The paper studies the hairpin completion distance problem, which is the minimum number of hairpin completion operations required to transform one string into another. Hairpin completion is a string operation derived from the hairpin formation observed in DNA biochemistry, and it is particularly useful in DNA computing.
The key insights and highlights of the paper are:
The authors show that for any ε > 0, there is no O(n^(2-ε))-time algorithm for computing the hairpin completion distance between two strings of length at most n, unless the Strong Exponential Time Hypothesis (SETH) is false. This provides a conditional lower bound that matches the previously known O(n^2) upper bound, up to sub-polynomial factors.
The authors prove this lower bound by reducing the Longest Common Subsequence (LCS) problem on ternary strings to the hairpin completion distance problem. Specifically, they construct two binary strings x and y such that the hairpin completion distance from y to x can be used to infer the LCS of two ternary strings S and T in linear time.
The reduction relies on carefully designed gadgets, including left and right information gadgets, protector gadgets, and synchronizer gadgets. The authors analyze the structure of optimal paths in the hairpin deletion graph Gx and show that there exists an optimal well-behaved path that follows a specific structure.
The authors provide a detailed analysis of the cost of different types of deletion steps in the well-behaved paths, including non-synchronized deletions, synchronized deletions of disagreeing mega gadgets, and synchronized deletions of agreeing mega gadgets.
By combining the reduction and the cost analysis, the authors prove that the hairpin completion distance can be used to compute the LCS, and thus any algorithm that computes the hairpin completion distance in O(n^(2-ε)) time would also solve the LCS problem in the same time, contradicting the known conditional lower bound for LCS.
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by Itai Boneh,D... ที่ arxiv.org 04-19-2024
https://arxiv.org/pdf/2404.11673.pdfสอบถามเพิ่มเติม