2,237 research outputs found

    Rectilinear Link Diameter and Radius in a Rectilinear Polygonal Domain

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    We study the computation of the diameter and radius under the rectilinear link distance within a rectilinear polygonal domain of nn vertices and hh holes. We introduce a \emph{graph of oriented distances} to encode the distance between pairs of points of the domain. This helps us transform the problem so that we can search through the candidates more efficiently. Our algorithm computes both the diameter and the radius in min{O(nω),O(n2+nhlogh+χ2)}\min \{\,O(n^\omega), O(n^2 + nh \log h + \chi^2)\,\} time, where ω<2.373\omega<2.373 denotes the matrix multiplication exponent and χΩ(n)O(n2)\chi\in \Omega(n)\cap O(n^2) is the number of edges of the graph of oriented distances. We also provide a faster algorithm for computing the diameter that runs in O(n2logn)O(n^2 \log n) time

    Clustering with Neighborhoods

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    In the standard planar kk-center clustering problem, one is given a set PP of nn points in the plane, and the goal is to select kk center points, so as to minimize the maximum distance over points in PP to their nearest center. Here we initiate the systematic study of the clustering with neighborhoods problem, which generalizes the kk-center problem to allow the covered objects to be a set of general disjoint convex objects C\mathscr{C} rather than just a point set PP. For this problem we first show that there is a PTAS for approximating the number of centers. Specifically, if roptr_{opt} is the optimal radius for kk centers, then in nO(1/ε2)n^{O(1/\varepsilon^2)} time we can produce a set of (1+ε)k(1+\varepsilon)k centers with radius ropt\leq r_{opt}. If instead one considers the standard goal of approximating the optimal clustering radius, while keeping kk as a hard constraint, we show that the radius cannot be approximated within any factor in polynomial time unless P=NP\mathsf{P=NP}, even when C\mathscr{C} is a set of line segments. When C\mathscr{C} is a set of unit disks we show the problem is hard to approximate within a factor of 133236.99\frac{\sqrt{13}-\sqrt{3}}{2-\sqrt{3}}\approx 6.99. This hardness result complements our main result, where we show that when the objects are disks, of possibly differing radii, there is a (5+23)8.46(5+2\sqrt{3})\approx 8.46 approximation algorithm. Additionally, for unit disks we give an O(nlogk)+(k/ε)O(k)O(n\log k)+(k/\varepsilon)^{O(k)} time (1+ε)(1+\varepsilon)-approximation to the optimal radius, that is, an FPTAS for constant kk whose running time depends only linearly on nn. Finally, we show that the one dimensional version of the problem, even when intersections are allowed, can be solved exactly in O(nlogn)O(n\log n) time

    Discretization of Planar Geometric Cover Problems

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    We consider discretization of the 'geometric cover problem' in the plane: Given a set PP of nn points in the plane and a compact planar object T0T_0, find a minimum cardinality collection of planar translates of T0T_0 such that the union of the translates in the collection contains all the points in PP. We show that the geometric cover problem can be converted to a form of the geometric set cover, which has a given finite-size collection of translates rather than the infinite continuous solution space of the former. We propose a reduced finite solution space that consists of distinct canonical translates and present polynomial algorithms to find the reduce solution space for disks, convex/non-convex polygons (including holes), and planar objects consisting of finite Jordan curves.Comment: 16 pages, 5 figure

    An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures

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    We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints that are imposed by the manufacturing process, we obtain quite different results. In particular, we study the variant of the carpenter's ruler problem in which there is a restriction that only one joint can be modified at a time. For a linkage that does not self-intersect or self-touch, the recent results of Connelly et al. and Streinu imply that it can always be straightened, modifying one joint at a time. However, we show that for a linkage with even a single vertex degeneracy, it becomes NP-hard to decide if it can be straightened while altering only one joint at a time. If we add the restriction that each joint can be altered at most once, we show that the problem is NP-complete even without vertex degeneracies. In the special case, arising in wire forming manufacturing, that each joint can be altered at most once, and must be done sequentially from one or both ends of the linkage, we give an efficient algorithm to determine if a linkage can be straightened.Comment: 28 pages, 14 figures, Latex, to appear in Computational Geometry - Theory and Application

    Fast approximation of centrality and distances in hyperbolic graphs

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    We show that the eccentricities (and thus the centrality indices) of all vertices of a δ\delta-hyperbolic graph G=(V,E)G=(V,E) can be computed in linear time with an additive one-sided error of at most cδc\delta, i.e., after a linear time preprocessing, for every vertex vv of GG one can compute in O(1)O(1) time an estimate e^(v)\hat{e}(v) of its eccentricity eccG(v)ecc_G(v) such that eccG(v)e^(v)eccG(v)+cδecc_G(v)\leq \hat{e}(v)\leq ecc_G(v)+ c\delta for a small constant cc. We prove that every δ\delta-hyperbolic graph GG has a shortest path tree, constructible in linear time, such that for every vertex vv of GG, eccG(v)eccT(v)eccG(v)+cδecc_G(v)\leq ecc_T(v)\leq ecc_G(v)+ c\delta. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of GG, the smaller its eccentricity is. We also show that the distance matrix of GG with an additive one-sided error of at most cδc'\delta can be computed in O(V2log2V)O(|V|^2\log^2|V|) time, where c<cc'< c is a small constant. Recent empirical studies show that many real-world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating centrality and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.Comment: arXiv admin note: text overlap with arXiv:1506.01799 by other author

    On the Line-Separable Unit-Disk Coverage and Related Problems

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    Given a set PP of nn points and a set SS of mm disks in the plane, the disk coverage problem asks for a smallest subset of disks that together cover all points of PP. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of PP by a line \ell. We present an m2/3n2/32O(log(m+n))+O((n+m)log(n+m))m^{2/3}n^{2/3}2^{O(\log^*(m+n))} + O((n+m)\log (n+m)) time algorithm for the problem. This improves the previously best result of O(nm+nlogn)O(nm+ n\log n) time. Our techniques also solve the line-constrained version of the problem, where centers of all disks of SS are located on a line \ell while points of PP can be anywhere in the plane. Our algorithm runs in O(mn+(n+m)log(n+m))O(m\sqrt{n} + (n+m)\log(n+m)) time, which improves the previously best result of O(nmlog(m+n))O(nm\log(m+n)) time. In addition, our results lead to an algorithm of n10/32O(logn)n^{10/3}2^{O(\log^*n)} time for a half-plane coverage problem (given nn half-planes and nn points, find a smallest subset of half-planes covering all points); this improves the previously best algorithm of O(n4logn)O(n^4\log n) time. Further, if all half-planes are lower ones, our algorithm runs in n4/32O(logn)n^{4/3}2^{O(\log^*n)} time while the previously best algorithm takes O(n2logn)O(n^2\log n) time.Comment: To appear in ISAAC 202

    Discriminating Codes in Geometric Setups

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    We study geometric variations of the discriminating code problem. In the \emph{discrete version} of the problem, a finite set of points PP and a finite set of objects SS are given in Rd\mathbb{R}^d. The objective is to choose a subset SSS^* \subseteq S of minimum cardinality such that for each point piPp_i \in P, the subset SiSS_i^* \subseteq S^* covering pip_i satisfies SiS_i^*\neq \emptyset, and each pair pi,pjPp_i,p_j \in P, iji \neq j, we have SiSjS_i^* \neq S_j^*. In the \emph{continuous version} of the problem, the solution set SS^* can be chosen freely among a (potentially infinite) class of allowed geometric objects. In the 1-dimensional case (d=1d=1), the points in PP are placed on a horizontal line LL, and the objects in SS are finite-length line segments aligned with LL (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. Still, for the 1-dimensional discrete version, we design a polynomial-time 22-approximation algorithm. We also design a PTAS for both discrete and continuous versions in one dimension, for the restriction where the intervals are all required to have the same length. We then study the 2-dimensional case (d=2d=2) for axis-parallel unit square objects. We show that both continuous and discrete versions are NP-complete, and design polynomial-time approximation algorithms that produce (16OPT+1)(16\cdot OPT+1)-approximate and (64OPT+1)(64\cdot OPT+1)-approximate solutions respectively, using rounding of suitably defined integer linear programming problems. We show that the identifying code problem for axis-parallel unit square intersection graphs (in d=2d=2) can be solved in the same manner as for the discrete version of the discriminating code problem for unit square objects

    Probabilistic embeddings of the Fr\'echet distance

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    The Fr\'echet distance is a popular distance measure for curves which naturally lends itself to fundamental computational tasks, such as clustering, nearest-neighbor searching, and spherical range searching in the corresponding metric space. However, its inherent complexity poses considerable computational challenges in practice. To address this problem we study distortion of the probabilistic embedding that results from projecting the curves to a randomly chosen line. Such an embedding could be used in combination with, e.g. locality-sensitive hashing. We show that in the worst case and under reasonable assumptions, the discrete Fr\'echet distance between two polygonal curves of complexity tt in Rd\mathbb{R}^d, where d{2,3,4,5}d\in\lbrace 2,3,4,5\rbrace, degrades by a factor linear in tt with constant probability. We show upper and lower bounds on the distortion. We also evaluate our findings empirically on a benchmark data set. The preliminary experimental results stand in stark contrast with our lower bounds. They indicate that highly distorted projections happen very rarely in practice, and only for strongly conditioned input curves. Keywords: Fr\'echet distance, metric embeddings, random projectionsComment: 27 pages, 11 figure
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