Rectilinear path problems in restricted memory setup

• Published in: Discrete Applied Mathematics Vol. 228; pp. 80 - 87
• Main Authors: Bhattacharya, Binay K., De, Minati, Maheshwari, Anil, Nandy, Subhas C., Roy, Sasanka
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 10-09-2017; Elsevier BV
• Subjects: Algorithms; Barriers; Bends; Complexity; Computing time; Constant work-space; In-place and read-only model of computation; Linear codes; Links; Points; Preprocessing; Rectangles; Rectilinear path problem; Time; Constant work-space; Rectilinear path problem; In-place and read-only model of computation
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2016.05.031

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 R2. 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(n2s+nlogs) 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(n2s+nlogs+Mslogn) time with O(s) extra space, where Ms 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(n) links between a pair of query points, and the path can be reported in O(nn) 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(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(nlogn) preprocessing time and O(n3/4+χ) query time, where χ is the number of links (bends) in the path. Both the preprocessing and query answering need O(1) extra space.