Reflex Vertices Research Articles

Uniformly monotone partitioning of polygons

We study the problem of partitioning a simple polygon P with n vertices (including R reflex vertices) and no holes into a minimum number of uniformly monotone subpolygons using diagonals that lie inside P, each connecting two distinct vertices of P. A set of polygons is uniformly monotone if there exists a line such that all of the polygons are monotone with respect to the line. We present an O(nRlog⁡n+R5)-time algorithm for the problem. When Steiner points can be placed on the boundary of P and each Steiner point is considered as a vertex of P, we present an O(n+R5)-time algorithm for the problem. We present an O(n+R4)-time algorithm when Steiner points can be placed anywhere in P. Our algorithms improve upon the previously best ones. We also present simple and efficient 2-approximation algorithms.

A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

Given a closed simple polygon $P$, we say two points $p,q$ see each other if the segment $pq$ is fully contained in $P$. The art gallery problem seeks a minimum size set $G\subset P$ of guards that sees $P$ completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods and is attributed to Sharir. As the art gallery problem is ER-complete, it seems unlikely to avoid algebraic methods, without additional assumptions. In this paper, we introduce the notion of vision stability. In order to describe vision stability consider an enhanced guard that can see "around the corner" by an angle of $\delta$ or a diminished guard whose vision is by an angle of $\delta$ "blocked" by reflex vertices. A polygon $P$ has vision stability $\delta$ if the optimal number of enhanced guards to guard $P$ is the same as the optimal number of diminished guards to guard $P$. We will argue that most relevant polygons are vision stable. We describe a one-shot vision stable algorithm that computes an optimal guard set for visionstable polygons using polynomial time and solving one integer program. It guarantees to find the optimal solution for every vision stable polygon. We implemented an iterative visionstable algorithm and show its practical performance is slower, but comparable with other state of the art algorithms. Our iterative algorithm is inspired and follows closely the one-shot algorithm. It delays several steps and only computes them when deemed necessary. Given a chord $c$ of a polygon, we denote by $n(c)$ the number of vertices visible from $c$. The chord-width of a polygon is the maximum $n(c)$ over all possible chords $c$. The set of vision stable polygons admits an FPT algorithm when parametrized by the chord-width. Furthermore, the one-shot algorithm runs in FPT time, when parameterized by the number of reflex vertices.

Clique-Based Separators for Geometric Intersection Graphs

Let F be a set of n objects in the plane and let mathcal {G}^{times }(F) be its intersection graph. A balanced clique-based separator of mathcal {G}^{times }(F) is a set mathcal {mathcal {S}} consisting of cliques whose removal partitions mathcal {G}^{times }(F) into components of size at most delta n, for some fixed constant delta <1. The weight of a clique-based separator is defined as sum _{Cin mathcal {mathcal {S}}}log (|C|+1). Recently De Berg et al. (SIAM J. Comput. 49: 1291-1331. 2020) proved that if S consists of convex fat objects, then mathcal {G}^{times }(F) admits a balanced clique-based separator of weight O(sqrt{n}). We extend this result in several directions, obtaining the following results. (i) Map graphs admit a balanced clique-based separator of weight O(sqrt{n}), which is tight in the worst case. (ii) Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n^{2/3}log n). If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to O(sqrt{n}log n). (iii) Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^{2/3}log n). (iv) Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight O(sqrt{n}+rlog (n/r)), which is tight in the worst case. These results immediately imply sub-exponential algorithms for Maximum Independent Set (and, hence, Vertex Cover), for Feedback Vertex Set, and for q-Coloring for constant q in these graph classes.

Piercing pairwise intersecting geodesic disks by five points

Given a simple polygon P on n vertices and a set D of m pairwise intersecting geodesic disks in P, we show that five points in P are always sufficient to pierce all the disks in D. The points can be computed in O((n+m)log⁡nr) time, where nr is the number of the reflex vertices of P. This improves the previous bound of 14, obtained by Bose, Carmi, and Shermer [1].

Vertex Guarding for Dynamic Orthogonal Art Galleries

We devise an algorithm for surveying a dynamic orthogonal polygonal domain by placing one guard at each vertex in a subset of its vertices, i.e., whenever an orthogonal polygonal domain [Formula: see text] is modified to result in another orthogonal polygonal domain [Formula: see text], our algorithm updates the set of vertex guards surveying [Formula: see text] so that the updated guard set surveys [Formula: see text]. Our algorithm modifies the guard placement in [Formula: see text] amortized time, while ensuring the updated orthogonal polygonal domain with [Formula: see text] holes and [Formula: see text] vertices is guarded using at most [Formula: see text] vertex guards. For the special case of the initial orthogonal polygon being hole-free and each update resulting in a hole-free orthogonal polygon, our guard update algorithm takes [Formula: see text] worst-case time. Here, [Formula: see text] and [Formula: see text] are the number of vertices of the orthogonal polygon before and after the update, respectively; and, [Formula: see text] is the sum of [Formula: see text] and the number of updates to a few structures maintained by our algorithm. Further, by giving a construction, we show it suffices for the algorithm to consider only the case in which the parity of the number of reflex vertices of both [Formula: see text] and [Formula: see text] are equal.

Maximum-area and maximum-perimeter rectangles in polygons

We study the problem of finding maximum-area and maximum-perimeter rectangles that are inscribed in polygons in the plane. There has been a fair amount of work on this problem when the rectangles have to be axis-aligned or when the polygons are convex. We consider this problem in polygons with n vertices that are not necessarily convex, possibly with holes, and with no restriction on the orientation of the rectangles. We present an algorithm that computes a maximum-area rectangle and a maximum-perimeter rectangle in O(n3log⁡n) time using O(kn2+n) space, where k is the number of reflex vertices of the polygon. Our algorithm can report all maximum-area rectangles in the same time using O(n3) space. We also present a simple algorithm that finds a maximum-area rectangle inscribed in a convex polygon with n vertices in O(n3) time using O(n) space.

Local Routing in Convex Subdivisions

In various wireless networking settings, node locations determine a network’s topology, allowing the network to be modelled by a geometric graph drawn in the plane. Without any additional information, local geometric routing algorithms can guarantee delivery to the target node only in restricted classes of geometric graphs, such as triangulations. In order to guarantee delivery on more general classes of geometric graphs (e.g., convex subdivisions or planar subdivisions), previous local geometric routing algorithms required [Formula: see text] state bits to be stored and passed with the message. We present the first local geometric routing algorithm using only one state bit to guarantee delivery on convex subdivisions and, when the algorithm has knowledge of the incoming port (the preceding node on the route), the first stateless local geometric routing algorithm that guarantees delivery on edge-augmented monotone subdivisions (including all convex subdivisions). We also show that [Formula: see text] state bits are necessary in planar subdivisions in which faces may have three or more reflex vertices.

A space–time trade-off for computing the visibility polygon in the multi-pass model

We study the problem of computing the visibility polygon of a simple polygon $${\mathcal {P}}$$ from a viewpoint inside it, when $${\mathcal {P}}$$ resides in a read-only memory without random accessibility. This model of stored data is called the multi-pass model in the literature. We present an algorithm to compute the visibility polygon in this model in $$O(nr/t + nt)$$ time, where n is the size of input data, r is the number of reflex vertices of input data, and t is the number of additional variables which we use to compute the visibility polygon where $$2<t<r$$.

Optimally guarding 2-reflex orthogonal polyhedra by reflex edge guards

Let an orthogonal polyhedron be the union of a finite set of boxes in R3 (i.e., cuboids with edges parallel to the coordinate axes), whose surface is a connected 2-manifold. We study the NP-complete problem of guarding a non-convex orthogonal polyhedron having reflex edges in just two directions (as opposed to three, in the general case) by placing the minimum number of edge guards on reflex edges only.We show that⌊r−g2⌋+1 reflex edge guards are sufficient, where r is the number of reflex edges and g is the polyhedron's genus. This bound is tight for g=0. We thereby generalize a classic planar Art Gallery theorem of O'Rourke, which states that the same upper bound holds for vertex guards in an orthogonal polygon with r reflex vertices and g holes.Then we give a similar upper bound in terms of m, the total number of edges in the polyhedron. We prove that⌊m−48⌋+g reflex edge guards are sufficient, whereas the previous best known bound was ⌊11m/72+g/6⌋−1 edge guards (not necessarily reflex).We also consider the setting in which guards are open (i.e., they are segments without the endpoints), proving that the same results hold even in this more challenging case.Finally, we show how to compute guard locations matching the above bounds in O(nlog⁡n) time.

Polygon simplification by minimizing convex corners

Let P be a polygon with r>0 reflex vertices and possibly with holes and islands (i.e., components of the polygon inside holes). A subsuming polygon of P is a polygon P′ such that P⊆P′, each connected component R of P is a subset of a distinct connected component R′ of P′, and the reflex corners of R coincide with those of R′. A subsuming chain of P′ is a minimal path on the boundary of P′ whose two end edges coincide with two edges of P. Aichholzer et al. proved that every polygon P has a subsuming polygon with O(r) vertices, and posed an open problem to determine the computational complexity of computing subsuming polygons with the minimum number of convex vertices.In this paper, we prove that the problem of computing an optimal subsuming polygon is NP-hard (but the complexity remains open for simple polygons; i.e., polygons without holes). Our NP-hardness result holds even when the subsuming chains are restricted to have constant length and lie on the arrangement of lines determined by the edges of the input polygon. We show that this restriction makes the problem polynomial-time solvable for simple polygons.

A quasi-visibility graph based clique-extraction heuristic model for partitioning of planar shape

This paper presents a graph theoretical model to partition polygonal approximation of a shape into visually meaningful constituent parts based on a heuristic approach. The proposed model introduces a new concept of approximated vertex-visibility graph termed as quasi-visibility graph to generate different viable cuts for partitioning the shape. In the shape representative graph, a maximal-clique perceptually corresponds to a distinguishable part. Based on this notion, we propose a heuristic based clique extraction strategy to decompose the shape exploring its quasi-visibility graph. A few refinement strategies are also attempted by exploring the options of: a) merging correlated parts for better visual interpretation; b) inserting antipodal points of reflex vertices in polygonal approximation for more possible viable cuts. The performance of the proposed model is evaluated by comparing partition-graphs of similar shapes. The partitioning based on the proposed model appears to be coherent with human observation and comparable with existing algorithms.

A quasi-visibility graph based clique-extraction heuristic model for partitioning of planar shape

This paper presents a graph theoretical model to partition polygonal approximation of a shape into visually meaningful constituent parts based on a heuristic approach. The proposed model introduces a new concept of approximated vertex-visibility graph termed as quasi-visibility graph to generate different viable cuts for partitioning the shape. In the shape representative graph, a maximal-clique perceptually corresponds to a distinguishable part. Based on this notion, we propose a heuristic based clique extraction strategy to decompose the shape exploring its quasi-visibility graph. A few refinement strategies are also attempted by exploring the options of: a) merging correlated parts for better visual interpretation; b) inserting antipodal points of reflex vertices in polygonal approximation for more possible viable cuts. The performance of the proposed model is evaluated by comparing partition-graphs of similar shapes. The partitioning based on the proposed model appears to be coherent with human observation and comparable with existing algorithms.

Minimizing the solid angle sum of orthogonal polyhedra

In this paper we characterize the set of orthogonal polyhedra in R3 that minimize the sum of their internal solid angles. To prove our result, we first generalize the well known result that any orthogonal polygon in the plane with n vertices has (n+4)/2 convex and (n−4)/2 reflex vertices; see [1,2]. We prove that an orthogonal polyhedron of genus g with n vertices, k of which have degree greater than or equal to 4, has (n+8−8g+3k)/2 convex vertices and (n−8+8g−3k)/2 reflex vertices. We also prove that the sum of the solid angles of an orthogonal polyhedron is at least (n−4+4g)π, and at most (3n−24−4g)π. These inequalities are minimized and maximized, respectively, when g=0.

Delta DLP 3-D Printing of Large Models

This paper presents a 3-D printing system that uses a low-cost off-the-shelf consumer projector to fabricate large models. Compared with traditional digital light processing (DLP) 3-D printers using a single vertical carriage, the platform of our DLP 3-D printer using delta mechanism can also move horizontally in the plane. We show that this system can print 3-D models much larger than traditional DLP 3-D printers. The major challenge to realize 3-D printing of large models in our system comes from how to cover a planar polygonal domain by a minimum number of rectangles with fixed size, which is NP-hard. We propose a simple yet efficient approximation algorithm to solve this problem. The key idea is to segment a polygonal domain using its medial axis and afterward merge small parts in the segmentation. Given an arbitrary polygon $\Omega $ with $n$ generators (i.e., line segments and reflex vertices in $\Omega $ ), we show that the time complexity of our algorithm is $O(n^{2}\log ^{2}~n)$ and the number of output rectangles covering $\Omega $ is $O(Kn)$ , where $K$ is an input-polygon-dependent constant. A physical prototype system is built and several large 3-D models with complex geometric structures have been printed as examples to demonstrate the effectiveness of our approach. Note to Practitioners —Low-cost 3-D printers and 3-D printing of large models are two important but often conflicting goals in manufacturing industry. Usually low-cost devices such as DLP 3-D printer using an off-the-shelf consumer projector cannot print large models, due to the small projection area of the projector. In this paper, we propose to horizontally move the platform such that a large area can be printed by the composition of multiple small projection areas. Based on this new mechanism, we propose a simple yet efficient algorithm to cover an arbitrary polygonal shape (possibly with holes or multiple disjoint polygons) by a small set of rectangles with fixed size. Our algorithm is theoretically sound and can be easily implemented. We built a physical prototype system of the proposed low-cost Delta DLP 3-D printer, which successfully prints several large models with satisfied mechanical properties.

The Fortress Problem in Terms of the Number of Reflex and Convex Vertices. A 3D objects scanning application

This work focuses on the visibility of exterior of a polygon. We show a lower bound for the Fortress Problem applied to polygons P having n vertices in terms of the number of reflex vertices, the number of convex vertices and the number of pockets that are found when the convex hull is made on P. The results are related to the task of the geometric data acquisition for architectural surveys through techniques such as laser scanner.

Computing positively weighted straight skeletons of simple polygons based on a bisector arrangement

We extend the work by Huber and Held (IJCGA 2012) on straight-skeleton computation based on motorcycle graphs to positively weighted skeletons. Resorting to a line arrangement induced by the r reflex vertices of a simple n-vertex polygon P allows to compute the weighted straight skeleton of P in O(n2+r3k+nrlog⁡n) time and O(n+kr) space, for an arbitrary positive integer k≤r.

Visibility-Based Target-Tracking Game: Bounds and Tracking Strategies

The art gallery problem or museum problem is a well-known visibility problem in computational geometry. It originates from a real-world problem of guarding an art gallery with the minimum number of guards who together can observe the whole gallery. This letter deals with a variant of the art gallery problem in which guards are deployed to track a mobile intruder. We consider a team of free guards equipped with omnidirectional cameras deployed to track a bounded speed intruder inside a simply connected polygonal environment. We present a partitioning technique for any simply connected polygon using its diagonals such that each partition is at most a nonagon. Next, we propose an event-triggered strategy for a single guard to track an intruder around a reflex vertex. Based on the proposed strategy, we formulate the problem of finding the minimum speed of the guard and its corresponding trajectory inside any partition as a convex-optimization problem. The solution to the convex-optimization problem provides an upper bound on the speed of the mobile guard required for persistent tracking of the mobile intruder. Furthermore, we show that the number of guards deployed by the proposed technique belongs to $\mathcal {O}(\frac{2n}{7})$ and $\Omega (\frac{n}{6})$ , which is strictly less than $\lfloor {\frac{n}{3}} \rfloor$ (sufficient and sometimes necessary number of guards required for coverage). Finally, we extend the analysis to orthogonal polygons, and show that the upper bound on the number of guards deployed for tracking is strictly less than $\lfloor {\frac{n}{4}} \rfloor$ which is sufficient and sometimes necessary for the coverage problem.

Flipping edge-labelled triangulations

Flips in triangulations have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the flipping process has not been addressed. We examine this question by attaching unique labels to the triangulation edges. We introduce the concept of the orbit of an edge e, which is the set of all edges reachable from e via flips.We establish the first upper and lower bounds on the diameter of the flip graph in this setting. Specifically, we prove tight Θ(nlog⁡n) bounds for edge-labelled triangulations of n-vertex convex polygons and combinatorial triangulations, contrasting with the Θ(n) bounds in their respective unlabelled settings. The Ω(nlog⁡n) lower bound for the convex polygon setting might be of independent interest, as it generalizes lower bounds on certain sorting models. When simultaneous flips are allowed, the upper bound for convex polygons decreases to O(log2⁡n), although we no longer have a matching lower bound.Moving beyond convex polygons, we show that edge-labelled triangulated polygons with a single reflex vertex can have a disconnected flip graph. This is in sharp contrast with the unlabelled case, where the flip graph is connected for any triangulated polygon. For spiral polygons, we provide a complete characterization of the orbits. This allows us to decide connectivity of the flip graph of a spiral polygon in linear time. We also prove an upper bound of O(n2) on the diameter of each connected component, which is optimal in the worst case. We conclude with an example of a non-spiral polygon whose flip graph has diameter Ω(n3).

A Faster Algorithm for Computing Straight Skeletons

We present a new algorithm for computing the straight skeleton of a polygon. For a polygon with n vertices, among which r are reflex vertices, we give a deterministic algorithm that reduces the straight skeleton computation to a motorcycle graph computation in O ( n (log n )log r ) time. It improves on the previously best known algorithm for this reduction, which is randomized, and runs in expected O ( n sqrt h + 1 log 2 n ) time for a polygon with h holes. Using known motorcycle graph algorithms, our result yields improved time bounds for computing straight skeletons. In particular, we can compute the straight skeleton of a nondegenerate polygon in O ( n (log n )log r + r 4/3 + ε ) time for any ε &gt; 0. On degenerate input, our time bound increases to O ( n (log n )log r + r 17/11 + ε ).

A Gradient Search Algorithm for the Maximal Visible Area Polygon Problem

This paper provides a gradient search algorithm for finding the maximal visible area polygon (VAP) viewed by an interior point in a simple polygon P. The algorithm is based on a natural partition of P into convex sets, such that each element of the partition is associated with a unique analytical form of the area function. We call this partition a back diagonal partition of P. Our maximal VAP algorithm converges in a finite number of steps, and is polynomial with a complexity of , for a simple polygon P with n vertices, and r reflex vertices. We use the maximal VAP algorithm as a basis for a greedy heuristic for the well known guardhouse problem with a computation complexity of .