19 research outputs found

    A Cover-Merging-Based Algorithm for the Longest Increasing Subsequence in a Sliding Window Problem

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    A longest increasing subsequence problem (LIS) is a well-known combinatorial problem with applications mainly in bioinformatics, where it is used in various projects on DNA sequences. Recently, a number of generalisations of this problem were proposed. One of them is to find an LIS among all fixed-size windows of the input sequence (LISW). We propose an algorithm for the LISW problem based on cover representation of the sequence that outperforms the existing methods for some class of the input sequences

    LIPIcs, Volume 248, ISAAC 2022, Complete Volume

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    LIPIcs, Volume 248, ISAAC 2022, Complete Volum

    Longest increasing subsequences in windows based on canonical antichain partition

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    Given a sequence Ļ€1Ļ€2... Ļ€n, a longest increasing subsequence (LIS) in a window Ļ€āŒ©l, rāŒŖ = Ļ€lĻ€l+1... Ļ€r is a longest subsequence Ļƒ = Ļ€i1Ļ€i2... Ļ€iT such that l ā‰¤ i1 < i2 < Ā· Ā· Ā· < iT ā‰¤ r and Ļ€i1 < Ļ€i2 < Ā· Ā· Ā· < Ļ€iT. We consider the Lisw problem, which is to find the longest increasing subsequences in a sliding window of fixed-size w over a sequence. Formally, it is to find a LIS for every window in a set SFIX = ļæ½ Ļ€āŒ©i + 1, i + w āŒŖ ļæ½ ļæ½ 0 ā‰¤ i ā‰¤ n āˆ’ w ļæ½ āˆŖ ļæ½ Ļ€āŒ©1, iāŒŖ, Ļ€āŒ©n āˆ’ i, n āŒŖ ļæ½ ļæ½ i < w ļæ½. By maintaining a canonical antichain partition in windows, we present an optimal output-sensitive algorithm to solve this problem in O(output) time, where output is the sum of the lengths of the n+w āˆ’1 LISs in those windows of SFIX. In addition, we propose a more generalized problem called Lisset problem, which is to find a LIS for every window in a set SVAR containing variable-size windows. By applying our algorithm, we provide an efficient solution for the Lisset problem to output a LIS (or all the LISs) in every window which is better than the straightforward generalization of classical LIS algorithms. An upper bound of our algorithm on the Lisset problem is discussed

    Longest increasing subsequences in windows based on canonical antichain partition

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    AbstractGiven a sequence Ļ€1Ļ€2ā€¦Ļ€n, a longest increasing subsequence (LIS) in a window Ļ€ć€ˆl,r怉=Ļ€lĻ€l+1ā€¦Ļ€r is a longest subsequence Ļƒ=Ļ€i1Ļ€i2ā€¦Ļ€iT such that lā‰¤i1<i2<ā‹Æ<iTā‰¤r and Ļ€i1<Ļ€i2<ā‹Æ<Ļ€iT. We consider the LiswĀ problem, which is to find the longest increasing subsequences in a sliding window of fixed-size w over a sequence. Formally, it is to find a LIS for every window in a set SFIX={Ļ€ć€ˆi+1,i+w怉āˆ£0ā‰¤iā‰¤nāˆ’w}āˆŖ{Ļ€ć€ˆ1,i怉,Ļ€ć€ˆnāˆ’i,n怉āˆ£i<w}. By maintaining a canonical antichain partition in windows, we present an optimal output-sensitive algorithm to solve this problem in O(output) time, where output is the sum of the lengths of the n+wāˆ’1 LISs in those windows of SFIX. In addition, we propose a more generalized problem called LissetĀ problem, which is to find a LIS for every window in a set SVAR containing variable-size windows. By applying our algorithm, we provide an efficient solution for the LissetĀ problem to output a LIS (or all the LISs) in every window which is better than the straightforward generalization of classical LIS algorithms. An upper bound of our algorithm on the LissetĀ problem is discussed

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

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    The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), SaarbrĀØucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), WĀØurzburg (1993), Caen (1994), MĀØunchen (1995), Grenoble (1996), LĀØubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..
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