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Register a new userSpherical Triangle Algorithm: A Fast Oracle for Convex Hull Membership Queries
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Added by
Bahman Kalantari
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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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 approxima
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Given a subset $mathbf{S}={A_1, dots, A_m}$ of $mathbb{S}^n$, the set of $n times n$ real symmetric matrices, we define its {it spectrahull} as the set $SH(mathbf{S}) = {p(X) equiv (Tr(A_1 X), dots, Tr(A_m X))^T : X in mathbf{Delta}_n}$, where ${bf Delta}_n$ is the {it spectraplex}, ${ X in mathbb{S}^n : Tr(X)=1, X succeq 0 }$. We let {it spectrahull membership} (SHM) to be the problem of testing if a given $b in mathbb{R}^m$ lies in $SH(mathbf{S})$. On the one hand when $A_i$s are diagonal matrices, SHM reduces to the {it convex hull membership} (CHM), a fundamental problem in LP. On the other hand, a bounded SDP feasibility is reducible to SHM. By building on the {it Triangle Algorithm} (TA) cite{kalchar,kalsep}, developed for CHM and its generalization, we design a TA for SHM, where given $varepsilon$, in $O(1/varepsilon^2)$ iterations it either computes a hyperplane separating $b$ from $SH(mathbf{S})$, or $X_varepsilon in mathbf{Delta}_n$ such that $Vert p(X_varepsilon) - b Vert leq varepsilon R$, $R$ maximum error over $mathbf{Delta}_n$. Under certain conditions iteration complexity improves to $O(1/varepsilon)$ or even $O(ln 1/varepsilon)$. The worst-case complexity of each iteration is $O(mn^2)$, plus testing the existence of a pivot, shown to be equivalent to estimating the least eigenvalue of a symmetric matrix. This together with a semidefinite version of Caratheodory theorem allow implementing TA as if solving a CHM, resorting to the {it power method} only as needed, thereby improving the complexity of iterations. The proposed Triangle Algorithm for SHM is simple, practical and applicable to general SDP feasibility and optimization. Also, it extends to a spectral analogue of SVM for separation of two spectrahulls.
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.
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.
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.
Let $mathcal{P}$ be a polygonal domain of $h$ holes and $n$ vertices. We study the problem of constructing a data structure that can compute a shortest path between $s$ and $t$ in $mathcal{P}$ under the $L_1$ metric for any two query points $s$ and $t$. To do so, a standard approach is to first find a set of $n_s$ gateways for $s$ and a set of $n_t$ gateways for $t$ such that there exist a shortest $s$-$t$ path containing a gateway of $s$ and a gateway of $t$, and then compute a shortest $s$-$t$ path using these gateways. Previous algorithms all take quadratic $O(n_scdot n_t)$ time to solve this problem. In this paper, we propose a divide-and-conquer technique that solves the problem in $O(n_s + n_t log n_s)$ time. As a consequence, we construct a data structure of $O(n+(h^2log^3 h/loglog h))$ size in $O(n+(h^2log^4 h/loglog h))$ time such that each query can be answered in $O(log n)$ time.
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