Abstract
We revisit classical geometric search problems under the assumption of rational coordinates. Our main result is a tight bound for point separation, ie, to determine whether n given points lie on one side of a query line.We show that with polynomial storage the query time is Θ(log b/ log log b), where b is the bit length of the rationals used in specifying the line and the points. The lower bound holds in Yao’s cell probe model with storage in n O(1) and word size in b O(1). By duality, this provides a tight lower bound on the complexity on the polygon point enclosure problem: given a polygon in the plane, is a query point in it?
This work was supported in part by NSF Grant CCR-96-23768, ARO Grant DAAH04-96-1- 0181, NEC Research Institute, Ecole Polytechnique, and INRIA.
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Chazelle, B. (1999). Geometric Searching over the Rationals. In: Nešetřil, J. (eds) Algorithms - ESA’ 99. ESA 1999. Lecture Notes in Computer Science, vol 1643. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48481-7_31
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DOI: https://doi.org/10.1007/3-540-48481-7_31
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