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 R of n axis-parallel rectangular obstacles in IR2. We need to preprocess the members in 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 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(n2/s + n log s + Ms log n) time, where Ms 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(n log n) and O(n3/4 + 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(log n) 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(log n) time