Lyndon Words and Short Superstrings
Lyndon Words and Short Superstrings
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林登词和短超弦
DOI：
10.1137/1.9781611973105.69
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
M. Mucha
中科院分区：
文献类型：
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作者：
M. Mucha
在最短的超声问题中，我们给出了一组字符串{s1，...，sk}，并希望找到一个包含所有SI作为子字符串且最小长度的字符串。这是一个经典的近似问题，最著名的近似因子是2 1/2，由Sweedyk [19]在1999年给出。此后，没有进行改进，Howereerer的另外两种方法产生了2 1/2 appproximation Algorithms的其他方法。由Kaplan等人提出。 [10]以及最近的Paluch等人。 [16] ---基于对最大不对称TSP路径（MAX-ATSP-Path）的减少和Breslauer等人的结构结果。 [5]。
在本文中，我们给出了一种达到2 11/23的近似值的算法，突破了2 1/2的长期结合。我们将最短苏格拉链的标准减少到Max-ATSP路径。新的，有些令人惊讶的算法想法是通过使用以下方式将获得的两种解决方案中的更好/2-Approximation算法。为了证明这确实会导致改善，我们进一步发展了弦绳重叠的理论，从而扩展了Breslauer等人的结果。 [5]。该理论基于Breslauer等人使用的lyndon单词的新颖用途，作为通用不良旋转和关键因素化的替代。
In the shortest superstring problem, we are given a set of strings {s1,...,sk} and want to find a string that contains all si as substrings and has minimum length. This is a classical problem in approximation and the best known approximation factor is 2 1/2, given by Sweedyk [19] in 1999. Since then no improvement has been made, howerever two other approaches yielding a 2 1/2-approximation algorithms have been proposed by Kaplan et al. [10] and recently by Paluch et al. [16] --- both based on a reduction to maximum asymmetric TSP path (Max-ATSP-Path) and structural results of Breslauer et al. [5].
In this paper we give an algorithm that achieves an approximation ratio of 2 11/23, breaking through the longstanding bound of 2 1/2. We use the standard reduction of Shortest-Superstring to Max-ATSP-Path. The new, somewhat surprising, algorithmic idea is to take the better of the two solutions obtained by using: (a) the currently best 2/3-approximation algorithm for Max-ATSP-Path and (b) a naive cycle-cover based 1/2-approximation algorithm. To prove that this indeed results in an improvement, we further develop a theory of string overlaps, extending the results of Breslauer et al. [5]. This theory is based on the novel use of Lyndon words, as a substitute for generic unbordered rotations and critical factorizations, as used by Breslauer et al.