Covering the Boundary of a Simple Polygon with Geodesic Unit Disks

We consider the problem of covering the boundary of a simple polygon on n vertices using the minimum number of geodesic unit disks. We present an O(n \log^2 n+k) time 2-approximation algorithm for finding the centers of the disks, with k denoting the number centers found by the algorithm

Point Set Isolation Using Unit Disks is NP-complete

We consider the situation where one is given a set S of points in the plane and a collection D of unit disks embedded in the plane. We show that finding a minimum cardinality subset of D such that any path between any two points in S is intersected by at least one disk is NP-complete. This settles an open problem raised by Matt Gibson et al[1]. Using a similar reduction, we show that finding a minimum cardinality subset D' of D such that R^2 - (D - D') consists of a single connected region is also NP-complete. Lastly, we show that the Multiterminal Cut Problem remains NP-complete when restricted to unit disk graphs

Geometric Separation and Packing Problems

The first part of this thesis investigates combinatorial and algorithmic aspects of geometric separation problems in the plane. In such a setting one is given a set of points and a set of separators such as lines, line segments or disks. The goal is to select a small subset of those separators such that every path between any two points is intersected by at least one separator. We first look at several problems which arise when one is given a set of points and a set of unit disks embedded in the plane and the goal is to separate the points using a small subset of the given disks. Next, we focus on a separation problem involving only one region: Given a region in the plane, bounded by a piecewise linear closed curve, such as a fence, place few guards inside the fenced region such that wherever an intruder cuts through the fence, the closest guard is at most a distance one away. Restricting the separating objects to be lines, we investigate combinatorial aspects which arise when we use them to pairwise separate a set of points in the plane; hereafter we generalize the notion of separability to arbitrary sets and present several enumeration results. Lastly, we investigate a packing problem with a non-convex shape in ℝ3. We show that ℝ3 can be packed at a density of 0.222 with tori of major radius one and minor radius going to zero. Furthermore, we show that the same torus arrangement yields the asymptotically optimal number of pairwise linked tori

On separating families of bipartitions

AbstractWe focus on families of bipartitions, i.e. set partitions consisting of at most two components. A family of bipartitions is a separating family for a set if every two elements in the set are separated by some bipartition. In this paper we enumerate separating families of arbitrary size. We furthermore enumerate inclusion-wise minimal separating families of minimum and maximum sizes

On isolating points using unit disks

Given a set of points in the plane and a set of disks (that we think of as wireless sensors) which separate the points, we consider the problem of selecting a minimum subset of the disks such that any path between any pair of points is intersected by at least one of the selected disks. We present a $(9 + \epsilon)$-approximation algorithm for this problem and show that it is NP-complete even if all disks have unit radius and no disk contains any points. Using a similar hardness reduction, we further show that the Multiterminal Cut problem remains NP-complete on unit disk graphs. Lastly, we prove that removing a minimum subset of a collection of unit disks, such that the plane minus the  arrangement of the remaining disks consists of a single connected region is also NP-complete

Range-aggregate queries for geometric extent problems

Let S be a set of n points in the plane. We present data structures that solve range-aggregate query problems on three geometric extent measure problems. Using these data structures, we can report, for any axis-parallel query rectangle Q, the area/perimeter of the convex hull, the width, and the radius of the smallest enclosing disk of the po

Fine-scale Spatiotemporal Mapping of Asymptomatic and Clinical Plasmodium falciparum Infections: Epidemiological Evidence for Targeted Malaria Elimination Interventions

International audienceAbstract Background A detailed understanding of the contribution of the asymptomatic Plasmodium reservoir to the occurrence of clinical malaria at individual and community levels is needed to guide effective elimination interventions. This study investigated the relationship between asymptomatic Plasmodium falciparum carriage and subsequent clinical malaria episodes in the Dielmo and Ndiop villages in Senegal. Methods The study used a total of 2792 venous and capillary blood samples obtained from asymptomatic individuals and clinical malaria datasets collected from 2013 to 2016. Mapping, spatial clustering of infections, and risk analysis were performed using georeferenced households. Results High incidences of clinical malaria episodes were observed to occur predominantly in households of asymptomatic P falciparum carriers. A statistically significant association was found between asymptomatic carriage in a household and subsequent episode of clinical malaria occurring in that household for each individual year (P values were 0.0017, 6 × 10–5, 0.005, and 0.008 for the years 2013, 2014, 2015, and 2016 respectively) and the combined years (P = 8.5 × 10–8), which was not found at the individual level. In both villages, no significant patterns of spatial clustering of P falciparum clinical cases were found, but there was a higher risk of clinical episodes <25 m from asymptomatic individuals in Ndiop attributable to clustering within households. Conclusion The findings provide strong epidemiological evidence linking the asymptomatic P falciparum reservoir to clinical malaria episodes at household scale in Dielmo and Ndiop villagers. This argues for a likely success of a mass testing and treatment intervention to move towards the elimination of malaria in the villages of Dielmo and Ndiop