Dynamic Point Location in General Subdivisions

Kurt Mehlhorn (joint work with Hanna Baumgarten, Hermann Jung) Max-Planck Institut für Informatik email: mehlhorn@mpi-sb.mpg.de Report B 92-10 March 92

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The {\em dynamic planar point location problem} is the task of maintaining a dynamic set $S$ of $n$ non-intersecting, except possibly at endpoints, line segments in the plane under the following operations: \begin{itemize} \item Locate($q$: point): Report the segment immediately above $q$, i.e., the first segment intersected by an upward vertical ray starting at $q$; \item Insert($s$: segment): Add segment $s$ to the collection $S$ of segments; \item Delete($s$: segment): Remove segment $s$ from the collection $S$ of segments. \end{itemize} We present a solution which requires space $O(n)$, has query and insertion time \instime\ and deletion time \deltime. A query time below \deltime\ was previously only known for monotone subdivisions and horizontal segments and required non-linear space.