An Efficient Algorithm for Generalized Polynomial Partitioning and Its Applications
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Guth showed that given a family S of n g-dimensional semi-algebraic sets in R^d and an integer parameter D ≥ 1, there is a d-variate partitioning polynomial P of degree at most D, so that each connected component of R^d ∖ Z(P) intersects O(n / D^d-g) sets from S. Such a polynomial is called a "generalized partitioning polynomial". We present a randomized algorithm that efficiently computes such a polynomial P. Specifically, the expected running time of our algorithm is only linear in |S|, where the constant of proportionality depends on d, D, and the complexity of the description of S. Our approach exploits the technique of "quantifier elimination" combined with that of "ϵ-samples". We present four applications of our result. The first is a data structure for answering point-location queries among a family of semi-algebraic sets in R^d in O( n) time; the second is data structure for answering range search queries with semi-algebraic ranges in O( n) time; the third is a data structure for answering vertical ray-shooting queries among semi-algebraic sets in R^d in O(^2 n) time; and the fourth is an efficient algorithm for cutting algebraic planar curves into pseudo-segments, i.e., into Jordan arcs, each pair of which intersect at most once.
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