On the Exact Convex Hull of IFS Fractals

The problem of finding the convex hull of an IFS fractal is relevant in both theoretical and computational settings. Various methods exist that approximate it, but our aim is its exact determination. The finiteness of extremal points is examined a priori from the IFS parameters, revealing some cases when the convex hull problem is solvable. Former results are detailed from the literature, and two new methods are introduced and crystallized for practical applicability -- one more general, the other more efficient. Focal periodicity in the address of extremal points emerges as the central idea.Comment: Some but not all of these results appeared in the author's doctoral dissertation, and were presented at the 2014 Winter Meeting of the Canadian Mathematical Society. This research was started in Fall 2009, and the results were finalized in Feb. 2015. Some formatting changes were made in 2017. The paper is to appear in Fractals, Vol. 26, No. 1 (2018) 1850002. (Contains 34 pages with 11 figures.

Solving Linear System of Equations Via A Convex Hull Algorithm

We present new iterative algorithms for solving a square linear system $Ax=b$ in dimension $n$ by employing the {\it Triangle Algorithm} \cite{kal12}, a fully polynomial-time approximation scheme for testing if the convex hull of a finite set of points in a Euclidean space contains a given point. By converting $Ax=b$ into a convex hull problem and solving via the Triangle Algorithm, together with a {\it sensitivity theorem}, we compute in $O(n^2\epsilon^{-2})$ arithmetic operations an approximate solution satisfying $\Vert Ax_\epsilon - b \Vert \leq \epsilon \rho$, where $\rho= \max \{\Vert a_1 \Vert,..., \Vert a_n \Vert, \Vert b \Vert \}$, and $a_i$ is the $i$-th column of $A$. In another approach we apply the Triangle Algorithm incrementally, solving a sequence of convex hull problems while repeatedly employing a {\it distance duality}. The simplicity and theoretical complexity bounds of the proposed algorithms, requiring no structural restrictions on the matrix $A$, suggest their potential practicality, offering alternatives to the existing exact and iterative methods, especially for large scale linear systems. The assessment of computational performance however is the subject of future experimentations.Comment: 15 pages, 3 figure

Rectilinear Convex Hull with minimum area

Let $P$ be a planar set of $n$ points in general position. We consider the problem of computing an orientation of the plane for which the Rectilinear Convex Hull of $P$ has minimum area. Bae et al. (Computational Geometry: Theory and Applications, Vol. 42, 2009) solved the problem in quadratic time and linear space. We describe an algorithm that reduces this time complexity to $\Theta(n \log n)$

Spherical Triangle Algorithm: A Fast Oracle for Convex Hull Membership Queries

The it Convex Hull Membership(CHM) problem is: Given a point $p$ and a subset $S$ of $n$ points in $\mathbb{R}^m$, is $p \in conv(S)$? CHM is not only a fundamental problem in Linear Programming, Computational Geometry, Machine Learning and Statistics, it also serves as a query problem in many applications e.g. Topic Modeling, LP Feasibility, Data Reduction. The {\it Triangle Algorithm} (TA) \cite{kalantari2015characterization} either computes an approximate solution in the convex hull, or a separating hyperplane. The {\it Spherical}-CHM is a CHM, where $p=0$ and each point in $S$ has unit norm. First, we prove the equivalence of exact and approximate versions of CHM and Spherical-CHM. On the one hand, this makes it possible to state a simple version of the original TA. On the other hand, we prove that under the satisfiability of a simple condition in each iteration, the complexity improves to $O(1/\varepsilon)$. The analysis also suggests a strategy for when the property does not hold at an iterate. This suggests the \textit{Spherical-TA} which first converts a given CHM into a Spherical-CHM before applying the algorithm. Next we introduce a series of applications of Spherical-TA. In particular, Spherical-TA serves as a fast version of vanilla TA to boost its efficiency. As an example, this results in a fast version of \emph{AVTA} \cite{awasthi2018robust}, called \emph{AVTA$^+$} for solving exact or approximate irredundancy problem. Computationally, we have considered CHM, LP and Strict LP Feasibility and the Irredundancy problem. Based on substantial amount of computing, Spherical-TA achieves better efficiency than state of the art algorithms. Leveraging on the efficiency of Spherical-TA, we propose AVTA$^+$ as a pre-processing step for data reduction which arises in such applications as in computing the Minimum Volume Enclosing Ellipsoid \cite{moshtagh2005minimum}.Comment: 21 pages, 8 figures, 9 table

Multirobot rendezvous with visibility sensors in nonconvex environments

This paper presents a coordination algorithm for mobile autonomous robots. Relying upon distributed sensing the robots achieve rendezvous, that is, they move to a common location. Each robot is a point mass moving in a nonconvex environment according to an omnidirectional kinematic model. Each robot is equipped with line-of-sight limited-range sensors, i.e., a robot can measure the relative position of any object (robots or environment boundary) if and only if the object is within a given distance and there are no obstacles in-between. The algorithm is designed using the notions of robust visibility, connectivity-preserving constraint sets, and proximity graphs. Simulations illustrate the theoretical results on the correctness of the proposed algorithm, and its performance in asynchronous setups and with sensor measurement and control errors.Comment: 21 page

Exact computation of projection regression depth and fast computation of its induced median and other estimators

Zuo (2019) (Z19) addressed the computation of the projection regression depth (PRD) and its induced median (the maximum depth estimator). Z19 achieved the exact computation of PRD via a modified version of regular univariate sample median, which resulted in the loss of invariance of PRD and the equivariance of depth induced median. This article achieves the exact computation without scarifying the invariance of PRD and the equivariance of the regression median. Z19 also addressed the approximate computation of PRD induced median, the naive algorithm in Z19 is very slow. This article modifies the approximation in Z19 and adopts Rcpp package and consequently obtains a much (could be $100$ times) faster algorithm with an even better level of accuracy meanwhile. Furthermore, as the third major contribution, this article introduces three new depth induced estimators which can run $300$ times faster than that of Z19 meanwhile maintaining the same (or a bit better) level of accuracy. Real as well as simulated data examples are presented to illustrate the difference between the algorithms of Z19 and the ones proposed in this article. Findings support the statements above and manifest the major contributions of the article.Comment: 21 pages, 1 figure, 4 tables. arXiv admin note: substantial text overlap with arXiv:1905.1184

Computing the hull number in toll convexity

A walk $W$ between vertices $u$ and $v$ of a graph $G$ is called a {\em tolled walk between $u$ and $v$} if $u$, as well as $v$, has exactly one neighbour in $W$. A set $S \subseteq V(G)$ is {\em toll convex} if the vertices contained in any tolled walk between two vertices of $S$ are contained in $S$. The {\em toll convex hull of $S$} is the minimum toll convex set containing~$S$. The {\em toll hull number of $G$} is the minimum cardinality of a set $S$ such that the toll convex hull of $S$ is $V(G)$. The main contribution of this work is an algorithm for computing the toll hull number of a general graph in polynomial time.Comment: 21 pages; 1 figur

Optimal Compression of a Polyline with Segments and Arcs

This paper describes an efficient approach to constructing a resultant polyline with a minimum number of segments and arcs. While fitting an arc can be done with complexity O(1) (see [1] and [2]), the main complexity is in checking that the resultant arc is within the specified tolerance. There are additional tests to check for the ends and for changes in direction (see [3, section 3] and [4, sections II.C and II.D]). However, the most important part in reducing complexity is the ability to subdivide the polyline in order to limit the number of arc fittings [2]. The approach described in this paper finds a compressed polyline with a minimum number of segments and arcs.Comment: 40 pages, 34 figures, 3 table

Relaxation, New Combinatorial and Polynomial Algorithms for the Linear Feasibility Problem

We consider the homogenized linear feasibility problem, to find an $x$ on the unit sphere, satisfying $n$ line ar inequalities $a_i^Tx\ge 0$. To solve this problem we consider the centers of the insphere of spherical simpl ices, whose facets are determined by a subset of the constraints. As a result we find a new combinatorial algor ithm for the linear feasibility problem. If we allow rescaling this algorithm becomes polynomial. We point out that the algorithm solves as well the more general convex feasibility problem. Moreover numerical experiments s how that the algorithm could be of practical interest

A Practical Algorithm for Enumerating Collinear Points

This paper studies the problem of enumerating all maximal collinear subsets of size at least three in a given set of $n$ points. An algorithm for this problem, besides solving degeneracy testing and the exact fitting problem, can also help with other problems, such as point line cover and general position subset selection. The classic \emph{topological sweeping} algorithm of Edelsbrunner and Guibas can find these subsets in $O(n^2)$ time in the dual plane. We present an alternative algorithm that, although asymptotically slower than their algorithm in the worst case, is simpler to implement and more amenable to parallelization. If the input points are decomposed into $m$ convex polygons, our algorithm has time complexity $O(n^2 \log m)$ and space complexity $O(n)$. Our algorithm can be parallelized on the CREW PRAM with time complexity $O(n \log m)$ using $n$ processors