arXiv subjects

Gill Barequet

Publications and source records attributed to Gill Barequet.

11 recordsLinked to original sources

arXiv2025

Polygon Containment and Translational Min-Hausdorff-Distance between Segment Sets are 3SUM-Hard

The 3SUM problem represents a class of problems conjectured to require $Ω(n^2)$ time to solve, where $n$ is the size of the input. Given two polygons $P$ and $Q$ in the plane, we show that some variants of the decision problem, whether there exists a transformation of $P$ that makes it contained in $Q$, are 3SUM-Hard. In the first variant $P$ and $Q$ are any simple polygons and the allowed transformations are translations only; in the second and third variants both polygons are convex and we allow either rotations only or any rigid motion. We also show that finding the translation in the plane that minimizes the Hausdorff distance between two segment sets is 3SUM-Hard.

cs.CG↗

arXiv2025

Efficiently Approximating the Minimum-Volume Bounding Box of a Point Set in Three Dimensions

$\renewcommand{\Re}{\mathbb{R}}$We present an efficient $O (n + 1/\varepsilon^{4.5})$-time algorithm for computing a $(1+\varepsilon$)-approximation of the minimum-volume bounding box of $n$ points in $\Re^3$. We also present a simpler algorithm (for the same purpose) whose running time is $O (n \log{n} + n / \varepsilon^3)$. We give some experimental results with implementations of various variants of the second algorithm. The implementation of the algorithm described in this paper is available online https://github.com/sarielhp/MVBB.

cs.CG↗

arXiv2022

Diamonds are Forever in the Blockchain: Geometric Polyhedral Point-Set Pattern Matching

Motivated by blockchain technology for supply-chain tracing of ethically sourced diamonds, we study geometric polyhedral point-set pattern matching as minimum-width polyhedral annulus problems under translations and rotations. We provide two $(1 + \varepsilon)$-approximation schemes under translations with $O(\varepsilon^{-d} n)$-time for $d$ dimensions and $O(n\log \varepsilon^{-1} + \varepsilon^{-2})$-time for two dimensions, and we give an $O(f^{d-1}\varepsilon^{1-2d}n)$-time algorithm when also allowing for rotations, parameterized on $f$, which we define as the slimness of the point set.

cs.CG↗

arXiv2022

Minimal-Perimeter Lattice Animals and the Constant-Isomer Conjecture

We consider minimal-perimeter lattice animals, providing a set of conditions which are sufficient for a lattice to have the property that inflating all minimal-perimeter animals of a certain size yields (without repetitions) all minimal-perimeter animals of a new, larger size. We demonstrate this result on the two-dimensional square and hexagonal lattices. In addition, we characterize the sizes of minimal-perimeter animals on these lattices that are not created by inflating members of another set of minimal-perimeter animals.

cs.CG↗

arXiv2020

Stable-Matching Voronoi Diagrams: Combinatorial Complexity and Algorithms

We study algorithms and combinatorial complexity bounds for \emph{stable-matching Voronoi diagrams}, where a set, $S$, of $n$ point sites in the plane determines a stable matching between the points in $\mathbb{R}^2$ and the sites in $S$ such that (i) the points prefer sites closer to them and sites prefer points closer to them, and (ii) each site has a quota or "appetite" indicating the area of the set of points that can be matched to it. Thus, a stable-matching Voronoi diagram is a solution to the well-known post office problem with the added (realistic) constraint that each post office has a limit on the size of its jurisdiction. Previous work on the stable-matching Voronoi diagram provided existence and uniqueness proofs, but did not analyze its combinatorial or algorithmic complexity. In this paper, we show that a stable-matching Voronoi diagram of $n$ point sites has $O(n^{2+\varepsilon})$ faces and edges, for any $\varepsilon>0$, and show that this bound is almost tight by giving a family of diagrams with $Θ(n^2)$ faces and edges. We also provide a discrete algorithm for constructing it in $O(n^3\log n+n^2f(n))$ time in the real-RAM model of computation, where $f(n)$ is the runtime of a geometric primitive (which we define) that can be approximated numerically, but cannot, in general, be performed exactly in an algebraic model of computation. We show, however, how to compute the geometric primitive exactly for polygonal convex distance functions.

cs.CG↗

arXiv2019

Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

A $d$-dimensional polycube is a facet-connected set of cells (cubes) on the $d$-dimensional cubical lattice $\mathbb{Z}^d$. Let $A_d(n)$ denote the number of $d$-dimensional polycubes (distinct up to translations) with $n$ cubes, and $λ_d$ denote the limit of the ratio $A_d(n{+}1)/A_d(n)$ as $n \to \infty$. The exact value of $λ_d$ is still unknown rigorously for any dimension $d \geq 2$; the asymptotics of $λ_d$, as $d \to \infty$, also remained elusive as of today. In this paper, we revisit and extend the approach presented by Klarner and Rivest in 1973 to bound $A_2(n)$ from above. Our contributions are: Using available computing power, we prove that $λ_2 \leq 4.5252$. This is the first improvement of the upper bound on $λ_2$ in almost half a century; We prove that $λ_d \leq (2d-2)e+o(1)$ for any value of $d \geq 2$, using a novel construction of a rational generating function which dominates that of the sequence $\left(A_d(n)\right)$; For $d=3$, this provides a subtantial improvement of the upper bound on $λ_3$ from 12.2071 to 9.8073; However, we implement an iterative process in three dimensions, which improves further the upper bound on $λ_3$to $9.3835$.

cs.DM↗

arXiv2016

The Flip Diameter of Rectangulations and Convex Subdivisions

We study the configuration space of rectangulations and convex subdivisions of $n$ points in the plane. It is shown that a sequence of $O(n\log n)$ elementary flip and rotate operations can transform any rectangulation to any other rectangulation on the same set of $n$ points. This bound is the best possible for some point sets, while $Θ(n)$ operations are sufficient and necessary for others. Some of our bounds generalize to convex subdivisions of $n$ points in the plane.

cs.DM↗

arXiv2015

Diffuse Reflection Diameter in Simple Polygons

We prove a conjecture of Aanjaneya, Bishnu, and Pal that the minimum number of diffuse reflections sufficient to illuminate the interior of any simple polygon with $n$ walls from any interior point light source is $\lfloor n/2 \rfloor - 1$. Light reflecting diffusely leaves a surface in all directions, rather than at an identical angle as with specular reflections.

cs.CG↗

arXiv2012

Orders induced by segments in floorplan partitions and (2-14-3,3-41-2)-avoiding permutations

A floorplan is a tiling of a rectangle by rectangles. There are natural ways to order the elements---rectangles and segments---of a floorplan. Ackerman, Barequet and Pinter studied a pair of orders induced by neighborhood relations between rectangles, and obtained a natural bijection between these pairs and (2-41-3, 3-14-2)-avoiding permutations, also known as (reduced) Baxter permutations. In the present paper, we first perform a similar study for a pair of orders induced by neighborhood relations between segments of a floorplan. We obtain a natural bijection between these pairs and another family of permutations, namely (2-14-3, 3-41-2)-avoiding permutations. Then, we investigate relations between the two kinds of pairs of orders---and, correspondingly, between (2-41-3, 3-14-2)- and (2-14-3, 3-41-2)-avoiding permutations. In particular, we prove that the superposition of both permutations gives a complete Baxter permutation (originally called w-admissible, by Baxter and Joichi in the sixties). In other words, (2-14-3, 3-41-2)-avoiding permutations are the hidden part of complete Baxter permutations. We enumerate these permutations. To our knowledge, the characterization of these permutations in terms of forbidden patterns and their enumeration are both new results. Finally, we also study the special case of the so-called guillotine floorplans.

math.CO↗

arXiv2011

On 2-Site Voronoi Diagrams under Geometric Distance Functions

We revisit a new type of a Voronoi diagram, in which distance is measured from a point to a pair of points. We consider a few more such distance functions, based on geometric primitives, and analyze the structure and complexity of the nearest- and furthest-neighbor Voronoi diagrams of a point set with respect to these distance functions.

cs.CG↗

arXiv2008

Straight Skeletons of Three-Dimensional Polyhedra

This paper studies the straight skeleton of polyhedra in three dimensions. We first address voxel-based polyhedra (polycubes), formed as the union of a collection of cubical (axis-aligned) voxels. We analyze the ways in which the skeleton may intersect each voxel of the polyhedron, and show that the skeleton may be constructed by a simple voxel-sweeping algorithm taking constant time per voxel. In addition, we describe a more complex algorithm for straight skeletons of voxel-based polyhedra, which takes time proportional to the area of the surfaces of the straight skeleton rather than the volume of the polyhedron. We also consider more general polyhedra with axis-parallel edges and faces, and show that any n-vertex polyhedron of this type has a straight skeleton with O(n^2) features. We provide algorithms for constructing the straight skeleton, with running time O(min(n^2 log n, k log^{O(1)} n)) where k is the output complexity. Next, we discuss the straight skeleton of a general nonconvex polyhedron. We show that it has an ambiguity issue, and suggest a consistent method to resolve it. We prove that the straight skeleton of a general polyhedron has a superquadratic complexity in the worst case. Finally, we report on an implementation of a simple algorithm for the general case.

cs.CG↗