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**11 - 13**of**13**### Towards an Optimal Method for Dynamic Planar Point Location

, 2015

"... We describe a fully dynamic linear-space data structure for point location in connected planar sub-divisions, or more generally vertical ray shooting among non-intersecting line segments, that supports queries in O(log n(log log n)2) time and updates in O(log n log log n) time. This is the first dat ..."

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We describe a fully dynamic linear-space data structure for point location in connected planar sub-divisions, or more generally vertical ray shooting among non-intersecting line segments, that supports queries in O(log n(log log n)2) time and updates in O(log n log log n) time. This is the first data structure that achieves close to logarithmic query and update time simultaneously, ignoring log log n factors. We further show how to reduce the query time to O(log n log log n) in the RAM model with randomization. Alternatively, the query time can be lowered to O(log n) if the update time is increased to O(log1+ε n) for any constant ε> 0, or vice versa.

### Efficient Range Searching for Categorical and Plain Data

"... In the orthogonal range searching problem we store a set of input points S in a data structure; the answer to a query Q is a piece of information about points in Q∩S, e.g., the list of all points in Q∩S or the number of points in Q. In the colored (or categorical) range searching problem the set of ..."

Abstract
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In the orthogonal range searching problem we store a set of input points S in a data structure; the answer to a query Q is a piece of information about points in Q∩S, e.g., the list of all points in Q∩S or the number of points in Q. In the colored (or categorical) range searching problem the set of input points is partitioned into categories; the answer to a query is a piece of information about categories of points in a query range. In this paper we describe several new results for one- and two-dimensional range searching problem. We obtain an optimal adaptive data structure for counting the number of objects in a three-sided range and for counting categories of objects in a one-dimensional range. We also obtain new results on color range reporting in two dimensions, approximate color counting in one dimension and some other related problems.