We consider subsequences with gap constraints, i.e., length-k subsequences p that can be embedded into a string w such that the induced gaps (i.e., the factors of w between the positions to which p is mapped to) satisfy given gap constraints $gc = (C_1, C_2, ..., C_{k-1})$; we call p a gc-subsequence of w. In the case where the gap constraints gc are defined by lower and upper length bounds $C_i = (L^-_i, L^+_i) \in \mathbb{N}^2$ and/or regular languages $C_i \in REG$, we prove tight (conditional on the orthogonal vectors (OV) hypothesis) complexity bounds for checking whether a given p is a gc-subsequence of a string w. We also consider the whole set of all gc-subsequences of a string, and investigate the complexity of the universality, equivalence and containment problems for these sets of gc-subsequences.
翻译：我们考虑的是存在差距限制的后继后果,即可以嵌入字符串的长度-k 子序列 p,这样引致的差距(即P所映射的位置之间的系数)能够满足特定差距限制$gc = (C_1, C_2,..., C ⁇ k-1}) $gc = (C_1, C_2,..., C ⁇ k-1}) 美元;我们称之为 gc 后继 w 。如果差距限制gc 被下下限和上限 $C_i = (L ⁇ __i, L ⁇ i)\ in\mathbb{N ⁇ 2$和/或普通语言 $C_i\ in REG$),我们证明(对正方矢量矢量(OV) 假设) 复杂程度的界限很紧,可以检查给定的p是否为字符串的g- 后继物。 我们还考虑整个字符串的所有 g- 后继物的整套组合,并调查这些序列的普遍性、等问题的复杂性。
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