Abstract
Let S be a set of n (possibly intersecting) line segments in the plane. We show that the arrangement of S can be stored implicitly into a data structure of size O(n log2 n) so that the following query can be answered in time O(n 1/2 log2 n): Given two query points, determine whether they lie in the same face of the arrangement of S and, if so, return a path between them that lies within the face. This version of the implicit point location problem is motivated by the following motion planning problem: Given a polygonal robot R with m vertices and a planar region bounded by polygonal obstacles with n vertices in total, preprocess them into a data structure so that, given initial and final positions of R, one can quickly determine whether there exists a continuous collision-free translationsal motion of R from the initial to the final position. We show that such a query can be answered in time O((mn)1/2 log2 mn) using O(mn log2 mn) storage.
Part of this work was done while the second author was visiting the first author at Duke University on a grant of the Netherlands Organization for Scientific Research (N.W.O.). The research of the first author was supported by National Science Foundation Grant CCR-91-06514. The research of the second author was supported by the ESPRIT Basic Research Action No. 3075 (project ALCOM).
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Agarwal, P.K., van Kreveld, M. (1992). Implicit point location in arrangements of line segments, with an application to motion planning. In: Shyamasundar, R. (eds) Foundations of Software Technology and Theoretical Computer Science. FSTTCS 1992. Lecture Notes in Computer Science, vol 652. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56287-7_96
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DOI: https://doi.org/10.1007/3-540-56287-7_96
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