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Boris Aronov

Publications and source records attributed to Boris Aronov.

At least 37 records · Page 2Linked to original sources
arXiv2019

Efficient Nearest-Neighbor Query and Clustering of Planar Curves

We study two fundamental problems dealing with curves in the plane, namely, the nearest-neighbor problem and the center problem. Let $\mathcal{C}$ be a set of $n$ polygonal curves, each of size $m$. In the nearest-neighbor problem, the goal is to construct a compact data structure over $\mathcal{C}$, such that, given a query curve $Q$, one can efficiently find the curve in $\mathcal{C}$ closest to $Q$. In the center problem, the goal is to find a curve $Q$, such that the maximum distance between $Q$ and the curves in $\mathcal{C}$ is minimized. We use the well-known discrete Frechet distance function, both under~$L_\infty$ and under $L_2$, to measure the distance between two curves. For the nearest-neighbor problem, despite discouraging previous results, we identify two important cases for which it is possible to obtain practical bounds, even when $m$ and $n$ are large. In these cases, either $Q$ is a line segment or $\mathcal{C}$ consists of line segments, and the bounds on the size of the data structure and query time are nearly linear in the size of the input and query curve, respectively. The returned answer is either exact under $L_\infty$, or approximated to within a factor of $1+\varepsilon$ under~$L_2$. We also consider the variants in which the location of the input curves is only fixed up to translation, and obtain similar bounds, under $L_\infty$. As for the center problem, we study the case where the center is a line segment, i.e., we seek the line segment that represents the given set as well as possible. We present near-linear time exact algorithms under $L_\infty$, even when the location of the input curves is only fixed up to translation. Under $L_2$, we present a roughly $O(n^2m^3)$-time exact algorithm.

cs.CG↗
arXiv2019

Eliminating Depth Cycles among Triangles in Three Dimensions

Given $n$ pairwise openly disjoint triangles in 3-space, their vertical depth relation may contain cycles. We show that, for any $\varepsilon>0$, the triangles can be cut into $O(n^{3/2+\varepsilon})$ connected semi-algebraic pieces, whose description complexity depends only on the choice of $\varepsilon$, such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. We are not aware of any previous study of this problem, in this full generality, with a subquadratic bound on the number of pieces. This work extends the recent study by two of the authors (Aronov, Sharir~2018) on eliminating depth cycles among lines in 3-space. Our approach is again algebraic, and makes use of a recent variant of the polynomial partitioning technique, due to Guth, which leads to a recursive procedure for cutting the triangles. In contrast to the case of lines, our analysis here is considerably more involved, due to the two-dimensional nature of the objects being cut, so additional tools, from topology and algebra, need to be brought to bear. Our result essentially settles a 35-year-old open problem in computational geometry, motivated by hidden-surface removal in computer graphics.

cs.CG↗
arXiv2018

More Turán-Type Theorems for Triangles in Convex Point Sets

We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact. This leads to 256 extremal Turán-type questions. We give nearly tight (within a $\log n$ factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein's longstanding tripod packing problem.

math.CO↗
arXiv2018

On Pseudo-disk Hypergraphs

Let $F$ be a family of pseudo-disks in the plane, and $P$ be a finite subset of $F$. Consider the hypergraph $H(P,F)$ whose vertices are the pseudo-disks in $P$ and the edges are all subsets of $P$ of the form $\{D \in P \mid D \cap S \neq \emptyset\}$, where $S$ is a pseudo-disk in $F$. We give an upper bound of $O(nk^3)$ for the number of edges in $H(P,F)$ of cardinality at most $k$. This generalizes a result of Buzaglo et al. (2013). As an application of our bound, we obtain an algorithm that computes a constant-factor approximation to the smallest _weighted_ dominating set in a collection of pseudo-disks in the plane, in expected polynomial time.

cs.CG↗
arXiv2017

Batched Point Location in SINR Diagrams via Algebraic Tools

The SINR model for wireless networks has been extensively studied recently. It tries to model whether a particular transmitter is heard at a specific location, with $n$ transmitting simultaneously. The SINR diagram consists of $n$ regions where each transmitter can be heard and the remaining space where no one can be heard. Efficient point location in the SINR diagram, i.e., building a data structure to determine, for a query point, whether any transmitter is heard there, and if so, which one, has been recently investigated. Previous such planar data structures are constructed in time at least quadratic in $n$ and support logarithmic-time approximate queries. Moreover, the performance of these depends not only on the number $n$ of transmitters and on the approximation parameter $\varepsilon$, but also on some geometric parameters that cannot be bounded a priori as a function of $n$ or $\varepsilon$. We address the question of batched point location queries, i.e., answering many queries simultaneously. Specifically, in one dimension, we can answer $n$ queries exactly in amortized polylogarithmic time per query, while in the plane we can do it approximately. All these results can handle arbitrary power assignments to the transmitters. Moreover, the amortized query time in these results depends only on $n$ and $\varepsilon$. We also show how to speed up the preprocessing in a previously proposed point-location structure in SINR diagram for uniform-power sites, by almost a full order of magnitude. For this we obtain results on the sensitivity of the reception regions to slight changes in the reception threshold, which are of independent interest. Finally, these results demonstrate the (so far underutilized) power of combining algebraic tools with those of computational geometry and other fields.

cs.CG↗
arXiv2017

Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams

We consider preprocessing a set $S$ of $n$ points in convex position in the plane into a data structure supporting queries of the following form: given a point $q$ and a directed line $\ell$ in the plane, report the point of $S$ that is farthest from (or, alternatively, nearest to) the point $q$ among all points to the left of line $\ell$. We present two data structures for this problem. The first data structure uses $O(n^{1+\varepsilon})$ space and preprocessing time, and answers queries in $O(2^{1/\varepsilon} \log n)$ time, for any $0 < \varepsilon < 1$. The second data structure uses $O(n \log^3 n)$ space and polynomial preprocessing time, and answers queries in $O(\log n)$ time. These are the first solutions to the problem with $O(\log n)$ query time and $o(n^2)$ space. The second data structure uses a new representation of nearest- and farthest-point Voronoi diagrams of points in convex position. This representation supports the insertion of new points in clockwise order using only $O(\log n)$ amortized pointer changes, in addition to $O(\log n)$-time point-location queries, even though every such update may make $Θ(n)$ combinatorial changes to the Voronoi diagram. This data structure is the first demonstration that deterministically and incrementally constructed Voronoi diagrams can be maintained in $o(n)$ amortized pointer changes per operation while keeping $O(\log n)$-time point-location queries.

cs.CG↗
arXiv2017

Convex Equipartitions: The Spicy Chicken Theorem

We show that, for any prime power n and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into n convex sets with equal volumes and equal surface areas. Similar results regarding equipartitions with respect to continuous functionals and absolutely continuous measures on convex bodies are also proven. These include a generalization of the ham-sandwich theorem to arbitrary number of convex pieces confirming a conjecture of Kaneko and Kano, a similar generalization of perfect partitions of a cake and its icing, and a generalization of the Gromov-Borsuk-Ulam theorem for convex sets in the model spaces of constant curvature. Most of the results in this paper appear in arxiv:1011.4762 and in arxiv:1010.4611. Since the main results and techniques there are essentially the same, we have merged the papers for journal publication. In this version we also provide a technical alternative to a part of the proof of the main topological result that avoids the use of compactly supported homology.

math.MG↗
arXiv2017

Time-Space Trade-off Algorithms for Triangulating a Simple Polygon

An $s$-workspace algorithm is an algorithm that has read-only access to the values of the input, write-only access to the output, and only uses $O(s)$ additional words of space. We present a randomized $s$-workspace algorithm for triangulating a simple polygon $P$ of $n$ vertices that runs in $O(n^2/s+n \log n \log^{5} (n/s))$ expected time using $O(s)$ variables, for any $s \leq n$. In particular, when $s \leq \frac{n}{\log n\log^{5}\log n}$ the algorithm runs in $O(n^2/s)$ expected time.

cs.CG↗
arXiv2016

Almost Tight Bounds for Eliminating Depth Cycles in Three Dimensions

Given $n$ non-vertical lines in 3-space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into $O(n^{3/2}\mathop{\mathrm{polylog}} n)$ pieces, such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. Previous results on this topic could only handle restricted cases of the problem (such as handling only triangular cycles, by Aronov, Koltun, and Sharir (2005), or only cycles in grid-like patterns, by Chazelle et al. (1992)), and the bounds were considerably weaker---much closer to the trivial quadratic bound. Our proof uses a recent variant of the polynomial partitioning technique, due to Guth, and some simple tools from algebraic geometry. It is much more straightforward than the previous "purely combinatorial" methods. Our technique can be extended to eliminating all cycles in the depth relation among segments, and among constant-degree algebraic arcs. We hope that a suitable extension of this technique could be used to handle the much more difficult case of pairwise-disjoint triangles. We also discuss several algorithms for constructing a small set of cuts so as to eliminate all depth-relation cycles among the lines (minimizing such a set, for the case of line segments, is known to be NP-complete). The performance of these algorithms improves due to our new bound, but, so far, none of them both (a) produce close to $n^{3/2}$ cuts, and (b) run in time close to $n^{3/2}$, in the worst case. Our results almost completely settle a 35-year-old open problem in computational geometry, motivated by hidden-surface removal in computer graphics.

cs.CG↗
arXiv2016

Nearest-Neighbor Searching Under Uncertainty II

Nearest-neighbor search, which returns the nearest neighbor of a query point in a set of points, is an important and widely studied problem in many fields, and it has wide range of applications. In many of them, such as sensor databases, location-based services, face recognition, and mobile data, the location of data is imprecise. We therefore study nearest-neighbor queries in a probabilistic framework in which the location of each input point is specified as a probability distribution function. We present efficient algorithms for - computing all points that are nearest neighbors of a query point with nonzero probability; and - estimating the probability of a point being the nearest neighbor of a query point, either exactly or within a specified additive error.

cs.CG↗
arXiv2016

Distance-Sensitive Planar Point Location

Let $\mathcal{S}$ be a connected planar polygonal subdivision with $n$ edges that we want to preprocess for point-location queries, and where we are given the probability $γ_i$ that the query point lies in a polygon $P_i$ of $\mathcal{S}$. We show how to preprocess $\mathcal{S}$ such that the query time for a point~$p\in P_i$ depends on~$γ_i$ and, in addition, on the distance from $p$ to the boundary of~$P_i$---the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time $O\left(\min \left(\log n, 1 + \log \frac{\mathrm{area}(P_i)}{γ_i Δ_{p}^2}\right)\right)$, where $Δ_{p}$ is the shortest Euclidean distance of the query point~$p$ to the boundary of $P_i$. Our structure uses $O(n)$ space and $O(n \log n)$ preprocessing time. It is based on a decomposition of the regions of $\mathcal{S}$ into convex quadrilaterals and triangles with the following property: for any point $p\in P_i$, the quadrilateral or triangle containing~$p$ has area $Ω(Δ_{p}^2)$. For the special case where $\mathcal{S}$ is a subdivision of the unit square and $γ_i=\mathrm{area}(P_i)$, we present a simpler solution that achieves a query time of $O\left(\min \left(\log n, \log \frac{1}{Δ_{p}^2}\right)\right)$. The latter solution can be extended to convex subdivisions in three dimensions.

cs.CG↗
arXiv2015

Fréchet Distance for Curves, Revisited

$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\eps}{\varepsilon} \newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)}$ We revisit the problem of computing Fréchet distance between polygonal curves under $L_1$, $L_2$, and $L_\infty$ norms, focusing on discrete Fréchet distance, where only distance between vertices is considered. We develop efficient algorithms for two natural classes of curves. In particular, given two polygonal curves of $n$ vertices each, a $\eps$-approximation of their discrete Fréchet distance can be computed in roughly $O(nκ^3\log n/\eps^3)$ time in three dimensions, if one of the curves is \emph{$κ$-bounded}. Previously, only a $κ$-approximation algorithm was known. If both curves are the so-called \emph{\backbone~curves}, which are widely used to model protein backbones in molecular biology, we can $\eps$-approximate their Fréchet distance in near linear time in two dimensions, and in roughly $O(n^{4/3}\log nm)$ time in three dimensions. In the second part, we propose a pseudo--output-sensitive algorithm for computing Fréchet distance exactly. The complexity of the algorithm is a function of a quantity we call the \emph{\bwnumber{}}, which is quadratic in the worst case, but tends to be much smaller in practice.

cs.CG↗
arXiv2015

The Number of Holes in the Union of Translates of a Convex Set in Three Dimensions

We show that the union of $n$ translates of a convex body in $\mathbb{R}^3$ can have $Θ(n^3)$ holes in the worst case, where a hole in a set $X$ is a connected component of $\mathbb{R}^3 \setminus X$. This refutes a 20-year-old conjecture. As a consequence, we also obtain improved lower bounds on the complexity of motion planning problems and of Voronoi diagrams with convex distance functions.

cs.CG↗
arXiv2014

How To Place a Point to Maximize Angles

We describe a randomized algorithm that, given a set $P$ of points in the plane, computes the best location to insert a new point $p$, such that the Delaunay triangulation of $P\cup\{p\}$ has the largest possible minimum angle. The expected running time of our algorithm is at most cubic, improving the roughly quartic time of the best previously known algorithm. It slows down to slightly super-cubic if we also specify a set of non-crossing segments with endpoints in $P$ and insist that the triangulation respect these segments, i.e., is the constrained Delaunay triangulation of the points and segments.

cs.CG↗
arXiv2011

Convex Equipartitions of volume and surface area

We show that, for any prime power p^k and any convex body K (i.e., a compact convex set with interior) in Rd, there exists a partition of K into p^k convex sets with equal volume and equal surface area. We derive this result from a more general one for absolutely continuous measures and continuous functionals on the space of convex bodies. This result was independently found by Roman Karasev generalizing work of Gromov and Memarian who proved it for the standard measure on the sphere and p=2. Our proof uses basics from the theory of optimal transport and equivariant topology. The topological ingredient is a Borsuk-Ulam type statement on configuration space with the standard action of the symmetric group. This result was discovered in increasing generality by Fuks, Vaseliev and Karasev. We include a detailed proof and discuss how it relates to Gromov's proof for the case p=2.

math.MG↗
arXiv2011

Complexity of a Single Face in an Arrangement of s-Intersecting Curves

Consider a face F in an arrangement of n Jordan curves in the plane, no two of which intersect more than s times. We prove that the combinatorial complexity of F is O(λ_s(n)), O(λ_{s+1}(n)), and O(λ_{s+2}(n)), when the curves are bi-infinite, semi-infinite, or bounded, respectively; λ_k(n) is the maximum length of a Davenport-Schinzel sequence of order k on an alphabet of n symbols. Our bounds asymptotically match the known worst-case lower bounds. Our proof settles the still apparently open case of semi-infinite curves. Moreover, it treats the three cases in a fairly uniform fashion.

cs.CG↗
arXiv2011

Witness Rectangle Graphs

In a witness rectangle graph (WRG) on vertex point set P with respect to witness points set W in the plane, two points x, y in P are adjacent whenever the open isothetic rectangle with x and y as opposite corners contains at least one point in W. WRGs are representative of a a larger family of witness proximity graphs introduced in two previous papers. We study graph-theoretic properties of WRGs. We prove that any WRG has at most two non-trivial connected components. We bound the diameter of the non-trivial connected components of a WRG in both the one-component and two-component cases. In the latter case, we prove that a graph is representable as a WRG if and only if each component is a connected co-interval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs. In addition, we prove that a WRG with no isolated vertices has domination number at most four. Moreover, we show that any combinatorial graph can be drawn as a WRG using a combination of positive and negative witnesses. Finally we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n point.

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