We investigate the function $L(h,p,q)$, called here the length function, such that $L(h,p,q)$ is the minimum length which guarantees that a natural extension of the periodicity lemma is valid for partial words with h holes and (so-called strong) periods $p,q$. In a series of papers, the formulae for the length function, in terms of p and q, were provided for each fixed $h\le 7$. We demystify the generic structure of such formulae and give a complete characterization of the length function for any parameter h expressed in terms of a piecewise-linear function with $\mathcal{O}(h)$ pieces. We also show how to evaluate the length function in $\mathcal{O}(\log p+\log q)$ time, which is an improvement upon the best previously known $\mathcal{O}(p+q)$-time algorithm.

我们研究函数$\mathrm{大号}(H,p,q)$，这里称为长度函数，这样$\mathrm{大号}(H,p,q)$是保证周期性引理的自然扩展对具有h个孔和（所谓的强）周期的部分单词有效的最小长度$p,q$. 在一系列论文中，为每个固定的长度函数提供了以p和q表示的公式$H\le 7$. 我们揭开了这些公式的一般结构的神秘面纱，并对任何参数h的长度函数进行了完整的表征，用分段线性函数表示$\mathcal{○}(H)$件。我们还展示了如何评估长度函数$\mathcal{○}(\mathrm{日志}p+\mathrm{日志}q)$时间，这是对以前最知名的改进$\mathcal{○}(p+q)$时间算法。