Solving SCS for bounded length strings in fewer than 2n steps
It is still not known whether a shortest common superstring (SCS) of n input strings can be found faster than in O⁎(2n) time (O⁎(⋅) suppresses polynomial factors of the input length). In this short note, we show that for any constant r, SCS for strings of length at most r can be solved in time O⁎(2(...
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| Vydané v: | Information processing letters Ročník 114; číslo 8; s. 421 - 425 |
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| Hlavní autori: | , , |
| Médium: | Journal Article |
| Jazyk: | English |
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Elsevier B.V
01.08.2014
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| ISSN: | 0020-0190, 1872-6119 |
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| Abstract | It is still not known whether a shortest common superstring (SCS) of n input strings can be found faster than in O⁎(2n) time (O⁎(⋅) suppresses polynomial factors of the input length). In this short note, we show that for any constant r, SCS for strings of length at most r can be solved in time O⁎(2(1−c(r))n) where c(r)=(1+2r2)−1. For this, we introduce so-called hierarchical graphs that allow us to reduce SCS on strings of length at most r to the directed rural postman problem on a graph with at most k=(1−c(r))n weakly connected components. One can then use a recent O⁎(2k) time algorithm by Gutin, Wahlström, and Yeo.
•We study the shortest common superstring problem on string of length r (r-SCS).•We introduce hierarchical graphs to reduce the r-SCS problem to the directed rural postman problem (DRPP).•We bound the number of weakly connected components in hierarchical graphs and call recent algorithm by Gutin et al.•The main result is a randomized 2n(1−Ω(r−2))-time algorithm for r-SCS. |
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| AbstractList | It is still not known whether a shortest common superstring (SCS) of n input strings can be found faster than in O⁎(2n) time (O⁎(⋅) suppresses polynomial factors of the input length). In this short note, we show that for any constant r, SCS for strings of length at most r can be solved in time O⁎(2(1−c(r))n) where c(r)=(1+2r2)−1. For this, we introduce so-called hierarchical graphs that allow us to reduce SCS on strings of length at most r to the directed rural postman problem on a graph with at most k=(1−c(r))n weakly connected components. One can then use a recent O⁎(2k) time algorithm by Gutin, Wahlström, and Yeo.
•We study the shortest common superstring problem on string of length r (r-SCS).•We introduce hierarchical graphs to reduce the r-SCS problem to the directed rural postman problem (DRPP).•We bound the number of weakly connected components in hierarchical graphs and call recent algorithm by Gutin et al.•The main result is a randomized 2n(1−Ω(r−2))-time algorithm for r-SCS. |
| Author | Golovnev, Alexander Mihajlin, Ivan Kulikov, Alexander S. |
| Author_xml | – sequence: 1 givenname: Alexander surname: Golovnev fullname: Golovnev, Alexander organization: New York University, United States – sequence: 2 givenname: Alexander S. surname: Kulikov fullname: Kulikov, Alexander S. organization: St. Petersburg Department of Steklov Institute of Mathematics, Russian Federation – sequence: 3 givenname: Ivan surname: Mihajlin fullname: Mihajlin, Ivan organization: St. Petersburg Academic University and St. Petersburg Department of Steklov Institute of Mathematics, Russian Federation |
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| Keywords | Exponential-time algorithm Traveling salesman problem Algorithms Shortest common superstring NP-hard problem Exact algorithm |
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| References | Bax, Franklin (br0010) November 1996; 60 Golovnev, Kulikov, Mihajlin (br0120) 2013; vol. 8087 Kulikov, Kutzkov (br0190) 2009; 19 Mucha (br0210) 2013 Bellman (br0030) January 1962; 9 Karp (br0160) 1982; 1 Vassilevska (br0230) 2005; vol. 3618 Christofides, Campos, Corberan, Mota (br0080) 1986; vol. 26 Williams (br0240) 2005; 348 Gallant, Maier, Storer (br0110) 1980; 20 Björklund, Husfeldt, Kaski, Koivisto (br0050) 2010; 47 Cygan, Pilipczuk (br0090) 2013; vol. 7965 Schöning (br0220) 2002; 32 Dantsin, Wolpert (br0100) 2006; vol. 4121 Kohn, Gottlieb, Kohn (br0170) 1977 Björklund, Husfeldt, Koivisto (br0060) 2009; 39 Bodlaender, Cygan, Kratsch, Nederlof (br0070) 2013 Björklund, Husfeldt, Kaski, Koivisto (br0040) 2008; vol. 5125 Gutin, Wahlström, Yeo (br0130) Koivisto (br0180) 2006; 98 Moser, Scheder (br0200) 2011 Hertli (br0150) Oct. 2011 Held, Karp (br0140) 1962; 10 Beigel, Eppstein (br0020) 2005; 54 |
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| Snippet | It is still not known whether a shortest common superstring (SCS) of n input strings can be found faster than in O⁎(2n) time (O⁎(⋅) suppresses polynomial... |
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| StartPage | 421 |
| SubjectTerms | Algorithms Exact algorithm Exponential-time algorithm NP-hard problem Shortest common superstring Traveling salesman problem |
| Title | Solving SCS for bounded length strings in fewer than 2n steps |
| URI | https://dx.doi.org/10.1016/j.ipl.2014.03.004 |
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