Fractional cascading is one of the influential techniques in data structures, as it provides a general framework for solving the important iterative search problem. In the problem, the input is a graph $G$ with constant degree and a set of values for every vertex of $G$. The goal is to preprocess $G$ such that when given a query value $q$, and a connected subgraph $\pi$ of $G$, we can find the predecessor of $q$ in all the sets associated with the vertices of $\pi$. The fundamental result of fractional cascading is that there exists a data structure that uses linear space and it can answer queries in $O(\log n + |\pi|)$ time [Chazelle and Guibas, 1986]. While this technique has received plenty of attention in the past decades, an almost quadratic space lower bound for "2D fractional cascading" [Chazelle and Liu, 2001] has convinced the researchers that fractional cascading is fundamentally a 1D technique. In 2D fractional cascading, the input includes a planar subdivision for every vertex of $G$ and the query is a point $q$ and a subgraph $\pi$ and the goal is to locate the cell containing $q$ in all the subdivisions associated with the vertices of $\pi$. In this paper, we show that it is possible to circumvent the lower bound of Chazelle and Liu for axis-aligned planar subdivisions. We present a number of upper and lower bounds which reveal that in 2D, the problem has a much richer structure. When $G$ is a tree and $\pi$ is a path, then queries can be answered in $O(\log{n}+|\pi|+\min\{|\pi|\sqrt{\log{n}},\alpha(n)\sqrt{|\pi|}\log{n}\})$ time using linear space where $\alpha$ is an inverse Ackermann function; surprisingly, we show both branches of this bound are tight, up to the inverse Ackermann factor. When $G$ is a general graph or when $\pi$ is a general subgraph, then the query bound becomes $O(\log n + |\pi|\sqrt{\log n})$ and this bound is once again tight in both cases.

中文翻译：

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轴对齐平面细分上的 2D 分数级联

分数级联是数据结构中一种有影响力的技术，因为它为解决重要的迭代搜索问题提供了一个通用框架。在这个问题中，输入是一个具有恒定度数的图 $G$ 和 $G$ 的每个顶点的一组值。目标是预处理 $G$，使得当给定查询值 $q$ 和 $G$ 的连通子图 $\pi$ 时，我们可以在与顶点关联的所有集合中找到 $q$ 的前驱$\pi$。分数级联的基本结果是存在一种使用线性空间的数据结构，它可以在 $O(\log n + |\pi|)$ 时间内回答查询 [Chazelle 和 Guibas，1986]。虽然这种技术在过去几十年中受到了大量关注，但“二维分数级联”的几乎二次空间下界 [Chazelle 和 Liu，2001] 使研究人员确信分数级联从根本上说是一种一维技术。在二维分数级联中，输入包括 $G$ 的每个顶点的平面细分，查询是点 $q$ 和子图 $\pi$，目标是在所有细分中定位包含 $q$ 的单元格与 $\pi$ 的顶点相关联。在本文中，我们表明，对于轴对齐的平面细分，可以绕过 Chazelle 和 Liu 的下界。我们提出了许多上限和下限，这表明在 2D 中，问题具有更丰富的结构。当 $G$ 是一棵树，$\pi$ 是一个路径，那么查询可以在 $O(\log{n}+|\pi|+\min\{|\pi|\sqrt{\log{ n}},\alpha(n)\sqrt{|\pi|}\log{n}\})$ 使用线性空间的时间，其中 $\alpha$ 是逆阿克曼函数；出奇，我们表明这个界限的两个分支都是紧密的，直到逆阿克曼因子。当 $G$ 是一般图或 $\pi$ 是一般子图时，则查询边界变为 $O(\log n + |\pi|\sqrt{\log n})$ 并且该边界再次变为在这两种情况下都很紧。