# 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 π of G, we can find the predecessor of q in all the sets associated with the vertices of π. 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 + |π|) 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 π and the goal is to locate the cell containing q in all the subdivisions associated with the vertices of π. 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 π is a path, then queries can be answered in O(logn+|π|+min{|π|√(logn),α(n)√(|π|)logn}) time using linear space where α 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 π is a general subgraph, then the query bound becomes O(log n + |π|√(log n)) and this bound is once again tight in both cases.