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38 Int'l Conf. Foundations of Computer Science | FCS'11 | Three Three Heuristic Heuristic Algorithms Algorithms Algorithms Algorithms for for for for Generation Generation Generation Generation of of of Random Random Polygons by by a a Given Given Set Set of of of Vertices Vertices A. Nourollah 1 Faculty of Electrical & Computer Engineering of Islamic Islamic Azad & Faculty of of Electrical Electrical & Computer Computer Engineering of of Shahid Shahid Rajaee Teacher Teacher Training Training University, University, Tehran, Tehran, Iran Iran 2 Department Department of Electrical & Computer Engineering of Islamic Azad University, Qazvin, Iran 1 , L.Mohammadi Faculty of Electrical & Computer Engineering of Islamic Azad Faculty of Electrical & Computer Engineering of Shahid Rajaee Teacher Training University, Tehran, Iran Electrical & Computer Engineering of Islamic Azad University, Qazvin, Iran 2 Faculty of Electrical & Computer Engineering of Islamic Azad University, Qazvin, Iran Faculty Faculty of of Electrical Electrical & & Computer Computer Engineering Engineering of of Shahid Shahid Rajaee Rajaee Teacher Teacher Training Training University, University, Tehran, Tehran, Tehran, Iran Iran Electrical & Computer Computer Engineering of of Islamic Islamic Azad Azad University, University, Qazvin, Iran Abstract - One One of of of the the discussed discussed issues issues in in computer computer graphics graphics graphics and and computational computational geometry geometry is is the the generation generation of of random random polygons. This problem is is is of of considerable considerable considerable importance importance importance in in the the practical practical evaluation evaluation of of algorithms algorithms algorithms that that that operate operate operate on on on polygons, polygons, polygons, where where it it is is necessary necessary to to check check the correctness and and to determine determine the the actual CPU-usage CPU CPU usage of an algorithm experimentally. experimentally. To To this this time, time, no no no polynomial polynomial-time polynomial time algorithm is known known for for the the random random generation generation of of polygons polygons with with a a uniform uniform distribution distribution. . In This This paper, paper, three three new new new inventive inventive algorithms algorithms are are presented resented for generating generating random random polygons polygons using using using a a set set of of random random vertices vertices using using time. Keywords: angular scanning, convex convex hull, random Simple polygon, star star-shaped shaped polygon, polygon, computer computer graphics. graphics. 1 Introduction The The computer computer graphics is the science of computing the images which would be be led to to generate generate them. them. One ne of the important important branches branches of of of computer computer graphics graphics is is the the computational computational geometry. geometry. The The computational computational geometry geometry executes executes executes some some some computing computing on on the the geometrical geometrical geometrical shapes shapes such such as as polygons. polygons. The The polyg polygons ons are appropriate shapes shapes for for for representing representing the the shapes shapes of of real real world world and and and every every object object in in the the nature nature nature nature would would be be be representable representable as as a a set set set of of polygons. polygons. For For the the reason reason of of this this issue issue importance, importance, one one one of of the the important important important discussed discussed issues issues in computer computer graphics and and comput computational comput ational geometry is the the problem problem of of generating generating random random polygons. polygons. The The problem of of random polygon polygon generati generation is defined as follows: follows: Given , a uniformly uniformly random polygon polygon on is a polygon polygon generated generated with with probability probability if there exist simple polygon on in total. Consider Consider two two two diffrenet diffrenet pairs pairs of of points points the the two two edges edges defined defined by by these these pairs. pairs. It It It is is is easy easy easy to to to observe observe observe that that that the the the number number number of of simple simple polygons polygons containing containing containing one one of of of these these edges edges by by by a a a dashed dashed line: line: For For the the first first edge edge there there exactly exactly two two simple simple simple polygons polygons (Fig 1-a, 1 Fig 1-b) b) which contain contain it,whereas it,whereas for for the the second second edge edge three three simple simple polygons (Fig 1-c, Fig 1-d, Fig 1-e) e) containing it exist. Unfortunately Unfortunately no algorithm has been presented presented yet yet for the problem problem of of generating generating random random simple simple polyg polygons with linear time time complexity. complexity. This motivated researchers to pursue pursue heuristics heuristics for for generating generating generating random random simple simple polygons. polygons. The The generation generation of of of random random random simple simple polygons polygons polygons has has two two main areas of applicati application: on: The The empirical empirical verification verification of of an an algorithm algorithm oprating oprating on on polygons. polygons. The The The practical practical practical testing testing of of its its CPU CPU-usage. Too Too many many algorithms algorithms have have been been presented presented for for solving solving the the problem problem of of of random random polygons polygons and and and some some some of of of them them are briefly explained here in this paper [1]. . The following sections of of this this paper paper has has been been organized in in this way way: section 2 has been allocated to related works for generating of random polygons polygons. In section 3, , three proposed algorithms would be offered and and section section 4 investigates the performance of the discussed algorithms and finally in section 5 conclusion conclusions would be discussed discussed. 2 Fig. 1. Two Two edges edges belonging belonging to to a a different different number number of of simple simple polygons The related work For For the the the reason reason mentioned mentioned in in section section ago, ago, the the random random generation generation of of of geometric geometric objects objects objects has has has received received received some some some attention attention attention by researchers: Epstein Epstein and Sack presented an an algorithm for generating generating of of simple simple polygons polygons polygons at at at random random random [ [2]. Devroy, Devroy, Epstein Epstein and and and Sack Sack studied studied studied the the the random random random generation generation of of intervals intervals and and hyperrectangles. hyperrectangles. They They They consider consider the the the problem problem of of generating generating a a a random random random hyperrectangle hyperrectangle hyperrectangle in in a a unit unit unit hypercube hypercube hypercube [3]. Atkinson and Sack Sack studied studied the the uniform uniform generation generation of of forests forests of restricted height. A -way way way tree tree is is either either empty empty or or consists consists of a a root node node and an an ordered ordered sequence sequence of subtrees. their
Int'l Conf. Foundations of Computer Science | FCS'11 | 39 algorithms runs in , where denotes the height of the tree[4]. An algorithm running in at most time for the random generation of -monotone polygons was described by Zhu et al [5]. An interesting approach for the generation of random polygons in the plane (but not on a given set of points) was researched by O'Rourke and Virmany[6]. In the next section three proposed heuristic algorithms are designed for this problem. 3 Proposed algorithms In this section, three proposed algorithms are discussed for generating of random polygons. each algorithm has time complexity . 3.1 Algorithm 1 Let be a set of random vertices and no three points are on the same line. The convex hull, the set of vertices in the plane ( ), of a point set in the plane is the enclosing convex polygon with smallest area. With this description, this algorithm performs in a way that first would be obtained. Let be a set of interior vertices of this convex hull. Then, for the set of those vertices which are inside it will perform in this way, i.e., : first the vertex with the least - coordinate is found. It is supposed that this vertex is . The vertices of is sorted according to the polar angle in the counter-clockwise around . By angular scanning of these vertices around vertex , visited vertices would be joined together. The last visited vertex is joined to vertex and in this way a star-shaped polygon would be obtained (that from a point inside it all vertices are visited). Now, from the set of vertices over , the closest vertex to is selected and it is supposed to be this vertex . If the vertex close to over interior star-shaped polygon in the counter-clockwise is and the vertex close to vertex over in the counterclockwise is (Fig 2-a), so the edges with the edges are replaced. Thus, a random polygon would be created (Fig 2-b). The procedure of this algorithm is as the following: • First, would be computed over the set of random vertices . • The set of interior vertices is considered as and the vertex with the least -coordinate is selected from the set of vertices (vertex ). • The vertices are visited according to the polar angle in the counter-clockwise and a star-shaped polygon is obtained with angular scanning of vertices around vertex and joining the sorted vertices. • The closest vertex to vertex is selected From the set of vertices over (vertex ). • The vertex close to and the vertex close to in the counter-clockwise is called and , respectively. • The edges re replaced by the edges . (a) (b) Fig. 2. (a) Creating a star-shaped polygon with the points inside the convex hull, (b) creating the random polygon with the replacement of edges with the edges . 3.2 Algorithm 2 In this algorithm like the previous algorithm, first is computed. Let be the vertex set inside of the convex hull. For this set of vertices, the existing convex bottom algorithm would be applied which has time and performs in this way that first two vertices which has the least and the most -coordinate is considered. These vertices is joined together using a presumptive line so that the set of vertices would be divided into two upper and lower half ( if the set of vertices in the lower half is empty, the presumptive line is considered as the convex hull of two points and the algorithm executes again).
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