Voronoi Diagram and Convex Hull

Transcription

1 Voronoi Diagram and Convex Hull The basic concept of Voronoi Diagram and Convex Hull along with their properties and applications are briefly explained in this chapter. A few common algorithms for generating voronoi diagram and convex hull are also discussed here. 3.1 Voronoi Diagram The Voronoi diagram of a set of sites (points) partitions space (plane) into regions (cells), one per site. For a set ofn sites, Voronoi diagram partitions the space into n region. In other word Voronoi diagram of a set of sites is a collection of regions that divide the plane. Each region corresponds to one of the sites (16]. The region for a site x consists of all points in the region that are closer to x than to any other site. All of the Voronoi regions are convex polygons. Technically Voronoi region is not a polygon since it may be unbounded if the plane is not bounded. The Voronoi diagram is named after the mathematician M. G. Voronoi who explored this geometric construction in 1908 [54]. An ordinary Voronoi diagram is formed by a set of points in the plane called the generators or generating seed. Let G be a set of N distinct generators or seeds in m.ll. Each generator corresponds to a center of one region. The set of all points S that are closer to a given generator Xi E G than any other generator Xj E G j == i is called Voronoi region of Xi. Every point in the plane is identified with the generator that is closest to it by some criteria. Let D(Xi, Xj) be a Euclidean distance function that determines the nearest neighbor (closest) criterion between two points. The Voronoi region is then defined as: V(i) = {x E ]R.ll : D(X,Xi) < D(x,xj) for all j -:;:. i } Varona; D;agram And Convex Hull 30

2 The union of the Voronoi regions of all Xi EGis called the Voronoi diagram ofg: Vor(G) = uv(i). The Euclidean distance between two points XI and X2 is calculated by the function D(XI, X2) using the formula given below: Xl ~. {Xl' rtd x~ ~. (:;1>~. tl'd. where.,.. and. ' are any two points in the plane. A line segment called Voronoi line separates two adjoining Voronoi regions [56]. The Voronoi line is a perpendicular bisector of a line segment joining two sites and IS terminated by two Voronoi points Properties ofvoronoi Diagram Some of the important properties of Voronoi diagram are mentioned below [56]. These properties are helpful in understating the algorithms for implementing Voronoi diagram. A Voronoi line consists of points that are equidistant to two sites in the plane. A Voronoi point is characterized by the equidistance to three different sites in the plane. When plane is bounded, Voronoi regions are always bounded. Therefore a Voronoi cell is considered to be a polygon. A Voronoi region is a convex polygon. Let P be a set of n sites in the plane. If all the sites are collinear then Vor(P) consists of n-1 parallel lines. Otherwise, Vor(P) is connected and its edges are either segments or half-lines [5]. " Voronoi Diagram And Convex HlIII 31

3 For n 2: 3, the number of vertices in the Voronoi diagram of a set of n sites in the plane is at most 2n-5 and the number of edges is at most 3n-6 [5] Example ofvoronoi Diagram Let us draw a Voronoi diagram (please see figure 3.1 similar to shown in [56]) of three sites (or generators) PI, P2, and P3 (three sites are taken for simplicity)). For three sites there will be three perpendicular bisectors i.e. Voronoi lines L1(PI, P2), L2(PI, P3), and L3(P2, P3) for three pairs of sites. These three Voronoi lines will divide a plane (rectangle/ box) into three regions one for each site. The bisectors will intersect at one point called Voronoi point. This point is equidistance form all three sites PI, P2, and P3. 1'2 1'3 Figure 3.1: Voronoi diagram for three sites Applications of Voronoi diagrams Voronoi diagram has number of applications in different areas of study. Scot [16] has listed some of the following applications where Voronoi diagrams are in common use. Robotics - Path planning in the presence of obstacles Networking - Routing in the network Voronoi Diagram And Convex Hull 32

4 Knuth's Post office problem - Given a set of locations for post offices, how to determine the closest post office to a given house? Closest Pairs - Given a set of points, which two are closest together All Nearest Neighbors- Given a set of points, find each point's nearest neighbor Euclidean Minimum Spanning Tree Algorithm for Voronoi Diagrams (Plane Intersect) There are more than hundreds of different algorithms for constructing various types of Voronoi diagrams [16]. In this section we discuss Plane Intersect for constructing Voronoi diagrams since it has been used in our implementation [5, 34, 55]. In this algorithm, construction of Voronoi diagram is based on the observation that a Voronoi region is generated by merging (intersecting) all half planes generated by bisecting lines (Voronoi lines) between the site and all other sites in the plane. For example let there are two sites p and q in a bounded plane, the perpendicular bisector of line segment pq divides the plane into two half planes. There will be only two half planes since there is only one bisector for one pair of sites p and q. Let the half plane that contain p is denoted by h(p,q) and other half plane that contain q by h(q,p). The Voronoi region for site p will be the intersection of h(p,q) and the complete plane since we have taken a bounded plane. Also note that any point y E h(p,q) if and only if D(y, p) < D(r, q). For generalization let P be a set of n sites in a bounded plane and VR(Pi) be a Voronoi region for a site Pi. l-'r(pi) \vill be intersection of n-1 half planes with in the bonded plane. T'7?(Pi) = n\ SjSn.j;ti h(pi, Pj) Voronoi diagram is a union of all Voronoi regions of n sites. Let TD(P) is Voronoi diagram for a set of sites P. Then VD(P) = u\ :ojsn TR(Pi) The time complexity of this algorithm is best - O(n log n) and worst - 0(n2) [34, 55]. Vorol1oi Diagram And Convex Hull

5 3.2 Convex Hull The convex hull of a set Q of points is the smallest convex polygon P for which each point in Q or a line segment joining any two points of Q is either on the boundary of P or in its interior. We denote the convex hull of Q by CH (Q). Intuitively, we can think of each point in Q as being a nail sticking out from a board. The convex hull is then a shape formed by a tight rubber band that surrounds all the nails (chapter 35 of [9]). Figure 3.1 (similar to fig of[9]) shows a set of points and its convex hull. Another equivalent definition states that CH(Q) equals the union of all triangles determined by points of Q. A point lying on the boundary of convex hull is called as boundary point and a point lying interior of convex hull is called as interior point. Boundary points that form the corner vertices of the convex hull are known as extreme points. \ Figure 3.2: Convex hull CH(Q) for set of points Q Applications of Convex Hull Convex hull are useful in a wide range of geometric and computer SCIence applications. A convex hui! provides a way to approximate a point set or other non- Voronoi Diagram And Convex Hull 34

6 convex set by a convex region. A few applications of convex hull are mentioned below: Pattern Recognition - an unknown shape may be represented by its convex hull or by a hierarchy of convex hulls, which is then matched to a database of known shapes [34]. Routing - a node holding a message may select boundary points of a convex hull of its neighbors as next hops to forward the message [48]. Robotics - in motion planning for a moving robot in a plane of obstacles [34] Algorithm for Convex Hull A number of algorithms have been developed using above methods. In this section we discuss the Jarvis' March algorithm (chapter 35 of (9], original algorithm proposed in [26]) that compute the convex hull of a set of n points. Its computational time is O(n*h), where h is the number of vertices of the convex hull This algorithm builds a sequence H = (PI, P2,..., Ph-I) of the vertices of CH(Q). We start with po since it is the lowest point. As shown in figure 3.3 given below (similar to figure in [9]), the next convex hull vertex PI has the least polar angle with respect to po. In case of tie we choose the point farthest from po. Similarly P2 has least polar angle with respect to PI, and so on. When we reach the highest vertex, say Pk, we have constructed the right chain of CH(Q). To construct the left chain, we start at Pk and choose Pk+1 as the point with the least polar angle with respect to Pk, but from the negative x-axis. We continue on, forming the left chain by taking polar angles from the negative x-axis, until we come back to our original vertex po For more detailed description refer chapter 35 of [9]. Voronoi Diagram And Convex Hull 35

7 left chain ro. -I~. right chain Figure 3.3: Construction of Convex Hull by Jarvis's March algorithm l'orolloi Diagram And Convex Hull 36