Flipturning Polygons

   Abstract. A flipturn transforms a nonconvex simple polygon into another simple polygon by rotating a concavity 180° around the midpoint of its bounding convex hull edge. Joss and Shannon proved in 1973 that a sequence of flipturns eventually transforms any simple polygon into a convex polygon. This paper describes several new results about such flipturn sequences. We show that any orthogonal polygon is convexified after at most n-5 arbitrary flipturns, or at most

Motion Planning for a Convex Polygon in a Polygonal Environment

We study the motion-planning problem for a convex m -gon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the three-dimensional space of all free placements of P in Q ) in time that is near-quadratic in mn , which is nearly optimal in the worst case. The algorithm is also conceptually simple. Previous solutions were incomplete, more expensive, or produced only part of the free configuration space. Combining our solution with parametric searching, we obtain an algorithm that finds the largest placement of P in Q in time that is also near-quadratic in mn . In addition, we describe an algorithm that preprocesses the computed free configuration space so that reachability queries can be answered in polylogarithmic time. Received September 9, 1997, and in revised form September 24, 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.     

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.     

Polygons Needing Many Flipturns

A flipturn on a polygon consists of reversing the order of edges inside a pocket of the polygon, without changing lengths or slopes. Any polygon with n edges must be convexified after at most (n − 1)! flipturns. A recent paper showed that in fact it will be convex after at most n(n−3)/2 flipturns.We give here lower bounds.We construct a polygon such that if pockets are chosen in a bad way, at least (n − 2)²/4 flipturns are needed to convexify the polygon. In another construction, (n −1)²/8 flipturns are needed, regardless of the order in which pockets are chosen. All our bounds are adaptive to a pre-specified number of distinct slopes of the edges.     

Morphing Simple Polygons

In this paper we investigate the problem of morphing (i.e., continuously deforming) one simple polygon into another. We assume that our two initial polygons have the same number of sides n , and that corresponding sides are parallel. We show that a morph is always possible through an interpolating simple polygon whose n sides vary but stay parallel to those of the two original ones. If we consider a uniform scaling or translation of part of the polygon as an atomic morphing step, then we show that O(n log n) such steps are sufficient for the morph. Furthermore, the sequence of steps can be computed in O(n log n) time. Received May 25, 1999, and in revised form November 15, 1999.     

Ray Shooting Amidst Convex Polygons in 2D

We consider the problem of ray shooting in a two-dimensional scene consisting ofmconvex polygons with a total ofnedges. We present a data structure that requiresO(mn log m) space and preprocessing time and that answers a ray shooting query inO(log² m log² n) time. If the polygons are pairwise disjoint, the space and preprocessing time can be improved toO((m²+n)log m) andO((m²+n log n)log m), respectively. Our algorithm also works for a collection of disjoint simple polygons. We also show that if we allow onlyO(n) space, a ray shooting query among a collection of disjoint simple polygons can be answered in timeO(m/[formula]¹⁺ log² n) time, for any >0.     

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.     

Voronoi diagrams for convex polygon-offset distance functions

In this paper we develop the concept of a convex polygon-offset distance function. Using offset as a notion of distance, we show how to compute the corresponding nearest- and furthest-site Voronoi diagrams of point sites in the plane. We provide near-optimal deterministic O (n (log n + log ² m) + m) -time algorithms, where n is the number of points and m is the complexity of the underlying polygon, for computing compact representations of both diagrams. Received January 8, 1999, and in revised form January 5, 2000, and June 12, 2000. Online publication December 4, 2000.     

Quadrilaterizing an Orthogonal Polygon in Parallel

We consider the problem of quadrilaterizing an orthogonal polygon P, that is to decompose P into nonoverlapping convex quadrangles without adding new vertices. In this paper we present a CREW-algorithm for this problem which runs in O(log N) time using Θ(N/log N) processors if the rectangle decomposition of P is given, and Θ(N) processors if not. Furthermore we will show that the latter result is optimal if the polygon is allowed to contain holes.     

A fast algorithm for computing minimum 3-way and 4-way cuts

For an edge-weighted graph G with n vertices and m edges, we present a new deterministic algorithm for computing a minimum k-way cut for k=3,4. The algorithm runs in O(n ^k-1 F(n,m))=O(mn ^k log(n ² /m)) time and O(n ²) space for k=3,4, where F(n,m) denotes the time bound required to solve the maximum flow problem in G. The bound for k=3 matches the current best deterministic bound ?(mn ³) for weighted graphs, but improves the bound ?(mn ³) to O(n ² F(n,m))=O(min{mn ^8/3,m ^3/2 n ²}) for unweighted graphs. The bound ?(mn ⁴) for k=4 improves the previous best randomized bound ?(n ⁶) (for m=o(n ²)). The algorithm is then generalized to the problem of finding a minimum 3-way cut in a symmetric submodular system. Received: April 1999 / Accepted: February 2000?Published online August 18, 2000     

Geometric Lower Bounds for Parametric Matroid Optimization

We relate the sequence of minimum bases of a matroid with linearly varying weights to three problems from combinatorial geometry: k -sets, lower envelopes of line segments, and convex polygons in line arrangements. Using these relations we show new lower bounds on the number of base changes in such sequences: Ω(nr ^1/3 ) for a general n -element matroid with rank r , and Ω(mα(n)) for the special case of parametric graph minimum spanning trees. The only previous lower bound was Ω(n log r) for uniform matroids; upper bounds of O(mn ^1/2 ) for arbitrary matroids and O(mn ^1/2 / log ^* n) for uniform matroids were also known. Received November 12, 1996, and in revised form February 19, 1997.     

Aperture-Angle and Hausdorff-Approximation of Convex Figures

The aperture angle α(x,Q) of a point x ∉ Q in the plane with respect to a convex polygon Q is the angle of the smallest cone with apex x that contains Q. The aperture angle approximation error of a compact convex set C in the plane with respect to an inscribed convex polygon Q⊂C is the minimum aperture angle of any x∈C∖Q with respect to Q. We show that for any compact convex set C in the plane and any k>2, there is an inscribed convex k-gon Q⊂C with aperture angle approximation error . This bound is optimal, and settles a conjecture by Fekete from the early 1990s. The same proof technique can be used to prove a conjecture by Brass: If a polygon P admits no approximation by a sub-k-gon (the convex hull of k vertices of P) with Hausdorff distance σ, but all subpolygons of P (the convex hull of some vertices of P) admit such an approximation, then P is a (k+1)-gon. This implies the following result: For any k>2 and any convex polygon P of perimeter at most 1 there is a sub-k-gon Q of P such that the Hausdorff-distance of P and Q is at most  . This research was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-311-D00763). NICTA is funded through the Australian Government’s Backing Australia’s Ability initiative, in part through the Australian Research Council.     

Isodiametric Problems for Polygons

The maximal area of a polygon with n = 2m edges and unit diameter is not known when m ≥ 5, nor is the maximal perimeter of a convex polygon with n = 2^m edges and unit diameter known when m ≥ 4. We construct improved polygons in both problems, and show that the values we obtain cannot be improved for large n by more than c₁/n³ in the area problem and c₂/n⁵ in the perimeter problem, for certain constants c₁ and c₂.     

A Faster Deterministic Maximum Flow Algorithm

Cheriyan and Hagerup developed a randomized algorithm to compute the maximum flow in a graph with n nodes and m edges in O(mn + n² log²n) expected time. The randomization is used to efficiently play a certain combinatorial game that arises during the computation. We give a version of their algorithm where a general version of their game arises. Then we give a strategy for the game that yields a deterministic algorithm for computing the maximum flow in a directed graph with n nodes and m edges that runs in time O(mn(log_{m/n log n}n)). Our algorithm gives an O(mn) deterministic algorithm for all m/n = Ω(n^ε) for any positive constant ε, and is currently the fastest deterministic algorithm for computing maximum flow as long as m/n = ω(log n).     

Computing Mixed Discriminants, Mixed Volumes, and Permanents

We construct a probabilistic polynomial time algorithm that computes the mixed discriminant of given n positive definite matrices within a 2 ^O(n) factor. As a corollary, we show that the permanent of an nonnegative matrix and the mixed volume of n ellipsoids in R ⁿ can be computed within a 2 ^O(n) factor by probabilistic polynomial time algorithms. Since every convex body can be approximated by an ellipsoid, the last algorithm can be used for approximating in polynomial time the mixed volume of n convex bodies in R ⁿ within a factor n ^O(n) . Received July 10, 1995, and in revised form May 20, 1996.     

Vertex-Rounding a Three-Dimensional Polyhedral Subdivision

Let P be a polyhedral subdivision in R ³ with a total of n faces. We show that there is an embedding σ of the vertices, edges, and facets of P into a subdivision Q , where every vertex coordinate of Q is an integral multiple of . For each face f of P , the Hausdorff distance in the L _∈fty metric between f and σ(f) is at most 3/2 . The embedding σ preserves or collapses vertical order on faces of P . The subdivision Q has O(n ⁴ ) vertices in the worst case, and can be computed in the same time. Received September 3, 1997, and in revised form March 29, 1999.     

A criterion for finite lattice coverings

For a centrally symmetric convex and a covering lattice L for K, a lattice polygon P is called a covering polygon, if . We prove that P is a covering polygon, if and only if its boundary bd(P) is covered by (L ∩ P) + K. Further we show that this characterization is false for non-symmetric planar convex bodies and in Euclidean d–space, d ≥ 3, even for the unit ball K = B ^d. This revised version was published online in June 2006 with corrections to the Cover Date.     

The two-convex-polygons TSP: A solvable case

In this paper, the Travelling Salesman Problem whenm points are on one convex polygonP, andn points are on another convex polygonQ, insideP, is polynomially solved. For this specific case, anO(m ³ n ³) time andO(m ² n ²) space algorithm to obtain the tour with minimum length is provided.