Rectilinear Path Problems in Restricted Memory Setup

Abstract

We study the rectilinear path problem in the presence of disjoint axis parallel rectangular obstacles in the in-place and read-only setup. The input to the problem is a set $\cal R$ of n axis-parallel rectangular obstacles in $I\!\!R^2$. We need to preprocess the members in $\cal R$ such that the following query can be answered efficiently.

Path-Query(p,q): Given a pair of points p and q, report an axis-parallel path from p to q avoiding the obstacles in $\cal R$.

In the read-only setup, we consider a restricted version of the Path-Query(p,q) problem, where the objective is to check the existence of an xy-monotone path between the given pair of points p and q avoiding the obstacles, and report it if such a path exists. Given O(s) extra space, the problem can be solved in $O(\frac{n^2}{s}+n\log s+ M_s\log n)$ time, where M _s is the time complexity for computing the median of n elements in read-only setup using O(s) extra space.

In the in-place setup, we preprocess the input rectangles in a data structure such that for any pair of query points p and q, the problem Path-Query(p,q) can be solved efficiently. The time complexities for the preprocessing and query are O(nlogn) and O(n ^3/4 + χ) respectively, where χ is the number of links (bends) in the path. The extra space requirement for both preprocessing and query answering are O(1). We also show that among a set of unit square obstacles, there always exists a path of O(logn) links between a pair of query points. Here, we use a different in-place data structure with same preprocessing time complexity to answer the query in O(logn) time.