Visibilty: Finding the Staircase Kernel in Orthogonal Polygons

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Polygon Placement Revisited: (Degree of Freedom + 1)-SUM Hardness and an Improvement via Offline Dynamic Rectangle UnionMarvin Künnemann and André NusserMarvin Künnemann and André Nusserpp.3181 - 3201Chapter DOI:https://doi.org/10.1137/1.9781611977073.124PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We revisit a classical problem in computational geometry: Determine the largest copy of a simple polygon P that can be placed into the simple polygon Q. Despite significant effort studying a number of settings, known algorithms require high polynomial running times, even for the interesting case when either P or Q have constant size. (Barequet and Har-Peled, 2001) give a conditional lower bound of n2–o(1) under the 3SUM conjecture when P and Q are (convex) polygons with Θ(n) vertices each. This leaves open whether we can establish (1) hardness beyond quadratic time and (2) any superlinear bound for constant-sized P or Q. In this paper, we affirmatively answer these questions under the higher-order kSUM conjecture, proving natural hardness results that increase with each degree of freedom (scaling, x-translation, y-translation, rotation): (scaling, x-translation:) Finding the largest copy of P that can be x-translated into Q requires time n2–o(1) under the 3SUM conjecture, even for orthogonal (rectilinear) polygons P, Q with O(1) and n vertices, respectively. (scaling, x-translation, y-translation:) Finding the largest copy of P that can be arbitrarily translatedinto Q requires time n2–o(1) under the 4SUM conjecture, even for orthogonal polygons P, Q with O(1) and n vertices, respectively. This establishes the same lower bound under the assumption that Subset Sum cannot be solved in time O((2–∊)n/2) for any ∊ > 0. The above lower bounds are almost tight when one of the polygons is of constant size: Using an offline dynamic algorithm for maintaining the area of a union of rectangles due to Overmars and Yap, we obtain an -time algorithm for orthogonal polygons P, Q with p and q vertices, respectively. This matches the lower bounds up to an n1/2+o(1)-factor when P, Q have O(1) and n vertices. (scaling, x-translation, y-translation, rotation:) Finally, finding the largest copy of P that can be arbitrarily rotated and translated into Q requires time n3–o(1) under the 5SUM conjecture. As in our reductions, each degree of freedom determines one summand in a kSUM instance, these lower bounds appear likely to be best possible under kSUM. We are not aware of any other such natural (degree of freedom + 1)-SUM hardness for a geometric optimization problem. Finally, we prove an additional tight OV hardness of the translations-only case. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771

Spatial data structures form a core ingredient of many geometric algorithms, both in theory and in practice. Many of these data structures, especially the ones used in practice, are based on partitioning the underlying space (examples are binary space partitions and decompositions of polygons) or partitioning the set of objects (examples are bounding-volume hierarchies). The efficiency of such data structures---and, hence, of the algorithms that use them---depends on certain characteristics of the partitioning. For example the performance of many algorithms that use binary space partitions (BSPs) depends on the size of the BSPs. Similarly, the performance of answering range queries using bounding-volume hierarchies (BVHs) depends on the so-called crossing number that can be associated with the partitioning on which the BVH is based. Much research has been done on the problem of computing partitioning whose characteristics are good in the worst case. In this thesis, we studied the problem from a different point of view, namely instance-optimality. In particular, we considered the following question: given a class of geometric partitioning structures---like BSPs, simplicial partitions, polygon triangulations, …---and a cost function---like size or crossing number---can we design an algorithm that computes a structure whose cost is optimal or close to optimal for any input instance (rather than only worst-case optimal). We studied the problem of finding optimal data structures for some of the most important spatial data structures. As an example having a set of n points and an input parameter r, It has been proved that there are input sets for which any simplicial partitions has crossing number ¿(vr). It has also been shown that for any set of n input points and the parameter r one can make a simplicial partition with stabbing number O(vr). However, there are input point sets for which one can make simplicial partition with lower stabbing number. As an example when the points are on a diagonal, one can always make a simplicial partition with stabbing number 1. We started our research by studying BSPs for line segments in the plane, where the cost function is the size of the BSPs. A popular type of BSPs for line segments are the so-called auto-partitions. We proved that finding an optimal auto-partition is NP-hard. In fact, finding out if a set of input segments admits an auto-partition without any cuts is already NP-hard. We also studied the relation between two other types of BSPs, called free and restricted BSPs, and showed that the number of cuts of an optimal restricted BSP for a set of segments in R2 is at most twice the number of cuts of an optimal free BSP for that set. The details are being represented in Chapter 1 of the thesis. Then we turned our attention to so-called rectilinear r-partitions for planar point sets, with the crossing number as cost function. A rectilinear r-partition of a point set P is a partitioning of P into r subsets, each having roughly |P|/r points. The crossing number of the partition is defined using the bounding boxes of the subsets; in particular, it is the maximum number of bounding boxes that can be intersected by any horizontal or vertical line. We performed some theoretical as well as experimental studies on rectilinear r-partitions. On the theoretical side, we proved that computing a rectilinear r-partition with optimal stabbing number for a given set of points and parameter r is NP-hard. We also proposed an exact algorithm for finding optimal rectilinear r-partitions whose running time is polynomial when r is a constant, and a faster 2-approximation algorithm. Our last theoretical result showed that considering only partitions whose bounding boxes are disjoint is not sufficient for finding optimal rectilinear r-partitions. On the experimental side, we performed a comparison between four different heuristics for constructing rectilinear r-partitions. The so-called windmill KD-tree gave the best results. Chapter 2 of the thesis describes all the details of our research on rectilinear r-partitions. We studied another spatial data structure in Chapter 3 of the thesis. Decomposition of the interior of polygons is one of the fundamental problems in computational geometry. In case of a simple polygon one usually wants to make a Steiner triangulation of it, and when we have a rectilinear polygon at hand, one typically wants to make a rectilinear decomposition for it. Due to this reason there are algorithms which make Steiner triangulations and rectangular decompositions with low stabbing number. These algorithms are worst-case optimal. However, similar to the two previous data structures, there are polygons for which one can make decompositions with lower stabbing numbers. In 3 we proposed a 3-approximation for finding an optimal rectangular decomposition of a rectilinear polygon. We also proposed an O(1)-approximation for finding optimal Steiner triangulation of a simple polygon. Finally, in Chapter 4 of the thesis, we considered another optimization problem, namely how to approximate a piecewise-linear function F: R ¿R with another piecewise-linear function with fewer pieces. Here one can distinguish two versions of the problem. The first one is called the min-k problem; the goal is then to approximate the function within a given error e such that the resulting function has the minimum number of links. The second one is called the min-e problem; here the goal is to find an approximation with at most k links (for a given k) such that the error is minimized. These problems have already been studied before. Our contribution is to consider the problem for so-called uncertain functions, where the value of the input function F at its vertices is given as a discrete set of different values, each with an associated probability. We show how to compute an approximation that minimizes the expected error.

Minimum rectilinear polygons for given angle sequences

Area bounds of rectilinear polygons realized by angle sequences

In this paper, we present an O(n3 log n) time and O(n) space algorithm to solve the problem of two-guarding a simple rectilinear polygon. The complexity of our algorithm is much faster than the O(n4) time and O(n2 log n) space algorithm for general simple polygons [Bel92]. It also sheds light on solving the open problem of computing the two-watchmen routes of a simple rectilinear polygon.

Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an n-point set $P$, and a collection $\mathcal{F} = \{F_1, ..., F_m\}$ of subsets of $P$, and our goal is color $P$ with two colors, red and blue, so that the maximum over the $F_i$ of the absolute difference between the number of red elements and the number of blue elements (the discrepancy) is minimized. Combinatorial discrepancy has many applications in mathematics and computer science, including constructions of uniformly distributed point sets, and lower bounds for data structures and private data analysis algorithms. We investigate the combinatorial discrepancy of geometrically defined systems, in which $P$ is an n-point set in $d$-dimensional space ,and $\mathcal{F}$ is the collection of subsets of $P$ induced by dilations and translations of a fixed convex polytope $B$. Such set systems include systems of sets induced by axis-aligned boxes, whose discrepancy is the subject of the well known Tusnady problem. We prove new discrepancy upper and lower bounds for such set systems by extending the approach based on factorization norms previously used by the author and Matousek. We improve the best known upper bound for the Tusnady problem by a logarithmic factor, using a result of Banaszczyk on signed series of vectors. We extend this improvement to any arbitrary convex polytope $B$ by using a decomposition due to Matousek. Using Fourier analytic techniques, we also prove a nearly matching discrepancy lower bound for sets induced by any fixed bounded polytope $B$ satisfying a certain technical condition. We also outline applications of our results to geometric discrepancy, data structure lower bounds, and differential privacy.

Random polygon Generation from a given set of points is very important for different applications. A heuristic termed as “GRP_AS” has been proposed here to generate a random simple polygon from a given set of `n' points in 2-Dimensional plane. The “2-Opt Move” heuristic with time complexity O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ) is the best known among the existing heuristics to generate a simple polygon. We recently designed another heuristic “GRP_CH” for the same problem with time complexity O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) which was shown to be better than that of “2-Opt Move” heuristic in terms of random behaviour and time complexity. In this work, our main objective is to design an algorithm with reduced time complexity. The proposed heuristic, “GRP_AS” is based on the principle of random selection of a line segment and angular sorting of the point set. This “GRP_AS” heuristic takes O(nlogn) time which is less than that of “2-Opt Move” heuristic as well as “GRP_CH” heuristic. The lower bound and upper bound on the number of polygons generated in “GRP_AS” heuristic has been computed.

We study the problem of covering a set of points or polyhedra in R3 with two axis-aligned boxes in order to minimize a function of the measures of the two boxes, such as the sum or the maximum of their volumes. This 2-box cover problem arises naturally in the construction of bounding volume hierarchies, as well as in shape approximation and clustering. Existing algorithms solve the min-max version of the exact problem in quadratic time. Our results are more general, addressing min-max, min-sum and other versions. Our results give the first approximation schemes for the problem, which run in nearly linear time, as well as some new exact algorithms. We give (1+e)-approximation algorithms for minimizing the maximum or sum of volumes (or surface areas, diameters, widths, or girths) of the two boxes in R3. We investigate also the problem of computing balanced coverings, in which each box covers at least a fraction of the input objects, and we discuss the application to constructing provably-good bounding volume hierarchies of polyhedra. We also generalize our results to higher dimension.

Let p and q be two points in a simple polygon Π. An open problem in computational geometry asks to devise a simple linear-time algorithm for computing a shortest path between p and q, which is contained in Π, such that the algorithm does not depend on a (complicated) linear-time triangulation algorithm. This report provides a contribution to the solution of this problem by applying the rubberband algorithm. The obtained solution has \({\cal O}\) (nlogn) time complexity (where the super-linear time complexity is only due to preprocessing, i.e. for the calculation of critical edges) and is, altogether, considerably simpler than the triangulation algorithm. It has applications in 2D pattern recognition, picture analysis, robotics, and so forth.Keywordsdigital geometrycomputational geometryrubberband algorithmsimple polygonEuclidean shortest path

We prove a new theorem for orthogonal art galleries in which the guards must guard one another in addition to guarding the polygonal gallery. A set of points G in a polygon Pn is a k-guarded guard set for Pn provided that (i) for every point x in Pn there exists a point w in G such that x is visible from w ; and (ii) every point in G is visible from at least k other points in G: The polygon Pn is orthogonal provided each interior angle is 90° or 270°. We prove that for k ? 1 and n ? 6 every orthogonal polygon with n sides has a k-guarded guard set of cardinality k?n/6? + ?(n+2)/6?; this bound is best possible. This result extends our recent theorem that treats the case k = 1.

Traditional geometric algorithms are often presented as if input imprecision does not exist, even though it is often unavoidable. We argue that in some cases, it may be desirable for geometric algorithms to treat this imprecision as an explicit component of the input, and to reflect this imprecision in the output. Starting with three problems from computational geometry whose inputs are planar point sets (Voronoi diagrams, convex hulls, and smallest bounding discs), we recast these as problems where each input point’s location is imprecise, but known to lie within a particular region of uncertainty. Where algorithms to solve each of the original problems produce a single geometric object as output, the algorithms that we present typically produce either guaranteed or possible output objects. A guaranteed object represents qualities that can be guaranteed for every point set that is consistent with the uncertain regions, and a possible object represents qualities that exist for at least one such point set. By dealing with input imprecision explicitly, these guaranteed and possible objects can represent a more accurate view of what can be reliably inferred from the input data.

The authors provide optimal parallel solutions to several fundamental link distance problems set in trapezoided rectilinear polygons. All parallel algorithms are deterministic, run in logarithmic time, have an optimal time-processor product and are designed to run on EREW PRAM. The authors develop techniques (e.g. rectilinear window partition) for solving link distance problems in parallel which are expected to find applications in the design of other parallel computational geometry algorithms. They employ these parallel techniques for example to compute the link diameter, link center, and central diagonal of a rectilinear polygon. Their results also imply an optimal linear-time sequential algorithm for constructing a data structure to support rectilinear link distance queries between points.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A simple polygon is said to be weakly internally visible from a line segment lying inside it (chord, if the endpoints of the segment lie on the boundary of the polygon), if every point on the boundary of the polygon is visible from some point of this line segment. In this paper we describe a simple linear time algorithm for computing such a chord, when it exists. We also computed all non-redundant components of P which must be intersected by any chord of P from which P is wiv. The most interesting aspects of this algorithm are that we have been able to dispense with the complicated triangulation algorithm of Chazelle [2] as well as the algorithm in [7] for computing all shortest paths from a given vertex in a triangulated polygon. As a consequence of this, we can now use this algorithm to derive an optimal algorithm for triangulating a weakly internally visible polygon. Also these results can now be used in [3] to find a shortest line segment from which a polygon is weakly internally visible in optimal linear time.

For a set T of n points in a metric space (X, d), a point y ∈ X is dominated by a point x ∈ X if d(x, t) ≤ d(y, t) for all t ∈ T and there exists t′ ∈ T such that d(x, t′) &lt; d(y, t′). The set of non-dominated points of X is called the Pareto envelope of T. H. Kuhn (1973) established that in Euclidean spaces, the Pareto envelopes and the convex hulls coincide. Chalmet et al. (1981) characterized the Pareto envelopes in the rectilinear plane (ℝ2, d1) and constructed them in O(n log n) time. In this note, we investigate the Pareto envelopes of point-sets in simple polygons P endowed with geodesic d2- or d1-metrics (i.e., Euclidean and Manhattan metrics). We show that Kuhn's characterization extends to Pareto envelopes in simple polygons with d2-metric, while that of Chalmet et al. extends to simple rectilinear polygons with d1-metric. These characterizations provide efficient algorithms for construction of these Pareto envelopes.

The Art Gallery Problem (AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer k , the goal is to decide if there exists a set G of k guards within 풫 such that every point p ∈ 풫 is seen by at least one guard g ∈ G . Each guard corresponds to a point in the polygon 풫, and we say that a guard g sees a point p if the line segment pg is contained in 풫. We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set S ⊆ [0, 1] 2 , there exists a polygon with corners at rational coordinates such that for every p ∈ [0, 1] 2 , there is a set of guards of minimum cardinality containing p if and only if p ∈ S . In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems.