On partitioning rectilinear polygons into star-shaped polygons

We present an algorithm for finding optimum partitions of simple monotone rectilinear polygons into star-shaped polygons. The algorithm may introduce Steiner points and its time complexity isO(n), wheren is the number of vertices in the polygon. We then use this algorithm to obtain anO(n logn) approximation algorithm for partitioning simple rectilinear polygons into star-shaped polygons with the size of the partition being at most six times the optimum.     

Linear-Time Algorithms for Hole-free Rectilinear Proportional Contact Graph Representations

In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we first study proportional contact representations that use rectilinear polygons without wasted areas (white space). In this setting, the best known algorithm for proportional contact representation of a maximal planar graph uses 12-sided rectilinear polygons and takes O(nlogn) time. We describe a new algorithm that guarantees 10-sided rectilinear polygons and runs in O(n) time. We also describe a linear-time algorithm for proportional contact representation of planar 3-trees with 8-sided rectilinear polygons and show that this is optimal, as there exist planar 3-trees that require 8-sided polygons. We then show that a maximal outer-planar graph admits a proportional contact representation using rectilinear polygons with 6 sides when the outer-boundary is a rectangle and with 4 sides otherwise. Finally we study maximal series-parallel graphs. Here we show that O(1)-sided rectilinear polygons are not possible unless we allow holes, but 6-sided polygons can be achieved with arbitrarily small holes.     

Note on covering monotone orthogonal polygons with star-shaped polygons

In 1986, Keil provided an O(n²) time algorithm for the problem of covering monotone orthogonal polygons with the minimum number of r-star-shaped orthogonal polygons. This was later improved to O(n) time and space by Gewali et al. in [L. Gewali, M. Keil, S.C. Ntafos, On covering orthogonal polygons with star-shaped polygons, Information Sciences 65 (1992) 45-63]. In this paper we simplify the latter algorithm—we show that with a little modification, the first step Sweep1 of the discussed algorithm—which computes the top ceilings of horizontal grid segments—can be omitted.In addition, for the minimum orthogonal guard problem in the considered class of polygons, our approach provides a linear time algorithm which uses O(k) additional space, where k is the size of the optimal solution—the algorithm in [L. Gewali, M. Keil, S.C. Ntafos, On covering orthogonal polygons with star-shaped polygons, Information Sciences 65 (1992) 45-63] uses both O(n) time and O(n) additional space.     

Visibility of disjoint polygons

Consider a collection of disjoint polygons in the plane containing a total ofn edges. We show how to build, inO(n ²) time and space, a data structure from which inO(n) time we can compute the visibility polygon of a given point with respect to the polygon collection. As an application of this structure, the visibility graph of the given polygons can be constructed inO(n ²) time and space. This implies that the shortest path that connects two points in the plane and avoids the polygons in our collection can be computed inO(n ²) time, improving earlierO(n ² logn) results.     

An optimal visibility graph algorithm for triangulated simple polygons

LetP be a triangulated simple polygon withn sides. The visibility graph ofP has an edge between every pair of polygon vertices that can be connected by an open segment in the interior ofP. We describe an algorithm that finds the visibility graph ofP inO(m) time, wherem is the number of edges in the visibility graph. Becausem can be as small asO(n), the algorithm improves on the more general visibility algorithms of Asanoet al. [AAGHI] and Welzl [W], which take Θ(n ²) time, and on Suri'sO(m logn) visibility graph algorithm for simple polygons [S].     

On approximation behavior of the greedy triangulation for convex polygons

We prove that the greedy triangulation heuristic for minimum weight triangulation of convex polygons yields solutions within a constant factor from the optimum. For interesting classes of convex polygons, we derive small upper bounds on the constant approximation factor. Our results contrast with Kirkpatrick's Ω(n) bound on the approximation factor of the Delaunay triangulation heuristic for minimum weight triangulation of convexn-vertex polygons. On the other hand, we present a straightforward implementation of the greedy triangulation heuristic for ann-vertex convex point set or a convex polygon takingO(n ²) time andO(n) space. To derive the latter result, we show that given a convex polygonP, one can find for all verticesv ofP a shortest diagonal ofP incident tov in linear time. Finally, we observe that the greedy triangulation for convex polygons having so-called semicircular property can be constructed in timeO(n logn).     

Approximating centroids for the maximum intersection of spherical polygons

This paper considers the problem of investigating the spherical regions owned by the maximum number of spherical polygons. We present a practical O(n(v+I)) time algorithm for finding the approximating centroids for the maximum intersection of spherical polygons, where n, v, and I are, respectively, the numbers of polygons, all vertices, and intersection points. In order to elude topological errors and handle geometric degeneracies, our algorithm takes the approach of edge-based partitioning of the sphere. Furthermore, the numerical complexity is avoided since the algorithm is completely spherical.     

A simple linear hidden-line algorithm for star-shaped polygons

A very simple, linear-running-time algorithm is presented for solving the hidden-line problem for star-shaped polygons. The algorithm first decomposes the visibility regions into edge-visible polygons and then solves the hidden-line problem for these simpler polygons. In addition to simplicity the algorithm possesses the virtue of affording a very easy proof of correctness. Some applications where this problem arises are mentioned.     

On geodesic properties of polygons relevant to linear time triangulation

Triangulating a simple polygon ofn vertices inO(n) time is one of the main open problems in computational geometry. The fastest algorithm to date, due to Tarjan and van Wyk, runs inO(n log logn), but several classes of simple polygons have been shown to admit linear time traingulation. Famous examples of such classes are: star-shaped, monotone, spiral, edge visible, and weakly externally visible polygons. The notion of geodesic paths is used here to characterize all classes of polygons for which linear time triangulation algorithms are known. First we introduce a new class of polygons,palm polygons, which subsumes many known classes of polygons for which linear time triangulation algorithms exist, and present a linear time algorithm for triangulating polygons in this class. Then a class of polygons,crab polygons, is defined and shown to contain all classes of existing polygons for which linear time triangulation algorithms are known. As a byproduct of this characterization, a new, very simple linear time algorithm for triangulating star-shaped polygons is obtained.Research supported by Faculty of Graduate Studies and Research (McGill University) and NSERC under grant OGP0036737Research supported by FCAR grant EQ-1678 and NSERC grant A9293     

The contour problem for rectilinear polygons

The union of a set of p, not necessarily disjoint, rectilinear polygons in the plane determines a set of disjoint rectilinear polygons. We present an O(n log n + e) time and O(n) space algorithm to compute the edges of the disjoint polygons, that is, the contour, where n is the total number of vertices in the original polygons and e the total number in the resulting set. This time-and space-optimal algorithm uses the scan-line paradigm as in two previous approaches to this problem for rectangles, but requires a simpler data structure. Moreover, if the given rectilinear polygons are rectilinear convex, the space requirement is reduced to O(p).     

Visibility of disjoint polygons

Consider a collection of disjoint polygons in the plane containing a total ofn edges. We show how to build, inO(n ²) time and space, a data structure from which inO(n) time we can compute the visibility polygon of a given point with respect to the polygon collection. As an application of this structure, the visibility graph of the given polygons can be constructed inO(n ²) time and space. This implies that the shortest path that connects two points in the plane and avoids the polygons in our collection can be computed inO(n ²) time, improving earlierO(n ² logn) results.     

Approximate Guarding of Monotone and Rectilinear Polygons

We show that vertex guarding a monotone polygon is NP-hard and construct a constant factor approximation algorithm for interior guarding monotone polygons. Using this algorithm we obtain an approximation algorithm for interior guarding rectilinear polygons that has an approximation factor independent of the number of vertices of the polygon. If the size of the smallest interior guard cover is OPT for a rectilinear polygon, our algorithm produces a guard set of size O(OPT ²).     

On approximation behavior of the greedy triangulation for convex polygons

We prove that the greedy triangulation heuristic for minimum weight triangulation of convex polygons yields solutions within a constant factor from the optimum. For interesting classes of convex polygons, we derive small upper bounds on the constant approximation factor. Our results contrast with Kirkpatrick's (n) bound on the approximation factor of the Delaunay triangulation heuristic for minimum weight triangulation of convexn-vertex polygons. On the other hand, we present a straightforward implementation of the greedy triangulation heuristic for ann-vertex convex point set or a convex polygon takingO(n ²) time andO(n) space. To derive the latter result, we show that given a convex polygonP, one can find for all verticesv ofP a shortest diagonal ofP incident tov in linear time. Finally, we observe that the greedy triangulation for convex polygons having so-called semicircular property can be constructed in timeO(n logn).     

Algorithms for computing diffuse reflection paths in polygons

Let s be a point source of light inside a polygon P of n vertices. A polygonal path from s to some point t inside P is called a diffuse reflection path if the turning points of the path lie on edges of?P. A?diffuse reflection path is said to be optimal if it has the minimum number of reflections on the path. The problem of computing a diffuse reflection path from s to t inside P has not been considered explicitly in the past. We present three different algorithms for this problem which produce suboptimal paths. For constructing such a path, the first algorithm uses a greedy method, the second algorithm uses a transformation of a minimum link path, and the third algorithm uses the edge–edge visibility graph of?P. The first two algorithms are for polygons without holes, and they run in O(n+klogn) time, where k denotes the number of reflections in the constructed path. The third algorithm is for polygons with or without holes, and it runs in O(n ²) time. The number of reflections in the path produced by this third algorithm can be at most three times that of an optimal diffuse reflection path. Though the combinatorial approach used in the third algorithm gives a better bound on the number of reflections on the path, the first and the second algorithms stand on the merit of their elegant geometric approaches based on local geometric information.     

An optimal visibility graph algorithm for triangulated simple polygons

LetP be a triangulated simple polygon withn sides. The visibility graph ofP has an edge between every pair of polygon vertices that can be connected by an open segment in the interior ofP. We describe an algorithm that finds the visibility graph ofP inO(m) time, wherem is the number of edges in the visibility graph. Becausem can be as small asO(n), the algorithm improves on the more general visibility algorithms of Asanoet al. [AAGHI] and Welzl [W], which take (n ²) time, and on Suri'sO(m logn) visibility graph algorithm for simple polygons [S].This work was supported in part by a U.S. Army Research Office fellowship under agreement DAAG29-83-G-0020.     

Optimal parallel algorithms for rectilinear link-distance problems

We provide optimal parallel solutions to several link-distance problems set in trapezoided rectilinear polygons. All our main parallel algorithms are deterministic and designed to run on the exclusive read exclusive write parallel random access machine (EREW PRAM). LetP be a trapezoided rectilinear simple polygon withn vertices. InO(logn) time usingO(n/logn) processors we can optimally compute:

1. Minimum réctilinear link paths, or shortest paths in theL ₁ metric from any point inP to all vertices ofP.
2. Minimum rectilinear link paths from any segment insideP to all vertices ofP.
3. The rectilinear window (histogram) partition ofP.
4. Both covering radii and vertex intervals for any diagonal ofP.
5. A data structure to support rectilinear link-distance queries between any two points inP (queries can be answered optimally inO(logn) time by uniprocessor).

Our solution to 5 is based on a new linear-time sequential algorithm for this problem which is also provided here. This improves on the previously best-known sequential algorithm for this problem which usedO(n logn) time and space.5 We develop techniques for solving link-distance problems in parallel which are expected to find applications in the design of other parallel computational geometry algorithms. We employ these parallel techniques, for example, to compute (on a CREW PRAM) optimally the link diameter, the link center, and the central diagonal of a rectilinear polygon.     

Rectilinear steiner tree heuristics and minimum spanning tree algorithms using geographic nearest neighbors

We study the application of the geographic nearest neighbor approach to two problems. The first problem is the construction of an approximately minimum length rectilinear Steiner tree for a set ofn points in the plane. For this problem, we introduce a variation of a subgraph of sizeO(n) used by YaO [31] for constructing minimum spanning trees. Using this subgraph, we improve the running times of the heuristics discussed by Bern [6] fromO(n ² log n) toO(n log² n). The second problem is the construction of a rectilinear minimum spanning tree for a set ofn noncrossing line segments in the plane. We present an optimalO(n logn) algorithm for this problem. The rectilinear minimum spanning tree for a set of points can thus be computed optimally without using the Voronoi diagram. This algorithm can also be extended to obtain a rectilinear minimum spanning tree for a set of nonintersecting simple polygons.The results in this paper are a part of Y. C. Yee's Ph.D. thesis done at SUNY at Albany. He was supported in part by NSF Grants IRI-8703430 and CCR-8805782. S. S. Ravi was supported in part by NSF Grants DCI-86-03318 and CCR-89-05296.