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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Vydáno v:Information processing letters Ročník 114; číslo 8; s. 421 - 425
Hlavní autoři: Golovnev, Alexander, Kulikov, Alexander S., Mihajlin, Ivan
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier B.V 01.08.2014
Témata:
Algorithms
Exact algorithm
Exponential-time algorithm
NP-hard problem
Shortest common superstring
Traveling salesman problem
Exponential-time algorithm
Traveling salesman problem
Algorithms
Shortest common superstring
NP-hard problem
Exact algorithm
ISSN:0020-0190, 1872-6119
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Popis
Shrnutí: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.
ISSN:0020-0190
1872-6119
DOI:10.1016/j.ipl.2014.03.004