Minimum convex partition of a constrained point set

Given an input consisting of an n-vertex convex polygon with k hole vertices or an n-vertex planar straight line graph (PSLG) with k holes and/or reflex vertices inside the convex hull, the parameterized minimum number convex partition (MNCP) problem asks for a partition into a minimum number of convex pieces. We give a fixed-parameter tractable algorithm for this problem that runs in the following time complexities: – linear time if k is constant, – time polynomial in n if $k=O(\frac{{\rm log}n}{{\rm log log}n})$, or, to be exact, in O(nk$^{\rm 6{\it k}-5}$ 216k) time.

Minimizing the total length has been one of the main optimality criteria for triangulations and other kinds of partitions. Indeed, the minimum weight triangulation or MWT (i.e. a triangulation of minimum total edge length) has frequently been referred to as the optimal This triangulation has some good properties [1] and is e.g. useful in numerical approximation of bivariate data [23]. Its complexity is one of the most intriguing problems in computational geometry. No proof has yet been presented showing that it is NP-hard. On the other hand, no polynomial time heuristic for it was shown to guarantee even a constant approximation factor. However, very recently we proved that what we called the quasi-greedy triangulation has length within a constant factor from the minimum [13]. A standard implementation of that heuristic using known results would yield a quadratic time algorithm. In this paper, we improve on this by describing an O(n log n) algorithm for quasi-greedy triangulation. Moreover, if the Delaunay triangulation is given, our algorithm runs in linear time. The minimum weight convex partition (MWCP) problem has been shown to be NP-complete [7, 17]. By a MWCP we mean a set of non-crossing diagonals (i.e. without Steiner points) of minimum total length which includes the convex hull and partitions it into empty convex polygons. In [13] it was shown that, given the quasi-greedy triangulation, a convex partition of length within a constant factor from the MWCP can be computed in linear time. Thus from the results in this paper we also obtain a fast heuristic for approximating the MWCP. The quasi-greedy triangulation is very closely related to the greedy triangulation (both are defined in the next section). In fact, we will describe an algorithm that, given the Delannay triangulation, computes the greedy triangulation in linear time. This algorithm behaves in such a way that it is very easily adapted to yield an algorithm for the quasi-greedy triangulation. Consequently, we also present the first O(n log n) algorithm for greedy triangulation. (By using a substantially different approach, an attempt to develop an O(n log n) algorithm for greedy triangulation was done in [21]. Even if that approach could be corrected

We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid). First, we prove that every such grid contains a convex polygon with Omega(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d in N), and obtain a tight lower bound of Omega(log^{d-1}n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the longest convex polygonal chain in a grid that contains no two points with the same x- or y-coordinate. We show that the maximum size of such a convex polygon can be efficiently approximated up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors.

This paper analyzes the number of polygons generated by map overlay. Overlay of maps yields a number of small polygons, and often causes problems in computation, handling, and storage of derived polygons. One method to deal with these problems is to estimate the number of derived polygons before executing overlay operation. To this end, a stochastic model is proposed by which the expected number of derived polygons is calculated. The results are summarized as follows: (1) the number of polygons generated by map overlay depends on the number of original polygons and their perimeters; (2) the number of derived polygons increases with the perimeters of original polygons; (3) McAlpine and Cook's (1971) earlier method underestimated the number of derived polygons; and (4) the number of polygons generated by the overlay of maps having the same lattice system is proportional to the square of the number of overlaid maps and the number of polygons on the map, and inversely proportional to the area‐perimeter ratio of the unit cell.

Extraction of filled-in information from document images in the presence of template poses challenges due to geometrical distortion. Filled-in document image consists of null background, general information foreground and vital information imposed layer. Template document image consists of null background and general information foreground layer. In this paper a novel document image registration technique has been proposed to extract imposed layer from input document image. A convex polygon is constructed around the content of the input and the template image using convex hull. The vertices of the convex polygons of input and template are paired based on minimum Euclidean distance. Each vertex of the input convex polygon is subjected to transformation for the permutable combinations of rotation and scaling. Translation is handled by tight crop. For every transformation of the input vertices, Minimum Hausdorff distance (MHD) is computed. Minimum Hausdorff distance identifies the rotation and scaling values by which the input image should be transformed to align it to the template. Since transformation is an estimation process, the components in the input image do not overlay exactly on the components in the template, therefore connected component technique is applied to extract contour boxes at word level to identify partially overlapping components. Geometrical features such as density, area and degree of overlapping are extracted and compared between partially overlapping components to identify and eliminate components common to input image and template image. The residue constitutes imposed layer. Experimental results indicate the efficacy of the proposed model with computational complexity. Experiment has been conducted on variety of filled-in forms, applications and bank cheques. Data sets have been generated as test sets for comparative analysis.

The problem of cutting a convex polygon P out of a piece of paper Q with minimum total cutting length is a well studied problem in computational geometry. Researchers studied several variations of the problem, such as P and Q are convex or non-convex polygons and the cuts are line cuts or rays cuts. In this paper we consider yet another variation of the problem where Q is a circle and P is convex polygon such that P is bounded by a half circle of Q and all the cuts are line cuts. We give a simple linear time O(log n)-approximation algorithm for this problem where n is the number of vertices of P.

The ocelot Leopardus pardalis is a medium-sized Neotropical spotted cat with a wide geographic range. The present study was conducted in a reserve in an Atlantic Forest area in southern Brazil, and provided information on ocelot ecology through photographic records. Density estimated by the half of the mean maximum distance moved method resulted in 0.04 ocelots per km2. The minimum home range sizes were obtained by the minimum convex polygon method and were in agreement with other studies for the males recorded. Our female’s home range size was smaller than ranges reported elsewhere. The nocturnal pattern recorded is probably related with prey activity, since it is suggested that ocelots adjust their movements to the probabilities of encountering local prey, or with an evolutionary factor, aiming to avoid competition/predation with larger cats. The low density observed could be a consequence of the conditions of the study area, small and isolated, which indicates the necessity for larger reserves and mechanisms to connect isolated populations.

Using species distribution models to inform IUCN Red List assessments

Radio-tracking squirrels: Performance of home range density and linkage estimators with small range and sample size

In this paper we present some efficient and a few optimal algorithms to compute the radius, diameter, incenter, circumcenter, width and k-dimensional enclosing cylinder for convex polyhedral and convex polyhedral offset distance functions in plane and in \(\mathfrak {R}^d\). The radius, incenter and circumcenter of a convex polygon can be optimally computed in linear time, i.e. \(O(n+m)\), in \(\mathfrak {R}^2\) and in time \(O(n+m+|R|)\), if the incenter or circumcenter is additionally constrained in a convex polygon R, where n is the size of input polygon and m is the size of polygon in convex polygonal or convex polygonal offset distance functions. In \(\mathfrak {R}^d\), the radius, incenter and circumcenter of a convex polyhedron as well as of a set of convex polyhedra, unconstrained or constrained, can be computed in O(nm) and \(O(nm+|R|)\) time respectively, for convex constraint polyhedron R. The diameter of a convex polygon in plane can be computed in \(O(n+m)\) time. The diameter of a polyhedron in \(\mathfrak {R}^d\) can be computed in O(nm) time by our methods. The width of a convex polyhedron can be computed in \(O(n + m)\) time and O(nm) time, for \(\mathfrak {R}^2\) and \(\mathfrak {R}^d\), respectively. The diameter and width of a set of convex polyhedra can be computed in O(nm)-time. We also show how the k-dimensional minimum enclosing cylinder can be computed in \(O(n^{d-k+1}nm^2)\) in \(\mathfrak {R}^d\) and in \(O(n^{d}m(n+m))\) in \(\mathfrak {R}^d\) for \(k=1\). The k-dimensional minimum enclosing cylinder for a set of convex polyhedra in \(\mathfrak {R}^d\) can be computed in \(O(n^{d-k+1}nm^2)\)-time. The diameter and width problems can also be solved for a set of points in \(\mathfrak {R}^d\), either in time O(nm) or in time O(mT(n)), where T(n) is the time complexity of the best convex hull computation algorithm for the given set of points. We also compute the minimum stabbing sphere for a set of convex polyhedra, unconstrained or constrained, for the above mentioned distance functions in O(nm)-time or \(O(nm+|R|)\)-time respectively for convex constraint polyhedron R.

Let S be a set of n points in the plane and CH(S) be the convex hull of S. We consider the problem of constructing an approximate convex hull which contains CH(S) with strong convexity. An ∊-convex δ-superhull of S is a convex polygon P satisfying the following conditions: (1) P has at most O(n) vertices, (2) P contains CH(S), (3) no vertex of P lies farther than δ outside CH(S), and (4) P remains convex even if its vertices are perturbed by as much as ∊. The parameters ∊ and δ represent the strength of convexity of P and the degree of approximation of P to CH(S), respectively. In this paper, we show that there exists an ∊-convex [Formula: see text]-superhull with at most n vertices for any S and any ∊ ≥ 0 and it can be constructed in O(n log n) time (in O(n) time is S is sorted).

A note on a QPTAS for maximum weight triangulation of planar point sets

Given a sequence of ordered convex polygons in which the adjacent polygons may intersect with each other, but the nonadjacent polygons do not intersect, a start point s, and an end point t in the plane, our goal is to obtain a shortest path that starts from s, visits each given polygon in order, and ends at t finally. We converted the touring polygons problem into the problem of computing the shortest path of visiting the disjoint line segments by analyzing the geometrical features of the given convex polygons, and preprocessing the intersection points of the jointed polygons, and using a forward partition process combined with a backward search process for finding the access edge of each convex polygon. Thus, we proposed an 0(max{n, klog⁁2k}) algorithm for solving the original problem, where n is the total number of vertices of the given polygons and k is the total number of polygons.

This study examines various algorithms for computing the convex hull of a set of n points in a d-dimensional space. Convex hulls are fundamental in computational geometry and are applied in computer graphics, pattern recognition, and computational biology. Such convex hulls can also be useful in symmetry problems. For instance, when points are arranged symmetrically, the convex hull is also likely to be symmetrically shaped, which can be useful for object recognition in computer vision or pattern recognition. The focus is primarily on two-dimensional algorithms, including well-known methods like Gift Wrapping, Graham Scan, Divide and Conquer, QuickHull, TORCH, Kirkpatrick–Sediel, and Chan’s algorithms. These algorithms vary in terms of time complexity and scalability to higher dimensions. This study is extended to three-dimensional convex hull algorithms, such as NAW, randomized insertion, and parallelized versions, such as CudaHull and CudaChain. This study aimed to elucidate the operational principles, step-by-step procedures, and comparative time complexities of each algorithm. The implementation in Python facilitates a detailed comparison of the algorithmic performance through stepwise analysis and graphical outputs. The ultimate goal is to provide insights into the strengths and weaknesses of each algorithm under various scenarios, thereby offering a comprehensive guide for practical implementation.

The morphology of convex polygons