We investigate the limits of one of the fundamental ideas in data structures: fractional cascading. This is an important data structure technique to speed up repeated searches for the same key in multiple lists and it has numerous applications. Specifically, the input is a "catalog" graph, $G$, of constant degree together with a list of values assigned to every vertex of $G$. The goal is to preprocess the input such that given a connected subgraph $H$ of $G$ and a single query value $q$, one can find the predecessor of $q$ in every list that belongs to $\scat$. The classical result by Chazelle and Guibas shows that in a pointer machine, this can be done in the optimal time of $\O(\log n + |\scat|)$ where $n$ is the total number of values. However, if insertion and deletion of values are allowed, then the query time slows down to $\O(\log n + |\scat| \log\log n)$. If only insertions (or deletions) are allowed, then once again, an optimal query time can be obtained but by using amortization at update time. We prove a lower bound of $\Omega( \log n \sqrt{\log\log n})$ on the worst-case query time of dynamic fractional cascading, when queries are paths of length $O(\log n)$. The lower bound applies both to fully dynamic data structures with amortized polylogarithmic update time and incremental data structures with polylogarithmic worst-case update time. As a side, this also roves that amortization is crucial for obtaining an optimal incremental data structure. This is the first non-trivial pointer machine lower bound for a dynamic data structure that breaks the $\Omega(\log n)$ barrier. In order to obtain this result, we develop a number of new ideas and techniques that hopefully can be useful to obtain additional dynamic lower bounds in the pointer machine model.

中文翻译：

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动态分数级联的下限

我们研究了数据结构中的基本思想之一的局限性：分数级联。这是一种重要的数据结构技术，可以加快在多个列表中重复搜索相同键的速度，它有很多应用。具体来说，输入是一个“目录”图，$G$，具有恒定度数，以及分配给 $G$ 的每个顶点的值列表。目标是对输入进行预处理，使得给定一个连接的 $G$ 子图 $H$ 和单个查询值 $q$，可以在每个属于 $\scat$ 的列表中找到 $q$ 的前驱。Chazelle 和 Guibas 的经典结果表明，在指针机中，这可以在 $\O(\log n + |\scat|)$ 的最佳时间完成，其中 $n$ 是值的总数。但是，如果允许插入和删除值，然后查询时间减慢到 $\O(\log n + |\scat| \log\log n)$。如果只允许插入（或删除），那么再一次，可以通过在更新时使用摊销来获得最佳查询时间。当查询是长度为 $O(\log n)$ 的路径时，我们证明了 $\Omega( \log n \sqrt{\log\log n})$ 在动态分数级联的最坏情况查询时间上的下界. 下限适用于具有摊销多对数更新时间的完全动态数据结构和具有多对数最坏情况更新时间的增量数据结构。另一方面，这也表明摊销对于获得最佳增量数据结构至关重要。这是打破 $\Omega(\log n)$ 障碍的动态数据结构的第一个非平凡指针机下界。为了得到这个结果，