Lower bounds for Arrangement-based Range-Free Localization in Sensor Networks

Colander are location aware entities that collaborate to determine approximate location of mobile or static objects when beacons from an object are received by all colanders that are within its distance $R$. This model, referred to as arrangement-based localization, does not require distance estimation between entities, which has been shown to be highly erroneous in practice. Colander are applicable in localization in sensor networks and tracking of mobile objects. A set $S \subset {\mathbb R}^2$ is an $(R,\epsilon)$-colander if by placing receivers at the points of $S$, a wireless device with transmission radius $R$ can be localized to within a circle of radius $\epsilon$. We present tight upper and lower bounds on the size of $(R,\epsilon)$-colanders. We measure the expected size of colanders that will form $(R, \epsilon)$-colanders if they distributed uniformly over the plane

Polychromatic Coloring for Half-Planes

We prove that for every integer $k$, every finite set of points in the plane can be $k$-colored so that every half-plane that contains at least $2k-1$ points, also contains at least one point from every color class. We also show that the bound $2k-1$ is best possible. This improves the best previously known lower and upper bounds of $\frac{4}{3}k$ and $4k-1$ respectively. We also show that every finite set of half-planes can be $k$ colored so that if a point $p$ belongs to a subset $H_p$ of at least $3k-2$ of the half-planes then $H_p$ contains a half-plane from every color class. This improves the best previously known upper bound of $8k-3$. Another corollary of our first result is a new proof of the existence of small size \eps-nets for points in the plane with respect to half-planes.Comment: 11 pages, 5 figure

Visualization of Big Spatial Data using Coresets for Kernel Density Estimates

The size of large, geo-located datasets has reached scales where visualization of all data points is inefficient. Random sampling is a method to reduce the size of a dataset, yet it can introduce unwanted errors. We describe a method for subsampling of spatial data suitable for creating kernel density estimates from very large data and demonstrate that it results in less error than random sampling. We also introduce a method to ensure that thresholding of low values based on sampled data does not omit any regions above the desired threshold when working with sampled data. We demonstrate the effectiveness of our approach using both, artificial and real-world large geospatial datasets

Побудова відокремлюваних Е-сіток двох множин

Іванчук М.А. Побудова відокремлюваних Е-сіток двох множин / М.А. Іванчук, І.В. Малик // Інформатика та системні науки (ІСН-2016): матеріали VІI Всеукраїнської науково-практичної конференції за міжнародною участю, (м. Полтава, 10–12 берез. 2016 р.). – Полтава: ПУЕТ, 2016.Ivanchuk M.A., Malyk I.V. Building the separated R^n-nets for two sets. In the article are discussed the method of the separation of two sets in the space , which is based on building and separation -nets of these sets in the range space w.r.t. hyperplanes

Побудова відокремлюваних Е-сіток двох множин

Іванчук М.А. Побудова відокремлюваних Е-сіток двох множин / М.А. Іванчук, І.В. Малик // Інформатика та системні науки (ІСН-2016): матеріали VІI Всеукраїнської науково-практичної конференції за міжнародною участю, (м. Полтава, 10–12 берез. 2016 р.). – Полтава: ПУЕТ, 2016.Ivanchuk M.A., Malyk I.V. Building the separated R^n-nets for two sets. In the article are discussed the method of the separation of two sets in the space , which is based on building and separation -nets of these sets in the range space w.r.t. hyperplanes

More about lower bounds for the number of k-facets

In this paper we present two results dealing with the number of (• k)-facets of a set of points: •² In R2, we use the notion of ²-net to give structural properties of sets that achieve the optimal lower bound 3¡k+22 ¢ of (• k)-edges for a ¯xed 0 • k • bn=3c ¡ 1;² •In Rd, we show that for k < bn=(d + 1)c the number of (• k)-facets of a set of n points in general position is at least (d + 1)¡k+dd ¢, and that this bound is tight in that range

On interference among moving sensors and related problems

We show that for any set of $n$ points moving along "simple" trajectories (i.e., each coordinate is described with a polynomial of bounded degree) in $\Re^d$ and any parameter $2 \le k \le n$, one can select a fixed non-empty subset of the points of size $O(k \log k)$, such that the Voronoi diagram of this subset is "balanced" at any given time (i.e., it contains $O(n/k)$ points per cell). We also show that the bound $O(k \log k)$ is near optimal even for the one dimensional case in which points move linearly in time. As applications, we show that one can assign communication radii to the sensors of a network of $n$ moving sensors so that at any given time their interference is $O(\sqrt{n\log n})$. We also show some results in kinetic approximate range counting and kinetic discrepancy. In order to obtain these results, we extend well-known results from $\varepsilon$-net theory to kinetic environments