparsification, Rabin-Karp Fingerprint, Probabilistic z-Fast Trie, Succinct Data-Structures}
}
```

 Keywords: Suffix Tree, Sparsification, Rabin-Karp Fingerprint, Probabilistic z-Fast Trie, Succinct Data-Structures Collection: 27th International Symposium on Algorithms and Computation (ISAAC 2016) Issue Date: 2016 Date of publication: 07.12.2016

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