Minimum constraint removal problem for line segments is NP-hard

In the minimum constraint removal ([Formula: see text]), there is no feasible path to move from a starting point towards the goal, and the minimum constraints should be removed in order to find a collision-free path. It has been proved that [Formula: see text] problem is NP-hard when constraints have arbitrary shapes or even they are in shape of convex polygons. However, it has a simple linear solution when constraints are lines and the problem is open for other cases yet. In this paper, using a reduction from Subset Sum problem, in three steps, we show that the problem is NP-hard for both weighted and unweighted line segments.

Spatial Behavior and Habitat Use of Two Sympatric Bat Species

Movement behavior and habitat use of the long-eared bat species Plecotus auritus and Plecotus austriacus were studied in the Havelland region in Brandenburg (Germany). Data collection included mist-netting, radiotelemetry, reconstruction of prey items, and monitoring of roosting sites. Body measurements confirm a high degree of phenotypic similarity between the two species. Total activity areas (100% Minimum Convex Polygons, MCPS) of Plecotus austriacus (2828.3 ± 1269.43 ha) were up to five-fold larger compared to Plecotus auritus (544.54 ± 295.89 ha). The activity areas of Plecotus austriacus contained up to 11 distinct core areas, and their mean total size (149.7 ± 0.07 ha) was approximately three-fold larger compared to core areas of Plecotus auritus (49.2 ± 25.6 ha). The mean distance between consecutive fixes per night was 12.72 ± 3.7 km for Plecotus austriacus and 4.23 ± 2.8 km for Plecotus auritus. While Plecotus austriacus was located most frequently over pastures (>40%) and meadows (>20%), P. auritus was located mostly within deciduous (>50%) and mixed forests (>30%) in close vicinity to its roosts. Roost site monitoring indicates that the activity of P. austriacus is delayed relative to P. auritus in spring and declined earlier in autumn. These phenological differences are probably related to the species’ respective diets. Levins’ measure of trophic niche breadth suggests that the prey spectrum for P. auritus is more diverse during spring (B = 2.86) and autumn (B = 2.82) compared to P. austriacus (spring: B = 1.7; autumn: B = 2.1). Our results give reason to consider these interspecific ecological variations and species-specific requirements of P. auritus and P. austriacus to develop adapted and improved conservation measures.

Physical and Cognitive Correlates of GPS-Derived Life-Space Characteristics in Older Adults

Life-space mobility, movement within one’s living environment, is important for functional independence in later life. It is unclear which life-space characteristics (i.e., space, duration, shape) are most affected by physical and cognitive limitations. GPS-derived measures mitigate recall bias and offer novel ways to characterize life-space. We examined associations between physical and cognitive performance and GPS-derived life-space characteristics. Participants were 164 community-dwelling adults (Age: M=77.3±6.5) from baseline data of a clinical trial to improve walking in older adults. Participants carried a portable GPS for 7 days, which passively collected real-time location. Standard deviational ellipses (SDEs) and minimum convex polygons (MCPs) were derived for each day. Area and compactness of these measures quantified activity space and shape, respectively. For each measure, 7-day medians and median absolute deviations (MAD) were computed to capture both central tendency and variability of weekly activity. Activity duration was quantified as percentage of time outside home. Adjusting for age and sex, percent time outside home was associated with lower mobility performance (i.e., 6-minute walk (6MWT), figure 8 walk, ρ’s=.17-.18, p’s<.05) and executive functioning (i.e., Trail Making Test, Part A: ρ=.16, p=.04, Part B: ρ=.19, p=.01). Median MCP and SDE areas, but not compactness, were associated with 6MWT performance (ρ’s=.18-.20, p’s<.05). MCP area MAD was associated with greater global cognition (3MSE, ρ=.15, p=.05). Life-space characteristics were differentially associated with performance measures, suggesting physical and cognitive limitations may constrain life-space mobility via different mechanisms. Variation in these associations by neighborhood walkability and active versus passive travel will also be examined.

Space: The Re-Visioning Frontier of Biological Image Analysis with Graph Theory, Computational Geometry, and Spatial Statistics

Every biological image contains quantitative data that can be used to test hypotheses about how patterns were formed, what entities are associated with one another, and whether standard mathematical methods inform our understanding of biological phenomena. In particular, spatial point distributions and polygonal tessellations are particularly amendable to analysis with a variety of graph theoretic, computational geometric, and spatial statistical tools such as: Voronoi polygons; Delaunay triangulations; perpendicular bisectors; circumcenters; convex hulls; minimal spanning trees; Ulam trees; Pitteway violations; circularity; Clark-Evans spatial statistics; variance to mean ratios; Gabriel graphs; and, minimal spanning trees. Furthermore, biologists have developed a number of empirically related correlations for polygonal tessellations such as: Lewis’s law (the number of edges of convex polygons are positively correlated with the areas of these polygons): Desch’s Law (the number of edges of convex polygons are positively correlated with the perimeters of these polygons); and Errara’s Law (daughter cell areas should be roughly half that of their parent cells’ areas). We introduce a new Pitteway Law that the number of sides of the convex polygons in a Voronoi tessellation of biological epithelia is proportional to the minimal interior angle of the convex polygons as angles less than 90 degrees result in Pitteway violations of the Delaunay dual of the Voronoi tessellation.

Measuring the Degree of Overlap and Segregation among Multiple Probabilistic Home Ranges: A New Index with Illustrative Application to the Lesser Kestrel Falco naumanni

Home range overlap/segregation has several important applications to wildlife conservation and management. In this work, we first address the issue of measuring the degree of overlap/segregation among an arbitrarily large number (i.e., n ≥ 2) of probabilistic animal home ranges (i.e., utilization distributions). This subject matter has recently been solved for home ranges measured as polygons (e.g., percent minimum convex polygons and multinuclear cores) but not yet for probabilistic ones. Accordingly, we introduce a novel index named the PGOI (probabilistic general overlap index), and its complement, the PGSI (probabilistic general segregation index), an index for computation of probabilistic home range overlap/segregation at individual, population and species levels. Whatever the number of probabilistic home ranges, the PGOI returns a single score ranging in the [0, 100] interval. We applied the PGOI to five lesser kestrels (Falco naumanni) at Santeramo in Colle (Apulia region; Southern Italy) as a case study. Our new index can be applied to any animal species and to home ranges derived from any type of probabilistic home range estimator.