Disjoint Convex Polygons Research Articles

A Shape-Newton Approach to the Problem of Covering with Identical Balls

The problem of covering a region of the plane with a fixed number of minimum-radius identical balls is studied in the present work. An explicit construction of bi-Lipschitz mappings is provided to model small perturbations of the union of balls. This allows us to obtain analytical expressions for first- and second-order derivatives of the cost functional using nonsmooth shape optimization techniques under appropriate regularity assumptions. For regions defined as the union of disjoint convex polygons, algorithms based on Voronoi diagrams that do not rely on approximations are given to compute the derivatives. Extensive numerical experiments illustrate the capabilities and limitations of the introduced approach.

Retraction Notice: A New Algorithm for the Shortest Path of Touring Disjoint Convex Polygons

Retraction Notice: A New Algorithm for the Shortest Path of Touring Disjoint Convex Polygons

A New Algorithm for the Shortest Path of Touring Disjoint Convex Polygons

Given a start point s, a target point t, and a sequence of k disjoint convex polygons in the plane, finding the shortest path from s to t which visits each convex polygon in the given order is our focus. In this paper, we present an improved method to compute the shortest path based on the last step shortest path maps by Dror et al. Instead of using of point location in previous algorithm, we propose an efficient method of locating the points in the path with linear query and make the data structures much simpler. Our improved algorithm gives the O(nk) running time which improves upon the time O(nklog(n/k)) by Dror et al., where n is the total number of vertices of all polygons. Furthermore, we have implemented this algorithm by programming. The result shows that our algorithm is correct and efficient..

On the minimum cardinality of a planar point set containing two disjoint convex polygons

Let N(k, l) be the smallest positive integer such that any set of N(k, l) points in the plane, no three collinear, contains both a convex k-gon and a convex l-gon with disjoint convex hulls. In this paper, we prove that N(3, 4) = 7, N(4, 4) = 9, N(3, 5) = 10 and N(4, 5) = 11.

A 2-approximation algorithm for the zookeeper's problem

Let P be a simple polygon, and let p be a set of disjoint convex polygons inside P, each sharing one edge with P. The zookeeper's route problem asks for a shortest route inside P that visits (but d...

A new fast algorithm for computing the distance between two disjoint convex polygons based on Voronoi diagram

Computing the distance between two convex polygons is often a basic step to the algorithms of collision detection and path planning. Now, the lowest time complexity algorithm takes O(logm+logn) time to compute the minimum distance between two disjoint convex polygons P and Q, where n and m are the number of the polygons’ edges respectively. This paper discusses the location relations of outer Voronoi diagrams of two disjoint convex polygons P and Q, and presents a new O(logm+logn) algorithm to compute the minimum distance between P and Q. The algorithm is simple and easy to implement, and does not need any preprocessing and extra data structures.

A 2-approximation algorithm for the zookeeper's problem

Let P be a simple polygon, and let P be a set of disjoint convex polygons inside P, each sharing one edge with P. The zookeeper's route problem asks for a shortest route inside P that visits (but does not enter) each polygon in P . We present an O ( n ) time algorithm for computing a zookeeper's route of length at most 2 times that of the shortest zookeeper's route. Our result improves upon the previous approximation factor 6.

A 2-approximation algorithm for the zookeeper's problem

Let P be a simple polygon, and let p be a set of disjoint convex polygons inside P, each sharing one edge with P. The zookeeper's route problem asks for a shortest route inside P that visits (but does not enter) each polygon in p. We present an O(n) time algorithm for computing a zookeeper's route of length at most 2 times that of the shortest zookeeper's route. Our result improves upon the previous approximation factor 6.

SEMI-BALANCED PARTITIONS OF TWO SETS OF POINTS AND EMBEDDINGS OF ROOTED FORESTS

Let m be a positive integer and let R1, R2 and B be three disjoint sets of points in the plane such that no three points of R1 ∪ R2 ∪ B lie on the same line and |B| = (m-1)|R1|+m|R2|. Put g = |R1∪R2|. Then there exists a subdivision X1∪X2∪⋯∪Xg of the plane into g disjoint convex polygons such that (i) |Xi ∩ (R1 ∪ R2)| = 1 for all 1 ≤ i ≤ g; and (ii) |Xi∩B| = m-1 if |Xi∩R1| = 1, and |Xi∩B| = m if |Xi∩R2| = 1. This partition is called a semi-balanced partition, and our proof gives an O(n4) time algorithm for finding the above semi-balanced partition, where n = |R1| + |R2| + |B|. We next apply the above result to the following theorem: Let T1,…,Tg be g disjoint rooted trees such that |Ti| ∈ {m,m+1} and vi is the root of Ti for all 1 ≤ i ≤ g. Let P be a set of |T1|+⋯+|Tg| points in the plane in general position that contains g specified points p1,…,pg. Then the rooted forest T1 ∪ ⋯ ∪ Tg can be straight-line embedded onto P so that each vi corresponds to pi for every 1 ≤ i ≤ g.

INNER-COVER OF NON-CONVEX SHAPES

We present an algorithm that for a given simple non-convex polygon P finds an approximate inner-cover by large convex polygons. The algorithm is based on an initial partitioning of P into a set C of disjoint convex polygons which are an exact tessellation of P. The algorithm then builds a set of large convex polygons contained in P by constructing the convex hulls of subsets of C. We discuss different strategies for selecting the subsets and we claim that in most cases our algorithm produces an effective approximation of P.

An approximative solution to the Zookeeper's Problem

Consider a simple polygon P containing disjoint convex polygons each of which shares an edge with P . The Zookeeper's Problem then asks for the shortest route in P that visits all convex polygons without entering their interiors. Existing algorithms that solve this problem run in time super-linear in the size of P and the convex polygons. They also suffer from numerical problems. In this paper, we shed more light on the problem and present a simple linear time algorithm for computing an approximate solution. The algorithm mainly computes shortest paths and intersections between lines using basic data structures. It does not suffer from numerical problems. We prove that the computed approximation route is at most 6 times longer than the shortest route in the exact solution.

Finding shortest safari routes in simple polygons

Let P be a simple polygon, and let P be a set of disjoint convex polygons inside P, each sharing one edge with P. The safari route problem asks for a shortest route inside P that visits each polygon in P . In this paper, we first present a dynamic programming algorithm with running time O( n 3) for computing the shortest safari route in the case that a starting point on the route is given, where n is the total number of vertices of P and polygons in P . (Ntafos in [Comput. Geom. 1 (1992) 149–170] claimed a more efficient solution, but as shown in Appendix A of this paper, the time analysis of Ntafos' algorithm is erroneous and no time bound is guaranteed for his algorithm.) The restriction of giving a starting point is then removed by a brute-force algorithm, which requires O( n 4) time. The solution of the safari route problem finds applications in watchman routes under limited visibility.

Erratum to: “Efficient algorithm for transversal of disjoint convex polygons”

Erratum to: “Efficient algorithm for transversal of disjoint convex polygons”

Some Aperture-Angle Optimization Problems

Abstract. Let P and Q be two disjoint convex polygons in the plane with m and n vertices, respectively. Given a point x in P , the aperture angle of x with respect to Q is defined as the angle of the cone that: (1) contains Q , (2) has apex at x and (3) has its two rays emanating from x tangent to Q . We present algorithms with complexities O(n log m) , O(n + n log (m/n)) and O(n + m) for computing the maximum aperture angle with respect to Q when x is allowed to vary in P . To compute the minimum aperture angle we modify the latter algorithm obtaining an O(n + m) algorithm. Finally, we establish an Ω(n + n log (m/n)) time lower bound for the maximization problem and an Ω(m + n) bound for the minimization problem thereby proving the optimality of our algorithms.

Efficient algorithm for transversal of disjoint convex polygons

Given a set S of n disjoint convex polygons { P i ∣1⩽ i⩽ n} in a plane, each with k i vertices, the transversal problem is to determine whether there exists a straight line that goes through every polygon in S . We show that the transversal problem can be solved in O( N+ nlog n) time, where N=∑ i=1 n k i is the total number of vertices of the polygons.

On the number of disjoint convex quadrilaterals for a planar point set

For a given planar point set P, consider a partition of P into disjoint convex polygons. In this paper, we estimate the maximum number of convex quadrilaterals in all partitions.

Shortest zookeeper's routes in simple polygons

Let P be a simple polygon, and let P be a set of disjoint convex polygons inside P, each sharing one edge with P. The zookeeper's route problem asks for a shortest route inside P that visits (but does not enter) each polygon in P . We present an O(n 2) time algorithm for computing a shortest zookeeper's route, on which no starting point is specified.

Generalizing Ham Sandwich Cuts to Equitable Subdivisions

We prove a generalization of the famous Ham Sandwich Theorem for the plane. Given gn red points and gm blue points in the plane in general position, there exists an equitable subdivision of the plane into g disjoint convex polygons, each of which contains n red points and m blue points. For g = 2 this problem is equivalent to the Ham Sandwich Theorem in the plane. We also present an efficient algorithm for constructing an equitable subdivision.

On a partition into convex polygons

In this paper we study the problem of partitioning point sets in the plane so that each equivalence class is a convex polygon with some conditions on the intersection properties of such sets. Let P be a set of n points in the plane. We define f( P) to be the minimum number of sets in a partition into disjoint convex polygons of P and F( n) as the maximum of f( P), over all sets P of n points. We establish lower and upper bounds for F( n). We also estimate the maximum of the minimum number of sets in a partition into empty convex polygons, over all sets of n points. Finally, we consider the maximum of the minimum number of convex polygons which cover the n points set P, over all sets P of n points.

Bisections and ham-sandwich cuts of convex polygons and polyhedra

We present a linear time sequential algorithm for finding a straight line that bisects two given disjoint convex polygons. The solution can be generalized to other measures (for instance, perimeter) and other proportions of cutting. Also, the problem is studied in three dimensional space: find a plane that cuts each of three given disjoint convex polyhedra into two parts of equal volume.