Coverage with k-transmitters in the presence of obstacles

For a fixed integer k ≥ 0, a k-transmitter is an omnidirectional wireless transmitter with an infinite broadcast range that is able to penetrate up to k “walls”, represented as line segments in the plane. We develop lower and upper bounds for the number of k-transmitters that are necessary and sufficient to cover a given collection of line segments, polygonal chains and polygons.Postprint (published version

Depth Explorer - A Software Tool for Analysis of Depth Measures

Data depth is an analysis method measuring how central a point is relative to a cloud of data points. Given a cloud of data points in R 2, the most central point, the median

Dynamic Ham-Sandwich Cuts for Two Point Sets with Bounded Convex-Hull-Peeling Depth

We provide an efficient data structure for dynamically maintaining a ham-sandwich cut of two (possibly overlapping) point sets in the plane, with a bounded number of convex-hull peeling layers. The ham-sandwich cut of S1 and S2 is a line that simultaneously bisects the area, perimeter or vertex count of both point sets. Our algorithm supports insertion and deletion of vertices in O(c log n) time, area and perimeter queries in O(log n) time and vertex-count queries in O(c 3 log 3 n) time, where n is the total number of points of S1 ∪ S2 and c is a bound on the number of convex hull peeling layers. Our algorithm considerably improves previous results [15, 1]. Stojmenović’s [15] static method finds a ham-sandwich cut for two separated point sets. The dynamic algorithm of Abbott et al. [1] maintains a ham-sandwich cut of two disjoint convex polygons in the plane. Our dynamic algorithm removes the restrictions based on the convex position and the separation of the points. It also solves an open problem from 1991 about finding the area and perimeter ham-sandwich cuts for overlapping convex point sets in the static setting [15].

Dynamic Ham-Sandwich Cuts for Two Overlapping Point Sets

We provide an efficient data structure for dynamically maintaining a ham-sandwich cut of two overlapping point sets in convex position in the plane. The ham-sandwich cut of S1 and S2 is a line that simultaneously bisects the area, perimeter or vertex count of both point sets. Our algorithm supports insertion and deletion of vertices in O(log n) time, and area, perimeter and vertex-count queries in O(log 3 n) time, where n is the total number of points of S1 ∪ S2. An extension of the algorithm for sets with a bounded number c of convex hull peeling layers enables area and perimeter queries, using O(c log n) time for insertions and deletions and O(log 3 n) query time. Our algorithm improves previous results [15, 1]. The static method of Stojmenović [15] and the dynamic algorithm of Abbott et al. [1] maintain a ham-sandwich cut of two disjoint convex polygons in the plane. Our dynamic algorithm removes the restrictions based on the separation of the points. It also solves an open problem from 1991 about finding the area and perimeter ham-sandwich cuts for overlapping convex point sets in the static setting [15].

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