3,231 research outputs found
Longest Common Subsequence on Weighted Sequences
We consider the general problem of the Longest Common Subsequence (LCS) on weighted sequences. Weighted sequences are an extension of classical strings, where in each position every letter of the alphabet may occur with some probability. Previous results presented a PTAS and noticed that no FPTAS is possible unless P=NP. In this paper we essentially close the gap between upper and lower bounds by improving both. First of all, we provide an EPTAS for bounded alphabets (which is the most natural case), and prove that there does not exist any EPTAS for unbounded alphabets unless FPT=W[1]. Furthermore, under the Exponential Time Hypothesis, we provide a lower bound which shows that no significantly better PTAS can exist for unbounded alphabets. As a side note, we prove that it is sufficient to work with only one threshold in the general variant of the problem
Why is it hard to beat for Longest Common Weakly Increasing Subsequence?
The Longest Common Weakly Increasing Subsequence problem (LCWIS) is a variant
of the classic Longest Common Subsequence problem (LCS). Both problems can be
solved with simple quadratic time algorithms. A recent line of research led to
a number of matching conditional lower bounds for LCS and other related
problems. However, the status of LCWIS remained open.
In this paper we show that LCWIS cannot be solved in strongly subquadratic
time unless the Strong Exponential Time Hypothesis (SETH) is false.
The ideas which we developed can also be used to obtain a lower bound based
on a safer assumption of NC-SETH, i.e. a version of SETH which talks about NC
circuits instead of less expressive CNF formulas
An Efficient Dynamic Programming Algorithm for the Generalized LCS Problem with Multiple Substring Exclusion Constrains
In this paper, we consider a generalized longest common subsequence problem
with multiple substring exclusion constrains. For the two input sequences
and of lengths and , and a set of constrains
of total length , the problem is to find a common subsequence of and
excluding each of constrain string in as a substring and the length of
is maximized. The problem was declared to be NP-hard\cite{1}, but we
finally found that this is not true. A new dynamic programming solution for
this problem is presented in this paper. The correctness of the new algorithm
is proved. The time complexity of our algorithm is .Comment: arXiv admin note: substantial text overlap with arXiv:1301.718
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