2D Fractional Cascading on Axis-aligned Planar Subdivisions

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.