4,726 research outputs found
A New Upper Bound for the VC-Dimension of Visibility Regions
In this paper we are proving the following fact. Let P be an arbitrary simple
polygon, and let S be an arbitrary set of 15 points inside P. Then there exists
a subset T of S that is not "visually discernible", that is, T is not equal to
the intersection of S with the visibility region vis(v) of any point v in P. In
other words, the VC-dimension d of visibility regions in a simple polygon
cannot exceed 14. Since Valtr proved in 1998 that d \in [6,23] holds, no
progress has been made on this bound. By epsilon-net theorems our reduction
immediately implies a smaller upper bound to the number of guards needed to
cover P.Comment: 25 pages, 18 Figures. An extended abstract of this paper appeared at
SoCG '1
Chemical Evolution in Hierarchical Models of Cosmic Structure II: The Formation of the Milky Way Stellar Halo and the Distribution of the Oldest Stars
This paper presents theoretical star formation and chemical enrichment
histories for the stellar halo of the Milky Way based on new chemodynamical
modeling. The goal of this study is to assess the extent to which metal-poor
stars in the halo reflect the star formation conditions that occurred in halo
progenitor galaxies at high redshift, before and during the epoch of
reionization. Simple prescriptions that translate dark-matter halo mass into
baryonic gas budgets and star formation histories yield models that resemble
the observed Milky Way halo in its total stellar mass, metallicity
distribution, and the luminosity function and chemical enrichment of dwarf
satellite galaxies. These model halos in turn allow an exploration of how the
populations of interest for probing the epoch of reionization are distributed
in physical and phase space, and of how they are related to lower-redshift
populations of the same metallicity. The fraction of stars dating from before a
particular time or redshift depends strongly on radius within the galaxy,
reflecting the "inside-out" growth of cold-dark-matter halos, and on
metallicity, reflecting the general trend toward higher metallicity at later
times. These results suggest that efforts to discover stars from z > 6 - 10
should select for stars with [Fe/H] <~ -3 and favor stars on more tightly bound
orbits in the stellar halo, where the majority are from z > 10 and 15 - 40% are
from z > 15. The oldest, most metal-poor stars - those most likely to reveal
the chemical abundances of the first stars - are most common in the very center
of the Galaxy's halo: they are in the bulge, but not of the bulge. These models
have several implications for the larger project of constraining the properties
of the first stars and galaxies using data from the local Universe.Comment: Submitted to ApJ, 22 pages emulateapj, 15 color figure
Visibility Domains and Complexity
Two problems in discrete and computational geometry are considered that are related to questions about the combinatorial complexity of arrangements of visibility domains and about the hardness of path planning under cost measures defined using visibility domains. The first problem is to estimate the VC-dimension of visibility domains. The VC-dimension is a fundamental parameter of every range space that is typically used to derive upper bounds on the size of hitting sets. Better bounds on the VC-dimension directly translate into better bounds on the size of hitting sets. Estimating the VC-dimension of visibility domains has proven to be a hard problem. In this thesis, new tools to tackle this problem are developed. Encircling arguments are combined with decomposition techniques of a new kind. The main ingredient of the novel approach is the idea of relativization that makes it possible to replace in the analysis of intersections the complicated visibility domains by simpler geometric ranges. The main result here is the new upper bound of 14 on the VC-dimension of visibility polygons in simple polygons that improves significantly upon the previously known best upper bound of 23. For the VC-dimension of perimeter visibility domains, the new techniques yield an upper bound of 7 that leaves only a very small gap to the best known lower bound of 5. The second problem considered is to compute the barrier resilience of visibility domains. In barrier resilience problems, one is given a set of barriers and two points s and t in R^d. The task is to find the minimum number of barriers one has to remove such that there is a way between s and t that does not cross a barrier. In the field of sensor networks, the barriers are interpreted as sensor ranges and the barrier resilience of a network is a measure for its vulnerability. In this thesis the very natural special case where the barriers are visibility domains is investigated. It can also be formulated in terms of finding a so-called minimum witness path. For visibility domains in simple polygons it is shown that one can find an optimal path efficiently. For polygons with holes an approximation hardness result is shown that is stronger than previous hardness results in geometric settings. Two different three-dimensional settings are considered and their respective relations to the Minimum Neighborhood Path problem and the Minimum Color Path problem in graphs are demonstrated. For one of the three-dimensional problems a 2-approximation algorithm is designed. For the general problem of finding minimum witness paths among polyhedral obstacles it turns out that it is not approximable in a strong sense
Geometric Crossing-Minimization - A Scalable Randomized Approach
We consider the minimization of edge-crossings in geometric drawings of graphs G=(V, E), i.e., in drawings where each edge is depicted as a line segment. The respective decision problem is NP-hard [Daniel Bienstock, 1991]. Crossing-minimization, in general, is a popular theoretical research topic; see Vrt\u27o [Imrich Vrt\u27o, 2014]. In contrast to theory and the topological setting, the geometric setting did not receive a lot of attention in practice. Prior work [Marcel Radermacher et al., 2018] is limited to the crossing-minimization in geometric graphs with less than 200 edges. The described heuristics base on the primitive operation of moving a single vertex v to its crossing-minimal position, i.e., the position in R^2 that minimizes the number of crossings on edges incident to v.
In this paper, we introduce a technique to speed-up the computation by a factor of 20. This is necessary but not sufficient to cope with graphs with a few thousand edges. In order to handle larger graphs, we drop the condition that each vertex v has to be moved to its crossing-minimal position and compute a position that is only optimal with respect to a small random subset of the edges. In our theoretical contribution, we consider drawings that contain for each edge uv in E and each position p in R^2 for v o(|E|) crossings. In this case, we prove that with a random subset of the edges of size Theta(k log k) the co-crossing number of a degree-k vertex v, i.e., the number of edge pairs uv in E, e in E that do not cross, can be approximated by an arbitrary but fixed factor delta with high probability. In our experimental evaluation, we show that the randomized approach reduces the number of crossings in graphs with up to 13 000 edges considerably. The evaluation suggests that depending on the degree-distribution different strategies result in the fewest number of crossings
VC-Dimension of Exterior Visibility
In this paper, we study the Vapnik-Chervonenkis (VC)-dimension of set systems arising in 2D polygonal and 3D polyhedral configurations where a subset consists of all points visible from one camera. In the past, it has been shown that the VC-dimension of planar visibility systems is bounded by 23 if the cameras are allowed to be anywhere inside a polygon without holes [1]. Here, we consider the case of exterior visibility, where the cameras lie on a constrained area outside the polygon and have to observe the entire boundary. We present results for the cases of cameras lying on a circle containing a polygon (VC-dimension= 2) or lying outside the convex hull of a polygon (VC-dimension= 5). The main result of this paper concerns the 3D case: We prove that the VC-dimension is unbounded if the cameras lie on a sphere containing the polyhedron, hence the term exterior visibility
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