Point-Location in The Arrangement of Curves

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 t...

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Hauptverfasser: Aghamolaei, Sepideh, Ghodsi, Mohammad
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Sprache:eng
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Zusammenfassung: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.
DOI:10.48550/arxiv.2007.11451