Rectilinear path problems in restricted memory setup

We study the rectilinear path problem in the presence of disjoint axis parallel rectangular obstacles in the read-only and in-place setup. The input to the problem is a set R of n axis-parallel rectangular obstacles in R-2. The objective is to answer the following query 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 show that Path-Query (p, q) problem can be solved in O(n(2)/s + n logs) time using O(s) extra space. We also show that the existence of an x-monotone path and reporting it, if it exists, can be done with the same asymptotic time complexity. If the objective is to test the existence of an xy-monotone path between the given pair of points p and q avoiding the obstacles, and report it if exists, then our proposed algorithm needs O(n(2)/s + n logs + M-s log n) time with O(s) extra space, where M-s is the time complexity for computing the median of n elements in the read-only setup using O(s) extra space. Finally, we show that when the obstacles are unit squares instead of rectangles of arbitrary size, then there always exists a path of O(root n) links between a pair of query points, and the path can be reported in O(n root n) time using O(1) extra work-space. It is also shown that there is an instance where the minimum number of links in a path between a pair of specified points is O(root n). The objective of the Path-Query (p, q) in the in-place setup is to preprocess the input rectangles in a data structure in the input array itself such that for any pair of query points p and q, a rectilinear path can be reported efficiently. Here we propose an algorithm with O(n log n) preprocessing time and O(n(3/4) + x) query time, where x is the number of links (bends) in the path. Both the preprocessing and query answering need O(1) extra space. (C) 2016 Elsevier B.V. All rights reserved