A Pedestrian Approach to Ray Shooting: Shoot a Ray, Take a Walk

We propose a very simple ray-shooting algorithm, whose only data structure is a triangulation. The query algorithm, after locating the triangle containing the origin of the ray, walks along the ray, advancing from one triangle to a neighboring one until the polygon boundary is reached. The key result of the paper is a Steiner triangulation of a simple polygon with the property that a ray can intersect at most O(log n) triangles before reaching the polygon boundary. We are able to compute such a triangulation in linear sequential time, or in O(log n) parallel time using O(n/log n) processors. This gives a simple, yet optimal, ray-shooting algorithm for a simple polygon. Using a well-known technique, we can extend our triangulation procedure to a multiconnected polygon with k components and n vertices, so that a ray intersects at most O(√k log n) triangles.     

Finding minimal nested polygons

An algorithm for finding a polygon with minimum number of edges nested in two simplen-sided polygons is presented. The algorithm solves the problem in at mostO(n logn) time, and improves the time complexity of two previousO(n ²) algorithms.The work was supported by NSERC grant OPG0041629.     

Decomposing a simple polygon into pseudo-triangles and convex polygons

In this paper we consider the problem of decomposing a simple polygon into subpolygons that exclusively use vertices of the given polygon. We allow two types of subpolygons: pseudo-triangles and convex polygons. We call the resulting decomposition PT-convex. We are interested in minimum decompositions, i.e., in decomposing the input polygon into the least number of subpolygons. Allowing subpolygons of one of two types has the potential to reduce the complexity of the resulting decomposition considerably.The problem of decomposing a simple polygon into the least number of convex polygons has been considered. We extend a dynamic-programming algorithm of Keil and Snoeyink for that problem to the case that both convex polygons and pseudo-triangles are allowed. Our algorithm determines such a decomposition in O(n³) time and space, where n is the number of the vertices of the polygon.     

Finding the Shortest Watchman Route in a Simple Polygon

We present the first polynomial time algorithm that finds the shortest route in a simple polygon such that all points of the polygon are visible from the route. This route is called the shortest watchman route, and we do not assume any restrictions on the route or on the simple polygon. Our algorithm runs in worst case O(n ⁶ ) time, but it is adaptive, making it run faster on polygons with a simple structure. Received December 12, 1997, and in revised form September 30, 1998.     

Polygon triangulation inO(n log logn) time with simple data structures

We give a newO(n log logn)-time deterministic algorithm for triangulating simplen-vertex polygons, which avoids the use of complicated data structures. In addition, for polygons whose vertices have integer coordinates of polynomially bounded size, the algorithm can be modified to run inO(n log*n) time. The major new techniques employed are the efficient location of horizontal visibility edges that partition the interior of the polygon into regions of approximately equal size, and a linear-time algorithm for obtaining the horizontal visibility partition of a subchain of a polygonal chain, from the horizontal visibility partition of the entire chain. The latter technique has other interesting applications, including a linear-time algorithm to convert a Steiner triangulation of a polygon into a true triangulation.This research was partially supported by the following grants: NSERC 583584, NSERC 580485, ONR-N00014-87-0467, and by DIMACS, an NSF Science and Technology Center (NSF-STC88-09648).     

The orthogonal convex skull problem

We give a combinatorial definition of the notion of a simple orthogonal polygon beingk-concave, wherek is a nonnegative integer. (A polygon is orthogonal if its edges are only horizontal or vertical.) Under this definition an orthogonal polygon which is 0-concave is convex, that is, it is a rectangle, and one that is 1-concave is orthoconvex in the usual sense, and vice versa. Then we consider the problem of computing an orthoconvex orthogonal polygon of maximal area contained in a simple orthogonal polygon. This is the orthogonal version of the potato peeling problem. AnO(n ²) algorithm is presented, which is a substantial improvement over theO(n ⁷) time algorithm for the general problem.The work of the first author was supported under a Natural Sciences and Engineering Research Council of Canada Grant No. A-5692 and the work of the second author was partially supported by NSF Grants Nos. DCR-84-01898 and DCR-84-01633.     

A polynomial solution for the potato-peeling problem

The potato-peeling problem asks for the largest convex polygon contained inside a given simple polygon. We give anO(n ⁷) time algorithm to this problem, answering a question of Goodman. We also give anO(n ⁶) time algorithm if the desired polygon is maximized with respect to perimeter.Work in this paper has been supported in part by NSF grants #DCR-84-01898 and #DCR-84-01633, the Office of Naval Research Grant N00014-82-K-0381, and by grants from Digital Equipment Corporation, the Sloan Foundation, the System Development Foundation, and the IBM Corporation. This paper contains the main results of the paper A Polynomial Solution for Potato-Peeling and other Polygon Inclusion and Enclosure Problems presented in the 25th Foundation of Computer Science Conference, 1984, Florida. The second half of that paper is submitted for publication elsewhere [1].     

New Similarity Measures between Polylines with Applications to Morphing and Polygon Sweeping

Abstract. We introduce two new related metrics, the geodesic width and the link width , for measuring the ``distance' between two nonintersecting polylines in the plane. If the two polylines have n vertices in total, we present algorithms to compute the geodesic width of the two polylines in O(n <sup>2</sup> log n) time using O(n <sup>2</sup> ) space and the link width in O(n <sup>3</sup> log n) time using O(n <sup>2</sup> ) working space where n is the total number of edges of the polylines. Our computation of these metrics relies on two closely related combinatorial strutures: the shortest-path diagram and the link diagram of a simple polygon. The shortest-path (resp., link) diagram encodes the Euclidean (resp., link) shortest path distance between all pairs of points on the boundary of the polygon. We use these algorithms to solve two problems: • Compute a continuous transformation that ``morphs' one polyline into another polyline. Our morphing strategies ensure that each point on a polyline moves as little as necessary during the morphing, that every intermediate polyline is also simple and disjoint to any other intermediate polyline, and that no portion of the polylines to be morphed is stretched or compressed by more than a user-defined parameter during the entire morphing. We present an algorithm that computes the geodesic width of the two polylines and utilizes it to construct a corresponding morphing strategy in O(n ² log ² n) time using O(n ² ) space. We also give an O(nlog n) time algorithm to compute a 2-approximation of the geodesic width and a corresponding morphing scheme. • Locate a continuously moving target using a group of guards moving inside a simple polygon. The guards always determine a simple polygonal chain within the polygon, with consecutive guards along the chain being mutually visible. We compute a strategy that sweeps such a chain of guards through the polygon in order to locate a target. We compute in O(n ³ ) time and O(n ² ) working space the minimum number r ^* of guards needed to sweep an n -vertex polygon. We also give an approximation algorithm, using O(n log n) time and O(n) space, to compute an integer r such that max(r - 16, 2)≤ r ^* ≤ r and P can be swept with a chain of r guards.     

New Similarity Measures between Polylines with Applications to Morphing and Polygon Sweeping

   Abstract. We introduce two new related metrics, the geodesic width and the link width , for measuring the ``distance' between two nonintersecting polylines in the plane. If the two polylines have n vertices in total, we present algorithms to compute the geodesic width of the two polylines in O(n <sup>2</sup> log n) time using O(n <sup>2</sup> ) space and the link width in O(n <sup>3</sup> log n) time using O(n <sup>2</sup> ) working space where n is the total number of edges of the polylines. Our computation of these metrics relies on two closely related combinatorial strutures: the shortest-path diagram and the link diagram of a simple polygon. The shortest-path (resp., link) diagram encodes the Euclidean (resp., link) shortest path distance between all pairs of points on the boundary of the polygon. We use these algorithms to solve two problems: • Compute a continuous transformation that ``morphs' one polyline into another polyline. Our morphing strategies ensure that each point on a polyline moves as little as necessary during the morphing, that every intermediate polyline is also simple and disjoint to any other intermediate polyline, and that no portion of the polylines to be morphed is stretched or compressed by more than a user-defined parameter during the entire morphing. We present an algorithm that computes the geodesic width of the two polylines and utilizes it to construct a corresponding morphing strategy in O(n ² log ² n) time using O(n ² ) space. We also give an O(nlog n) time algorithm to compute a 2-approximation of the geodesic width and a corresponding morphing scheme. • Locate a continuously moving target using a group of guards moving inside a simple polygon. The guards always determine a simple polygonal chain within the polygon, with consecutive guards along the chain being mutually visible. We compute a strategy that sweeps such a chain of guards through the polygon in order to locate a target. We compute in O(n ³ ) time and O(n ² ) working space the minimum number r ^* of guards needed to sweep an n -vertex polygon. We also give an approximation algorithm, using O(n log n) time and O(n) space, to compute an integer r such that max(r - 16, 2)≤ r ^* ≤ r and P can be swept with a chain of r guards.     

An Efficient Algorithm for Euclidean Shortest Paths Among Polygonal Obstacles in the Plane

We give an algorithm to compute a (Euclidean) shortest path in a polygon with h holes and a total of n vertices. The algorithm uses O(n) space and requires time. Received March 30, 1993, and in revised form August 26, 1994, and May 1, 1997.     

Finding the convex hull of a simple polygon

It is well known that the convex hull of a set of n points in the plane can be found by an algorithm having worst-case complexity O(n log n). A short linear-time algorithm for finding the convex hull when the points form the (ordered) vertices of a simple (i.e., non-self-intersecting) polygon is given.     

A linear-time algorithm for computing the voronoi diagram of a convex polygon

We present an algorithm for computing certain kinds of three-dimensional convex hulls in linear time. Using this algorithm, we show that the Voronoi diagram ofn sites in the plane can be computed in (n) time when these sites form the vertices of a convex polygon in, say, counterclockwise order. This settles an open problem in computational geometry. Our techniques can also be used to obtain linear-time algorithms for computing the furthest-site Voronoi diagram and the medial axis of a convex polygon and for deleting a site from a general planar Voronoi diagram.This research began while the first and fourth authors were visiting the Mathematical Sciences Research Institute in Berkeley, California. Work by the fourth author was supported in part by NSF Grant No. 8120790.     

Shortest watchman routes in simple polygons

In this paper we present an O(n ⁴, log logn) algorithm to find a shortest watchman route in a simple polygon through a point,s, in its boundary. A watchman route is a route such that each point in the interior of the polygon is visible from at least one point along the route. S. Ntafos was supported in part by a grant from Texas Instruments, Inc.     

Packing two disks into a polygonal environment

We consider the following problem. Given a polygon P, possibly with holes, and having n vertices, compute a pair of equal radius disks that do not intersect each other, are contained in P, and whose radius is maximized. Our main result is a simple randomized algorithm whose expected running time, on any input, is O(nlogn). This is optimal in the algebraic decision tree model of computation.     

Polygonizations of point sets in the plane

We examine the different ways a set ofn points in the plane can be connected to form a simple polygon. Such a connection is called apolygonization of the points. For some point sets the number of polygonizations is exponential in the number of points. For this reason we restrict our attention to star-shaped polygons whose kernels have nonempty interiors; these are callednondegenerate star-shaped polygons.We develop an algorithm and data structure for determining the nondegenerate star-shaped polygonizations of a set ofn points in the plane. We do this by first constructing an arrangement of line segments from the point set. The regions in the arrangement correspond to the kernels of the nondegenerate star-shaped polygons whose vertices are the originaln points. To obtain the data structure representing this arrangement, we show how to modify data structures for arrangements of lines in the plane. This data structure can be computed inO(n ⁴) time and space. By visiting the regions in this data structure in a carefully chosen order, we can compute the polygon associated with each region inO(n) time, yielding a total computation time ofO(n ⁵) to compute a complete list ofO(n ⁴) nondegenerate star-shaped polygonizations of the set ofn points.     

Packing and covering the plane with translates of a convex polygon

A covering of the Euclidean plane by a polygon P is a system of translated copies of P whose union is the plane, and a packing of P in the plane is a system of translated copies of P whose interiors are disjoint. A lattice covering is a covering in which the translates are defined by the points of a lattice, and a lattice packing is defined similarly. We show that, given a convex polygon P with n vertices, the densest lattice packing of P in the plane can be found in O(n) time. We also show that the sparsest lattice covering of the plane by a centrally symmetric convex polygon can be solved in O(n) time. Our approach utilizes results from classical geometry that reduce these packing and covering problems to the problems of finding certain extremal enclosed figures within the polygon.     

On separating two simple polygons by a single translation

LetP andQ be two disjoint simple polygons havingn sides each. We present an algorithm which determines whetherQ can be moved by a single translation to a position sufficiently far fromP, and which produces all such motions if they exist. The algorithm runs in timeO(t(n)) wheret(n) is the time needed to triangulate ann-sided polygon. Since Tarjan and Van Wyk have recently shown thatt(n)=O(n log logn) this improves the previous best result for this problem which wasO(n logn) even after triangulation.This research was supported by NSERC Grant A9293, FCAR Grant EQ-1678, and a Killam Fellowship from the Canada Council.     

Computing the maximum clique in the visibility graph of a simple polygon

In this paper, we present an algorithm for computing the maximum clique in the visibility graph G of a simple polygon P in O(n²e) time, where n and e are number of vertices and edges of G respectively. We also present an O(ne) time algorithm for computing the maximum hidden vertex set in the visibility graph G of a convex fan P. We assume in both algorithms that the Hamiltonian cycle in G that corresponds to the boundary of P is given as an input along with G.     

On the Area Bisectors of a Polygon

We consider the family of lines that are area bisectors of a polygon (possibly with holes) in the plane. We say that two bisectors of a polygon P are combinatorially distinct if they induce different partitionings of the vertices of P . We derive an algebraic characterization of area bisectors. We then show that there are simple polygons with n vertices that have Ω(n ² ) combinatorially distinct area bisectors (matching the obvious upper bound), and present an output-sensitive algorithm for computing an explicit representation of all the bisectors of a given polygon. Received September 11, 1997, and in revised form April 8, 1998.     

Polygon containment under translation

The polygon containment problem is the problem of deciding whether a polygon P can be translated to fit inside another polygon P'. We present algorithms for two cases of the polygon containment problem: when both P and P' are rectilinearly convex, and when P is convex and P' is arbitrary. In both cases the algorithms run in time O(n²logn), where n is the sum of the number of bounding edges of the two polygons. Applications to an inspection problem and a stock-cutting problem are discussed, as is the containment problem when both P and P' are nonconvex.