Rectilinear path problems in restricted memory setup

Article Type

Research Article

Publication Title

Discrete Applied Mathematics


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([Formula presented]+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([Formula presented]+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.

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Open Access, Bronze

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