scholarly journals A Convex Cover for Closed Unit Curves has Area at Least 0.0975

2021 ◽  
pp. 2050006
Author(s):  
Bogdan Grechuk ◽  
Sittichoke Som-am
Keyword(s):  
Lower Bound ◽  
Convex Hull ◽  
Convex Set ◽  
Search Algorithm ◽  
Unit Length ◽  
Plane Curve ◽  
Geometric Methods ◽  
Convex Cover ◽  
Minimal Area ◽  
Closed Plane

We combine geometric methods with a numerical box search algorithm to show that the minimal area of a convex set in the plane which can cover every closed plane curve of unit length is at least [Formula: see text]. This improves the best previous lower bound of [Formula: see text]. In fact, we show that the minimal area of the convex hull of circle, equilateral triangle, and rectangle of perimeter [Formula: see text] is between [Formula: see text] and [Formula: see text].

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

scholarly journals A Unifying View on Individual Bounds and Heuristic Inaccuracies in Bidirectional Search

2020 ◽  
Vol 34 (03) ◽  
pp. 2327-2334
Author(s):  
Vidal Alcázar ◽  
Pat Riddle ◽  
Mike Barley
Keyword(s):  
Lower Bound ◽  
Heuristic Search ◽  
Search Algorithm ◽  
Substantial Improvement ◽  
Full Potential ◽  
The Past ◽  
Heuristic Search Algorithms ◽  
Bidirectional Search ◽  
Order Of Magnitude ◽  
The Cost

In the past few years, new very successful bidirectional heuristic search algorithms have been proposed. Their key novelty is a lower bound on the cost of a solution that includes information from the g values in both directions. Kaindl and Kainz (1997) proposed measuring how inaccurate a heuristic is while expanding nodes in the opposite direction, and using this information to raise the f value of the evaluated nodes. However, this comes with a set of disadvantages and remains yet to be exploited to its full potential. Additionally, Sadhukhan (2013) presented BAE∗, a bidirectional best-first search algorithm based on the accumulated heuristic inaccuracy along a path. However, no complete comparison in regards to other bidirectional algorithms has yet been done, neither theoretical nor empirical. In this paper we define individual bounds within the lower-bound framework and show how both Kaindl and Kainz's and Sadhukhan's methods can be generalized thus creating new bounds. This overcomes previous shortcomings and allows newer algorithms to benefit from these techniques as well. Experimental results show a substantial improvement, up to an order of magnitude in the number of necessarily-expanded nodes compared to state-of-the-art near-optimal algorithms in common benchmarks.

Iterative computation of the convex hull of a rational plane curve

2015 ◽  
Vol 49 (2) ◽  
pp. 52-52
Author(s):  
Juan G. Alcázar ◽  
Carlos Hermoso ◽  
Jorge Caravantes ◽  
Gema M. Díaz-Toca
Keyword(s):  
Convex Hull ◽  
Plane Curve ◽  
Iterative Computation ◽  
Rational Plane ◽  
Rational Plane Curve

On resampling schemes for polytopes

2019 ◽  
Vol 56 (4) ◽  
pp. 959-980
Author(s):  
Weinan Qi ◽  
Mahmoud Zarepour
Keyword(s):  
Convex Hull ◽  
Asymptotic Theory ◽  
Convex Set ◽  
Real Life ◽  
Convex Hulls ◽  
Asymptotic Results ◽  
Underlying Distribution ◽  
Asymptotic Consistency ◽  
Computational Intractability ◽  
Practical Implications

AbstractThe convex hull of a sample is used to approximate the support of the underlying distribution. This approximation has many practical implications in real life. To approximate the distribution of the functionals of convex hulls, asymptotic theory plays a crucial role. Unfortunately most of the asymptotic results are computationally intractable. To address this computational intractability, we consider consistent bootstrapping schemes for certain cases. Let $S_n=\{X_i\}_{i=1}^{n}$ be a sequence of independent and identically distributed random points uniformly distributed on an unknown convex set in $\mathbb{R}^{d}$ ($d\ge 2$ ). We suggest a bootstrapping scheme that relies on resampling uniformly from the convex hull of $S_n$ . Moreover, the resampling asymptotic consistency of certain functionals of convex hulls is derived under this bootstrapping scheme. In particular, we apply our bootstrapping technique to the Hausdorff distance between the actual convex set and its estimator. For $d=2$ , we investigate the asymptotic consistency of the suggested bootstrapping scheme for the area of the symmetric difference and the perimeter difference between the actual convex set and its estimate. In all cases the consistency allows us to rely on the suggested resampling scheme to study the actual distributions, which are not computationally tractable.

TIME-OPTIMAL DIGITAL GEOMETRY ALGORITHMS ON MESHES WITH MULTIPLE BROADCASTING

1995 ◽  
Vol 09 (04) ◽  
pp. 601-613
Author(s):  
V. BOKKA ◽  
H. GURLA ◽  
S. OLARIU ◽  
J.L. SCHWING ◽  
I. STOJMENOVIĆ
Keyword(s):  
Lower Bound ◽  
Convex Hull ◽  
Lower Bounds ◽  
Minimum Distance ◽  
Fundamental Problem ◽  
Optimal Solution ◽  
The Other ◽  
Digital Geometry ◽  
Black Pixel ◽  
Time Optimal

The main contribution of this work is to show that a number of digital geometry problems can be solved elegantly on meshes with multiple broadcasting by using a time-optimal solution to the leftmost one problem as a basic subroutine. Consider a binary image pretiled onto a mesh with multiple broadcasting of size [Formula: see text] one pixel per processor. Our first contribution is to prove an Ω(n1/6) time lower bound for the problem of deciding whether the image contains at least one black pixel. We then obtain time lower bounds for many other digital geometry problems by reducing this fundamental problem to all the other problems of interest. Specifically, the problems that we address are: detecting whether an image contains at least one black pixel, computing the convex hull of the image, computing the diameter of an image, deciding whether a set of digital points is a digital line, computing the minimum distance between two images, deciding whether two images are linearly separable, computing the perimeter, area and width of a given image. Our second contribution is to show that the time lower bounds obtained are tight by exhibiting simple O(n1/6) time algorithms for these problems. As previously mentioned, an interesting feature of these algorithms is that they use, directly or indirectly, an algorithm for the leftmost one problem recently developed by one of the authors.

Layered Tabu Search Algorithm for Large-MIMO Detection and a Lower Bound on ML Performance

2011 ◽  
Vol 59 (11) ◽  
pp. 2955-2963 ◽  
Author(s):  
N. Srinidhi ◽  
Tanumay Datta ◽  
A. Chockalingam ◽  
B. Sundar Rajan
Keyword(s):  
Lower Bound ◽  
Tabu Search ◽  
Search Algorithm ◽  
Tabu Search Algorithm ◽  
Mimo Detection

A New Method for Crosstalk Prediction Between Triple-twisted Strand (Uniform and Non-uniform) and Signal Wire based on CDBAS-BPNN Algorithm

2021 ◽  
Vol 36 (1) ◽  
pp. 1-9
Author(s):  
Yanxing Ji ◽  
Wei Yan ◽  
Yang Zhao ◽  
Chao Huang ◽  
Shiji Li ◽  
...  
Keyword(s):  
Neural Network ◽  
Search Algorithm ◽  
Geometric Model ◽  
Unit Length ◽  
Back Propagation ◽  
Prediction Method ◽  
Parameter Method ◽  
Cross Sectional ◽  
Parameter Matrix ◽  
Signal Wire

This paper proposes a novel crosstalk prediction method between the triple-twisted strand (uniform and non-uniform) and the signal wire, that is, using back-propagation neural network optimized by the beetle antennae search algorithm based on chaotic disturbance mechanism (CDBAS-BPNN) to extract the per unit length (p.u.l) parameter matrix, and combined with the chain parameter method to obtain crosstalk. Firstly, the geometric model and cross-sectional model between the uniform triple-twisted strand and the signal wire are established, and the corresponding model between the non-uniform triple-twisted strand and the signal wire is obtained by the Monte Carlo (MC) method. Then, the beetle antennae search algorithm based on chaotic disturbance mechanism (CDBAS) and backpropagation neural network (BPNN) are combined to construct a new extraction network of the p.u.l parameter matrix, and the chain parameter method is combined to predict crosstalk. Finally, in the verification and analysis part of the numerical experiments, comparing the crosstalk results of CDBAS-BPNN, BAS-BPNN and Transmission Line Matrix (TLM) algorithms, it is verified that the proposed method has better accuracy for the prediction of the model.

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