I/O-efficient map overlay and point location in low-density subdivisions

We present improved and simplified i/o-efficient algorithms for two problems on planar low-density subdivisions, namely map overlay and point location. More precisely, we show how to preprocess a lowdensity subdivision with n edges in O(sort(n)) i/o’s into a compressed linear quadtree such that one can: (i) compute the overlay of two such preprocessed subdivisions in O(scan(n)) i/o’s, where n is the total number of edges in the two subdivisions, (ii) answer a single point location query in O(logB n) i/o’s and k batched point location queries in O(scan(n) + sort(k)) i/o’s. For the special case where the subdivision is a fat triangulation, we show how to obtain the same bounds with an ordinary (uncompressed) quadtree, and we show how to make the structure fully dynamic using O(logB n) i/o’s per update. Our algorithms and data structures improve on the previous best known bounds for general subdivisions both in the number of i/o’s and storage usage, they are significantly simpler, and several of our algorithms are cache-oblivious

I/O-efficient map overlay and point location in low-density subdivisions

We present improved and simplified i/o-efficient algorithms for two problems on planar low-density subdivisions, namely map overlay and point location. More precisely, we show how to preprocess a lowdensity subdivision with n edges in O(sort(n)) i/o’s into a compressed linear quadtree such that one can: (i) compute the overlay of two such preprocessed subdivisions in O(scan(n)) i/o’s, where n is the total number of edges in the two subdivisions, (ii) answer a single point location query in O(logB n) i/o’s and k batched point location queries in O(scan(n) + sort(k)) i/o’s. For the special case where the subdivision is a fat triangulation, we show how to obtain the same bounds with an ordinary (uncompressed) quadtree, and we show how to make the structure fully dynamic using O(logB n) i/o’s per update. Our algorithms and data structures improve on the previous best known bounds for general subdivisions both in the number of i/o’s and storage usage, they are significantly simpler, and several of our algorithms are cache-oblivious

Short photoperiod-induced decrease of histamine H3 receptors facilitates activation of hypothalamic neurons in the Siberian Hamster

Nonhibernating seasonal mammals have adapted to temporal changes in food availability through behavioral and physiological mechanisms to store food and energy during times of predictable plenty and conserve energy during predicted shortage. Little is known, however, of the hypothalamic neuronal events that lead to a change in behavior or physiology. Here we show for the first time that a shift from long summer-like to short inter-like photoperiod, which induces physiological adaptation to winter in the Siberian hamster, including a body weight decrease of up to 30%, increases neuronal activity in the dorsomedial region of the arcuate nucleus (dmpARC) assessed by electro physiological patch-clamping recording. Increased neuronal activity in short days is dependent on a photoperiod-driven down-regulation of H3 receptor expression and can be mimicked in long-day dmpARC neurons by the application of the H3 receptor antagonist, clobenproprit. Short-day activation of dmpARC neurons results in increased c-Fos expression. Tract tracing with the trans-synaptic retrograde tracer, pseudorabies virus, delivered into adipose tissue reveals a multisynaptic neuronal sympathetic outflow from dmpARC to white adipose tissue. These data strongly suggest that increased activity of dmpARC neurons, as a consequence of down-regulation of the histamine H3 receptor, contributes to the physiological adaptation of body weight regulation in seasonal photoperiod

Distance-Sensitive Planar Point Location

Let $\mathcal{S}$ be a connected planar polygonal subdivision with $n$ edges that we want to preprocess for point-location queries, and where we are given the probability $\gamma_i$ that the query point lies in a polygon $P_i$ of $\mathcal{S}$. We show how to preprocess $\mathcal{S}$ such that the query time for a point~$p\in P_i$ depends on~$\gamma_i$ and, in addition, on the distance from $p$ to the boundary of~$P_i$---the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time $O\left(\min \left(\log n, 1 + \log \frac{\mathrm{area}(P_i)}{\gamma_i \Delta_{p}^2}\right)\right)$, where $\Delta_{p}$ is the shortest Euclidean distance of the query point~$p$ to the boundary of $P_i$. Our structure uses $O(n)$ space and $O(n \log n)$ preprocessing time. It is based on a decomposition of the regions of $\mathcal{S}$ into convex quadrilaterals and triangles with the following property: for any point $p\in P_i$, the quadrilateral or triangle containing~$p$ has area $\Omega(\Delta_{p}^2)$. For the special case where $\mathcal{S}$ is a subdivision of the unit square and $\gamma_i=\mathrm{area}(P_i)$, we present a simpler solution that achieves a query time of $O\left(\min \left(\log n, \log \frac{1}{\Delta_{p}^2}\right)\right)$. The latter solution can be extended to convex subdivisions in three dimensions

Orthogonal Range Reporting and Rectangle Stabbing for Fat Rectangles

In this paper we study two geometric data structure problems in the special case when input objects or queries are fat rectangles. We show that in this case a significant improvement compared to the general case can be achieved. We describe data structures that answer two- and three-dimensional orthogonal range reporting queries in the case when the query range is a \emph{fat} rectangle. Our two-dimensional data structure uses $O(n)$ words and supports queries in $O(\log\log U +k)$ time, where $n$ is the number of points in the data structure, $U$ is the size of the universe and $k$ is the number of points in the query range. Our three-dimensional data structure needs $O(n\log^{\varepsilon}U)$ words of space and answers queries in $O(\log \log U + k)$ time. We also consider the rectangle stabbing problem on a set of three-dimensional fat rectangles. Our data structure uses $O(n)$ space and answers stabbing queries in $O(\log U\log\log U +k)$ time.Comment: extended version of a WADS'19 pape

Constant-Factor Approximation for TSP with Disks

We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of $n$ disks in the plane, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in time that is polynomial in $n$. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and arbitrary intersections. In order to achieve a $O(1)$-approximation, we reduce the traveling salesman problem with disks, up to constant factors, to a minimum weight hitting set problem in a geometric hypergraph. The connection between TSPN and hitting sets in geometric hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure

The Skip Quadtree: A Simple Dynamic Data Structure for Multidimensional Data

We present a new multi-dimensional data structure, which we call the skip quadtree (for point data in R^2) or the skip octree (for point data in R^d, with constant d>2). Our data structure combines the best features of two well-known data structures, in that it has the well-defined "box"-shaped regions of region quadtrees and the logarithmic-height search and update hierarchical structure of skip lists. Indeed, the bottom level of our structure is exactly a region quadtree (or octree for higher dimensional data). We describe efficient algorithms for inserting and deleting points in a skip quadtree, as well as fast methods for performing point location and approximate range queries.Comment: 12 pages, 3 figures. A preliminary version of this paper appeared in the 21st ACM Symp. Comp. Geom., Pisa, 2005, pp. 296-30