Elsevier

Journal of Algorithms

Volume 3, Issue 2, June 1982, Pages 160-176
Journal of Algorithms

Rectilinear line segment intersection, layered segment trees, and dynamization

https://doi.org/10.1016/0196-6774(82)90016-5Get rights and content

Abstract

The aim of the present paper is to provide an efficient solution to the following problem: “Given a family of n rectilinear line segments in two-space report all intersections in the family with a query consisting of an arbitrary rectilinear line segment.” We provide an algorithm which takes O(nlog2n) preprocessing time, o(nlog2n) space and O(log2n + k) query time, where k is the number of reported intersections. This solution serves to introduce a powerful new data structure, the layered segment tree, which is of independent interest. Second it yields, by way of recent dynamization techniques, a solution to the on-line version of the above problem, that is the operations INSERT and DELETE and QUERY with a line segment are allowed. Third it also yields a new nonscanning solution to the batched version of the above problem. Finally we apply these techniques to the problem obtained by replacing “line segment” by “rectangle” in the above problem, giving an efficient solution in this case also.

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Work carried out under a Natural Sciences and Engineering Research Council of Canada Grant A-7700, while the first author was a postdoctoral fellow at McMaster University.

From July 1st 1982: Department of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.

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