We consider the longest common subsequence problem in the context of subsequences with gap constraints. In particular, following Day et al. 2022, we consider the setting when the distance (i. e., the gap) between two consecutive symbols of the subsequence has to be between a lower and an upper bound (which may depend on the position of those symbols in the subsequence or on the symbols bordering the gap) as well as the case where the entire subsequence is found in a bounded range (defined by a single upper bound), considered by Kosche et al. 2022. In all these cases, we present effcient algorithms for determining the length of the longest common constrained subsequence between two given strings.

翻译：我们考虑带间隙约束的子序列框架下的最长公共子序列问题。具体而言，遵循Day等人(2022)的研究，我们探讨了子序列中连续两个符号之间的距离（即间隙）须处于下界与上界之间（该界限可能依赖于这些符号在子序列中的位置或间隙边界的符号）的设置，以及Kosche等人(2022)所考虑的整个子序列出现在有界范围（由单一上界定义）内的情形。针对所有这些情况，我们提出了高效算法，用于确定两个给定字符串之间最长公共约束子序列的长度。