Finding the Shortest Watchman Route in a Simple Polygon

We use the concept of pointed pseudo-triangulations to establish new upper and lower bounds on a well known problem from the area of art galleries: What is the worst case optimal number of vertex π-guards that collectively monitor a simple polygon with n vertices? Our results are as follows: (1) Any simple polygon with n vertices can be monitored by at most \lfloor n/2 \rfloor general vertex π-guards. This bound is tight up to an additive constant of 1. (2) Any simple polygon with n vertices, k of which are convex, can be monitored by at most \lfloor (2n – k)/3 \rfloor edge-aligned vertexπ-guards. This is the first non-trivial upper bound for this problem and it is tight for the worst case families of polygons known so far.

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.

We use the concept of pointed pseudo-triangulations to establish new upper and lower bounds on a well known problem from the area of art galleries: What is the worst case optimal number of vertex π-guards that collectively monitor a simple polygon with n vertices?Our results are as follows:1. Any simple polygon with n vertices can be mon- itored by at most [n/2] general vertex π-guards. This bound is tight up to an additive constant of 1.2. Any simple polygon with n vertices, k of which are convex, can be monitored by at most [(2n -- k)/3] edge-aligned vertex π-guards. This is the first non- trivial upper bound for this problem and it is tight for the worst case families of polygons known so far.

Maximizing the area of an axially symmetric polygon inscribed in a simple polygon

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

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 investigate the $$L_1$$ geodesic farthest neighbors in a simple polygon P, and address several fundamental problems related to farthest neighbors. Given a subset $$S \subseteq P$$ , an $$L_1$$ geodesic farthest neighbor of $$p \in P$$ from S is one that maximizes the length of $$L_1$$ shortest path from p in P. Our list of problems include: computing the diameter, radius, center, farthest-neighbor Voronoi diagram, and two-center of S under the $$L_1$$ geodesic distance. We show that all these problems can be solved in linear or near-linear time based on our new observations on farthest neighbors and extreme points. Among them, the key observation shows that there are at most four extreme points of any compact subset $$S \subseteq P$$ with respect to the $$L_1$$ geodesic distance after removing redundancy.

We consider the problem of finding the staircase kernel in orthogonal polygons, with or without holes, in the plane. Orthogonal polygon is a simple polygon in the plane whose sides are either horizontal or vertical. We generalize the notion of visibility in the following way: We say that two points a and b in an orthogonal polygon P are visible to each other via staircase paths if and only if there exist an orthogonal chain connecting a and b and lying entirely in the interior of P. Furthermore, the orthogonal chain should have the property that the angles between the consecutive segments in the chain are either +90 ∘ or −90 ∘ , and these should alternate along the chain. There are two principal types of staircases, NW-SE and NE-SW. The notion of staircase visibility has been studied in the literature for the last three decades. Based on this notion we can generalize the notion of star-shapedness. A polygon P is called star-shaped under staircase visibility, or simply s-star if and only if there is nonempty set of points S in the interior of P, such that any point of S sees any point of P via staircase path. The largest such set of points is called the staircase kernel of P and denoted ker P. Our work is motivated by the work of Breen (1). She proves that the staircase kernel of an orthogonal polygon without holes is the intersection of all maximal orthogonally convex polygons contained in it. We extend Breen's results for the case when the orthogonal polygon has holes. We prove the necessary geometric properties, and use them to derive a quadratic time, O(���� 2 ) algorithm for computing the staircase kernel of an orthogonal polygon with holes, having n vertices in total, including the holes' vertices. The algorithm is based on the plane sweep technique, widely used in Computational Geometry(4). Our result is optimal in the case of orthogonal polygon with holes, since the kernel (as proven) can consist of quadratic number of disjoint regions. In the case of polygon without holes, there is a linear time algorithm by Gewali(3), that is specific to the case of a polygon without holes. We present ex- amples of our algorithm's results.

Let P be a simple planar polygon. We present a linear worst-case time algorithm for constructing the bounded Voronoi diagram of P in the Manhattan metric, where each point z in P belongs to the region of the closest vertex of P that is visible from z. Among other consequences, the minimal spanning tree of the vertices in the Manhattan metric that is contained in P can be computed within optimal linear time.

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

Characterizing LR-visibility polygons and related problems

In "A linear-time algorithm for triangulating a simple polygon" [Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing (1986), 380-388. 486], the analysis showing that the authors' triangulation algorithm runs in linear time is incorrect, and indeed the algorithm does not run in linear time in the worst case. So far they have been unable to obtain a linear-time algorithm for the triangulation problem. They have been able to obtain an O(n loglogn)-time algorithm, however. The details are described in "An O(n loglogn)-Time Algorithm for Triangulating a Simple Polygon," SIAM Journal on Computing 17, 1 (February, 1988), to appear.

A 2-approximation algorithm for the zookeeper's problem

We present an O(nrG) time algorithm for computing and maintaining a minimum length shortest watchman tour that sees a simple polygon under monotone visibility in direction \(\theta \), while \(\theta \) varies in \([0,180^{\circ })\), obtaining the directions for the tour to be the shortest one over all tours, where n is the number of vertices, r is the number of reflex vertices, and \(G\le r\) is the maximum number of gates of the polygon used at any time in the algorithm.

High resolution 3D range scanning as well as iso-surface extraction have introduced densely and uniformly sampled models that are difficult to render at an interactive rate. To remove excessive details and produce meshes of various resolutions for different kinds of applications, the study of fast and high quality polygonal mesh simplification algorithms has become important. In this paper, we propose a new linear time algorithm that can achieve fast and high quality mesh simplification. In the new algorithm, we pipeline the cost computation, optimization, and edge collapse, and use a small constant-sized replacement selection min-heap instead of a large greedy queue to effectively reduce the runtime complexity to linear complexity. Compared to previous works, our new algorithm has at least three advantages. First, the new algorithm is runtime efficient. Second, the new algorithm is memory efficient. Third, the algorithm is capable of generating competitive high quality outputs.