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Register a new userThe convex hull of a regular set of integer vectors is polyhedral and effectively computable
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Added by
Jerome Leroux
Publication date
2008
fields
Informatics Engineering
and research's language is
English
Authors
Alain Finkel
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No Arabic abstract
Number Decision Diagrams (NDD) provide a natural finite symbolic representation for regular set of integer vectors encoded as strings of digit vectors (least or most significant digit first). The convex hull of the set of vectors represented by a NDD is proved to be an effectively computable convex polyhedron.
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In this paper, we first consider the subpath convex hull query problem: Given a simple path $pi$ of $n$ vertices, preprocess it so that the convex hull of any query subpath of $pi$ can be quickly obtained. Previously, Guibas, Hershberger, and Snoeyink [SODA 90] proposed a data structure of $O(n)$ space and $O(log nloglog n)$ query time; reducing the query time to $O(log n)$ increases the space to $O(nloglog n)$. We present an improved result that uses $O(n)$ space while achieving $O(log n)$ query time. Like the previous work, our query algorithm returns a compact interval tree representing the convex hull so that standard binary-search-based queries on the hull can be performed in $O(log n)$ time each. Our new result leads to improvements for several other problems. In particular, with the help of the above result, we present new algorithms for the ray-shooting problem among segments. Given a set of $n$ (possibly intersecting) line segments in the plane, preprocess it so that the first segment hit by a query ray can be quickly found. We give a data structure of $O(nlog n)$ space that can answer each query in $(sqrt{n}log n)$ time. If the segments are nonintersecting or if the segments are lines, then the space can be reduced to $O(n)$. All these are classical problems that have been studied extensively. Previously data structures of $widetilde{O}(sqrt{n})$ query time (the notation $widetilde{O}$ suppresses a polylogarithmic factor) were known in early 1990s; nearly no progress has been made for over two decades. For all problems, our results provide improvements by reducing the space of the data structures by at least a logarithmic factor while the preprocessing and query times are the same as before or even better.
A subset $A$ of a Banach space is called Banach-Saks when every sequence in $A$ has a Ces{`a}ro convergent subsequence. Our interest here focusses on the following problem: is the convex hull of a Banach-Saks set again Banach-Saks? By means of a combinatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result.
Given a finite set of points $P subseteq mathbb{R}^d$, we would like to find a small subset $S subseteq P$ such that the convex hull of $S$ approximately contains $P$. More formally, every point in $P$ is within distance $epsilon$ from the convex hull of $S$. Such a subset $S$ is called an $epsilon$-hull. Computing an $epsilon$-hull is an important problem in computational geometry, machine learning, and approximation algorithms. In many real world applications, the set $P$ is too large to fit in memory. We consider the streaming model where the algorithm receives the points of $P$ sequentially and strives to use a minimal amount of memory. Existing streaming algorithms for computing an $epsilon$-hull require $O(epsilon^{-(d-1)/2})$ space, which is optimal for a worst-case input. However, this ignores the structure of the data. The minimal size of an $epsilon$-hull of $P$, which we denote by $text{OPT}$, can be much smaller. A natural question is whether a streaming algorithm can compute an $epsilon$-hull using only $O(text{OPT})$ space. We begin with lower bounds that show that it is not possible to have a single-pass streaming algorithm that computes an $epsilon$-hull with $O(text{OPT})$ space. We instead propose three relaxations of the problem for which we can compute $epsilon$-hulls using space near-linear to the optimal size. Our first algorithm for points in $mathbb{R}^2$ that arrive in random-order uses $O(log ncdot text{OPT})$ space. Our second algorithm for points in $mathbb{R}^2$ makes $O(log(frac{1}{epsilon}))$ passes before outputting the $epsilon$-hull and requires $O(text{OPT})$ space. Our third algorithm for points in $mathbb{R}^d$ for any fixed dimension $d$ outputs an $epsilon$-hull for all but $delta$-fraction of directions and requires $O(text{OPT} cdot log text{OPT})$ space.
Let $K in R^d$ be a convex body, and assume that $L$ is a randomly rotated and shifted integer lattice. Let $K_L$ be the convex hull of the (random) points $K cap L$. The mean width $W(K_L)$ of $K_L$ is investigated. The asymptotic order of the mean width difference $W(l K)-W((l K)_L)$ is maximized by the order obtained by polytopes and minimized by the order for smooth convex sets as $l to infty$.
Let $x_1,ldots ,x_N$ be independent random points distributed according to an isotropic log-concave measure $mu $ on ${mathbb R}^n$, and consider the random polytope $$K_N:={rm conv}{ pm x_1,ldots ,pm x_N}.$$ We provide sharp estimates for the quermass{}integrals and other geometric parameters of $K_N$ in the range $cnls Nlsexp (n)$; these complement previous results from cite{DGT1} and cite{DGT} that were given for the range $cnls Nlsexp (sqrt{n})$. One of the basic new ingredients in our work is a recent result of E.~Milman that determines the mean width of the centroid body $Z_q(mu )$ of $mu $ for all $1ls qls n$.
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