Point-Location in The Arrangement of Curves
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An arrangement of n curves in the plane is given. The query is a point q and the goal is to find the face of the arrangement that contains q. A data-structure for point-location, preprocesses the curves into a data structure of polynomial size in n, such that the queries can be answered in time polylogarithmic in n. We design a data structure for solving the point location problem queries in O(log C(n)+log S(n)) time using O(T(n)+S(n)log(S(n))) preprocessing time, if a polygonal subdivision of total size S(n), with cell complexity at most C(n) can be computed in time T(n), such that the order of the parts of the curves inside each cell has a monotone order with respect to at least one segment of the boundary of the cell. We call such a partitioning a curve-monotone polygonal subdivision.
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