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Approximation Algorithms for LCS and LIS with Truly Improved Running Times
SIAM Journal on Computing ( IF 1.6 ) Pub Date : 2023-11-17 , DOI: 10.1137/20m1316500 Aviad Rubinstein 1 , Saeed Seddighin 2 , Zhao Song 3 , Xiaorui Sun 4
SIAM Journal on Computing ( IF 1.6 ) Pub Date : 2023-11-17 , DOI: 10.1137/20m1316500 Aviad Rubinstein 1 , Saeed Seddighin 2 , Zhao Song 3 , Xiaorui Sun 4
Affiliation
SIAM Journal on Computing, Ahead of Print.
Abstract. Longest common subsequence (LCS) is a classic and central problem in combinatorial optimization. While LCS admits a quadratic time solution, recent evidence suggests that solving the problem may be impossible in truly subquadratic time. A special case of LCS wherein each character appears at most once in every string is equivalent to the longest increasing subsequence (LIS) problem which can be solved in quasilinear time. In this work, we present novel algorithms for approximating LCS in truly subquadratic time and LIS in truly sublinear time. Our approximation factors depend on the ratio of the optimal solution size to the input size. We denote this ratio by [math] and obtain the following results for LCS and LIS without any prior knowledge of [math]: a truly subquadratic time algorithm for LCS with approximation factor [math] and a truly sublinear time algorithm for LIS with approximation factor [math]. The triangle inequality was recently used by M. Boroujeni, S. Ehsani, M. Ghodsi, M. HajiAghayi, and S. Seddingham [Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, 2018, pp. 1170–1189] and D. Chakraborty, D. Das, E. Goldenberg, M. Koucky, and M. Saks [Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science, 2018, pp. 979–990] to present new approximation algorithms for edit distance. Our techniques for LCS extend the notion of the triangle inequality to nonmetric settings.
中文翻译：
LCS 和 LIS 的近似算法真正缩短了运行时间
SIAM 计算杂志，印刷前。
抽象的。最长公共子序列（LCS）是组合优化中的经典和核心问题。虽然 LCS 承认二次时间解决方案，但最近的证据表明，在真正的次二次时间中解决该问题可能是不可能的。LCS 的一种特殊情况，其中每个字符在每个字符串中最多出现一次，相当于可以在拟线性时间内解决的最长递增子序列（LIS）问题。在这项工作中，我们提出了在真正的次二次时间中逼近 LCS 和在真正的次线性时间中逼近 LIS 的新颖算法。我们的近似因子取决于最佳解决方案大小与输入大小的比率。我们用[数学]表示这个比率，并在没有任何[数学]先验知识的情况下获得LCS和LIS的以下结果：具有近似因子[数学]的LCS的真正次二次时间算法和具有近似因子的LIS的真正次线性时间算法[数学]。M. Boroujeni、S. Ehsani、M. Ghodsi、M. HajiAghayi 和 S. Seddingham 最近使用了三角不等式 [第 29 届 ACM-SIAM 离散算法年度研讨会论文集，2018 年，第 1170–1189 页] 和D. Chakraborty、D. Das、E. Goldenberg、M. Koucky 和 M. Saks [第 59 届 IEEE 计算机科学基础研讨会论文集，2018 年，第 979–990 页] 提出新的编辑距离近似算法。我们的 LCS 技术将三角不等式的概念扩展到非度量设置。
更新日期：2023-11-17
Abstract. Longest common subsequence (LCS) is a classic and central problem in combinatorial optimization. While LCS admits a quadratic time solution, recent evidence suggests that solving the problem may be impossible in truly subquadratic time. A special case of LCS wherein each character appears at most once in every string is equivalent to the longest increasing subsequence (LIS) problem which can be solved in quasilinear time. In this work, we present novel algorithms for approximating LCS in truly subquadratic time and LIS in truly sublinear time. Our approximation factors depend on the ratio of the optimal solution size to the input size. We denote this ratio by [math] and obtain the following results for LCS and LIS without any prior knowledge of [math]: a truly subquadratic time algorithm for LCS with approximation factor [math] and a truly sublinear time algorithm for LIS with approximation factor [math]. The triangle inequality was recently used by M. Boroujeni, S. Ehsani, M. Ghodsi, M. HajiAghayi, and S. Seddingham [Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, 2018, pp. 1170–1189] and D. Chakraborty, D. Das, E. Goldenberg, M. Koucky, and M. Saks [Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science, 2018, pp. 979–990] to present new approximation algorithms for edit distance. Our techniques for LCS extend the notion of the triangle inequality to nonmetric settings.
中文翻译：
LCS 和 LIS 的近似算法真正缩短了运行时间
SIAM 计算杂志，印刷前。
抽象的。最长公共子序列（LCS）是组合优化中的经典和核心问题。虽然 LCS 承认二次时间解决方案，但最近的证据表明，在真正的次二次时间中解决该问题可能是不可能的。LCS 的一种特殊情况，其中每个字符在每个字符串中最多出现一次，相当于可以在拟线性时间内解决的最长递增子序列（LIS）问题。在这项工作中，我们提出了在真正的次二次时间中逼近 LCS 和在真正的次线性时间中逼近 LIS 的新颖算法。我们的近似因子取决于最佳解决方案大小与输入大小的比率。我们用[数学]表示这个比率，并在没有任何[数学]先验知识的情况下获得LCS和LIS的以下结果：具有近似因子[数学]的LCS的真正次二次时间算法和具有近似因子的LIS的真正次线性时间算法[数学]。M. Boroujeni、S. Ehsani、M. Ghodsi、M. HajiAghayi 和 S. Seddingham 最近使用了三角不等式 [第 29 届 ACM-SIAM 离散算法年度研讨会论文集，2018 年，第 1170–1189 页] 和D. Chakraborty、D. Das、E. Goldenberg、M. Koucky 和 M. Saks [第 59 届 IEEE 计算机科学基础研讨会论文集，2018 年，第 979–990 页] 提出新的编辑距离近似算法。我们的 LCS 技术将三角不等式的概念扩展到非度量设置。
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