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Register a new userDynamic Distribution-Sensitive Point Location
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Added by
Siu-Wing Cheng
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We propose a dynamic data structure for the distribution-sensitive point location problem. Suppose that there is a fixed query distribution in $mathbb{R}^2$, and we are given an oracle that can return in $O(1)$ time the probability of a query point falling into a polygonal region of constant complexity. We can maintain a convex subdivision $cal S$ with $n$ vertices such that each query is answered in $O(mathrm{OPT})$ expected time, where OPT is the minimum expected time of the best linear decision tree for point location in $cal S$. The space and construction time are $O(nlog^2 n)$. An update of $cal S$ as a mixed sequence of $k$ edge insertions and deletions takes $O(klog^5 n)$ amortized time. As a corollary, the randomized incremental construction of the Voronoi diagram of $n$ sites can be performed in $O(nlog^5 n)$ expected time so that, during the incremental construction, a nearest neighbor query at any time can be answered optimally with respect to the intermediate Voronoi diagram at that time.
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We present self-adjusting data structures for answering point location queries in convex and connected subdivisions. Let $n$ be the number of vertices in a convex or connected subdivision. Our structures use $O(n)$ space. For any convex subdivision $S$, our method processes any online query sequence $sigma$ in $O(mathrm{OPT} + n)$ time, where $mathrm{OPT}$ is the minimum time required by any linear decision tree for answering point location queries in $S$ to process $sigma$. For connected subdivisions, the processing time is $O(mathrm{OPT} + n + |sigma|log(log^* n))$. In both cases, the time bound includes the $O(n)$ preprocessing time.
Let $S$ be a set of $n$ sites, each associated with a point in $mathbb{R}^2$ and a radius $r_s$ and let $mathcal{D}(S)$ be the disk graph on $S$. We consider the problem of designing data structures that maintain the connectivity structure of $mathcal{D}(S)$ while allowing the insertion and deletion of sites. For unit disk graphs we describe a data structure that has $O(log^2n)$ amortized update time and $O((log n)/(loglog n))$ amortized query time. For disk graphs where the ratio $Psi$ between the largest and smallest radius is bounded, we consider the decremental and the incremental case separately, in addition to the fully dynamic case. In the fully dynamic case we achieve amortized $O(Psi lambda_6(log n) log^{9}n)$ update time and $O(log n)$ query time, where $lambda_s(n)$ is the maximum length of a Davenport-Schinzel sequence of order $s$ on $n$ symbols. This improves the update time of the currently best known data structure by a factor of $Psi$ at the cost of an additional $O(log log n)$ factor in the query time. In the incremental case we manage to achieve a logarithmic dependency on $Psi$ with a data structure with $O(alpha(n))$ query and $O(logPsi lambda_6(log n) log^{9}n)$ update time. For the decremental setting we first develop a new dynamic data structure that allows us to maintain two sets $B$ and $P$ of disks, such than at a deletion of a disk from $B$ we can efficiently report all disks in $P$ that no longer intersect any disk of $B$. Having this data structure at hand, we get decremental data structures with an amortized query time of $O((log n)/(log log n))$ supporting $m$ deletions in $O((nlog^{5}n + m log^{9}n) lambda_6(log n) + nlogPsilog^4n)$ overall time for bounded radius ratio $Psi$ and $O(( nlog^{6} n + m log^{10}n) lambda_6(log n))$ for general disk graphs.
Given a finite set $X subset mathbb{R}^d$ and a binary linear classifier $c: mathbb{R}^d to {0,1}$, how many queries of the form $c(x)$ are required to learn the label of every point in $X$? Known as textit{point location}, this problem has inspired over 35 years of research in the pursuit of an optimal algorithm. Building on the prior work of Kane, Lovett, and Moran (ICALP 2018), we provide the first nearly optimal solution, a randomized linear decision tree of depth $tilde{O}(dlog(|X|))$, improving on the previous best of $tilde{O}(d^2log(|X|))$ from Ezra and Sharir (Discrete and Computational Geometry, 2019). As a corollary, we also provide the first nearly optimal algorithm for actively learning halfspaces in the membership query model. En route to these results, we prove a novel characterization of Barthes Theorem (Inventiones Mathematicae, 1998) of independent interest. In particular, we show that $X$ may be transformed into approximate isotropic position if and only if there exists no $k$-dimensional subspace with more than a $k/d$-fraction of $X$, and provide a similar characterization for exact isotropic position.
We study how to dynamize the Trapezoidal Search Tree - a well known randomized point location structure for planar subdivisions of kinetic line segments. Our approach naturally extends incremental leaf-level insertions to recursive methods and allows adaptation for the online setting. Moreover, the dynamization carries over to the Trapezoidal Search DAG, offering a linear sized data structure with logarithmic point location costs as a by-product. On a set $S$ of non-crossing segments, each update performs expected ${mathcal O}(log^2|S|)$ operations. We demonstrate the practicality of our method with an open-source implementation, based on the Computational Geometry Algorithms Library, and experiments on the update performance.
We consider the problem of maintaining an approximate maximum independent set of geometric objects under insertions and deletions. We present data structures that maintain a constant-factor approximate maximum independent set for broad classes of fat objects in $d$ dimensions, where $d$ is assumed to be a constant, in sublinear textit{worst-case} update time. This gives the first results for dynamic independent set in a wide variety of geometric settings, such as disks, fat polygons, and their high-dimensional equivalents. For axis-aligned squares and hypercubes, our result improves upon all (recently announced) previous works. We obtain, in particular, a dynamic $(4+epsilon)$-approximation for squares, with $O(log^4 n)$ worst-case update time. Our result is obtained via a two-level approach. First, we develop a dynamic data structure which stores all objects and provides an approximate independent set when queried, with output-sensitive running time. We show that via standard methods such a structure can be used to obtain a dynamic algorithm with textit{amortized} update time bounds. Then, to obtain worst-case update time algorithms, we develop a generic deamortization scheme that with each insertion/deletion keeps (i) the update time bounded and (ii) the number of changes in the independent set constant. We show that such a scheme is applicable to fat objects by showing an appropriate generalization of a separator theorem. Interestingly, we show that our deamortization scheme is also necessary in order to obtain worst-case update bounds: If for a class of objects our scheme is not applicable, then no constant-factor approximation with sublinear worst-case update time is possible. We show that such a lower bound applies even for seemingly simple classes of geometric objects including axis-aligned rectangles in the plane.
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