Longest common subsequence ($\mathsf{LCS}$) is a classic and central problem in combinatorial optimization. While $\mathsf{LCS}$ admits a quadratic time solution, recent evidence suggests that solving the problem may be impossible in truly subquadratic time. A special case of $\mathsf{LCS}$ wherein each character appears at most once in every string is equivalent to the longest increasing subsequence problem ($\mathsf{LIS}$) which can be solved in quasilinear time. In this work, we present novel algorithms for approximating $\mathsf{LCS}$ in truly subquadratic time and $\mathsf{LIS}$ in truly sublinear time. Our approximation factors depend on the ratio of the optimal solution size over the input size. We denote this ratio by $\lambda$ and obtain the following results for $\mathsf{LCS}$ and $\mathsf{LIS}$ without any prior knowledge of $\lambda$. $\bullet$ A truly subquadratic time algorithm for $\mathsf{LCS}$ with approximation factor $\Omega(\lambda^3)$. $\bullet$A truly sublinear time algorithm for $\mathsf{LIS}$ with approximation factor $\Omega(\lambda^3)$. Triangle inequality was recently used by [Boroujeni, Ehsani, Ghodsi, HajiAghayi and Seddighin SODA 2018] and [Charkraborty, Das, Goldenberg, Koucky and Saks FOCS 2018] to present new approximation algorithms for edit distance. Our techniques for $\mathsf{LCS}$ extend the notion of triangle inequality to non-metric settings.
翻译:长期常见子序列 (mathsfsf{LCS}$) 是组合优化的一个经典和中心问题。 虽然 $\ mathsf{LCS} 接受了二次时间解决方案, 但最近的证据表明, 在真正的二次时间里, 问题不可能解决 $\ mathsf{LCS} 。 每个字符在每个字符串中最多出现一次的特例, 相当于在准线时间里可以解决的最长期增加的子序列问题 ($\ mathsf{LIS} $ 。 在这项工作中, 我们展示了用于在真正二次时间里使用 $\ maths\ half{LCS} 的新型算法 。 我们的近似系数取决于在投入大小上的最佳解决方案大小中的比例 $\ diembda$( mindalf), 并且以 $\ mexcial_ maxal_ riceral_ 美元来获取以下结果 $\ calsada_ lisax_ 美元 美元。 lix_ lix_ a requistrationals a listrations_ blam listrations a blam limals. 我们的近点点点数, 我们的比数=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxlxlxxxx