Voronoi Diagrams and their Applications

Transcription

1 XXVI. ASR '2001 Seminar, Instruments and Control, Ostrava, April 26-27, 2001 Paper 65 Voronoi Diagrams and their Applications ŠEDA, Miloš RNDr. Ing. Ph.D., Brno University of Technology, Faculty of Mechanical Engineering, Institute of Automation and Computer Science, Technická 2, Brno, , Abstract: The Voronoi diagram is a fundamental structure in computational geometry and arises naturally in various branches of science. This paper surveys basic properties of the Voronoi diagram, algorithms for its construction, and typical applications of the Voronoi diagram and its geometric dual, the Delaunay triangulation. Keywords: computational geometry, Voronoi diagram, Delaunay triangulation 1 Introduction Computational geometry emerged from the field of algorithms design and analysis in the late 1970s. It has many application domains including computer graphics, geographic information systems (GIS), robotics, and others in which geometric algorithms play a fundamental role. In this sense, Voronoi diagrams belong to most important structures with a surprising variety of applications. A typical task for introducing Voronoi diagrams is Knuth s Post Office Problem stated as follows: Given a set of locations for post offices, how do we determine the closest post office to a given house? If we would have a map showing a subdivision of the city into regions indicating the nearest post office all we need to do is figure out which region the house is in. This subdivision into regions is represented by Voronoi diagrams. More formally: Definition 1.1: A Voronoi diagram (or Voronoi tessellation) for a given set P={p 1, p 2,, p n } of points (or sites) is a partition of the plane into closed or open areas V(p 1 ), V(p 2 ),, V(p n ), called Voronoi regions (or Voronoi cells or Voronoi polytopes) such that p i V(p i ) for each p i, and any point q V(p i ) is closer to p i than to any other p j (points on the boundary of V(p i ) and V(p j ) are equidistant to p i and p j ). A Voronoi diagram is called degenerate if four or more of its Voronoi edges have a common endpoint. Evidently, the Voronoi region V(p i ) is the intersection of all halfplanes H(p i, p j ), i j, such that each halfplane contains the point p i and is bounded by the perpendicular to the segment p i p j at its middle points. Thus, V(p i ) is a convex polygon. The vertices of polygons V(p i ) are called the vertices of the Voronoi diagram, and their edges are called the edges of the Voronoi diagram. Clearly, each edge of the Voronoi diagram belongs to just two Voronoi regions. Now, let us list some important properties of the Voronoi diagrams [AURENHAMMER, 1991], [De BERG et al., 2000], [IVANOV and TUZHILIN, 1994], [SHAMOS and HOEY, 1975], [TAMASSIA, 1999]: Assume that Voronoi diagrams are non degenerate. Then it is satisfied: Every vertex of a Voronoi diagram V(P) is a common intersection of exactly three edges of the diagram. A point q is a vertex of V(P) if and only if its largest empty circle C p (q) contains three points on its boundary. The bisector between points p i and p j defines an edge of V(P) if and only if there is a point q such that C p (q) contains both p i and p j on its boundary but no other point

2 For any q in P, V(q) is convex. Voronoi diagram V(P) of P is planar. Polygon V(p i ) is unbounded if and only if p i is a point on the boundary of convex hull of the set P. (Convex hull CH(P) of a set P is the smallest convex set that contains P). Definition 1.2: A triangulation T is a collection of N triangles satisfying the following requirements: 1. The interiors of the triangles are pairwise disjoint. 2. Each edge of a triangle in T is either a common edge of two triangles in T or else it is on the boundary of the union D of all the triangles. 3. D is homeomorphic to a square (this requirement rules out holes, pinchpoints where just two triangles meet in a single point, and disjoint sets of triangles). Definition 1.3: The graph D(P) on P with an edge (p i, p j ) if V(p i ) and V(p j ) share a common side is called a Delaunay triangulation. Figure 1. Voronoi diagram and Delaunay triangulation 2 Algorithms for construction of Voronoi diagram The earliest diagrams were drawn with pencil and ruler. Now there are many different algorithms for constructing various types of Voronoi diagrams. Fundamental algorithms and their modifications [AMATO and RAMOS, 1995], [AURENHAMMER, 1991], [De BERG et al., 2000], [DEHNE and KLEIN, 1997], [SHAMOS and HOEY, 1975] are the incremental algorithm, random incremental algorithm, divide and conquer algorithm and plane sweep algorithm (or Fortune s algorithm). We will briefly describe two of them

3 2.1 Incremental algorithm The incremental algorithm inserts the points one at a time into the diagram. Figure 2 illustrates the insertion of a point p that involves three things. First, we need to find the current Voronoi region in which the new point p falls. Let q be the point defining this region; the separator of p and q then will contribute an edge, e, to p s region. Second, we need to "walk around" the boundary of the new point's Voronoi region (bold). This boundary is created edge by edge, starting with e. Finally, we delete all the old edges sticking into the new region (dashed). p Figure 2. Inserting a Voronoi region The second and third steps are the harder ones. It's possible that each new point's Voronoi region will touch all the old regions. Thus, in the worst case, we end up spending linear time on each region, or O(n 2 ) time overall. In [CHEW and FORTUNE, 1997], the question is studied, if the Voronoi diagram can be built faster when points are sorted before we start. One approach to speeding up insertion is randomization. It has been proven that inserting the points in random order yields an O(n log n)-time performance with high probability, regardless of which set of points is given. 2.1 Divide and conquer algorithm A widely used method to design fast algorithms is divide and conquer. This method was applied in [SHAMOS and HOEY, 1975] for computing Voronoi diagrams in the plane. The given set of n points is divided by a vertical line into two subsets S 1 and S 2 of approximately equal size. The Voronoi diagrams for these subsets are computed recursively and are then merged in the conquer step to form the total diagram. The merge step can be done in linear time by the "walking ant" method. An ant starts down at, walking upward along the path halfway between some S 1 -point and some S 2 -point (p is a S i -point iff p S i ). The ant wants to walk all the way up to +, staying as far away from the points as possible. Whenever the ant gets to an S 1 -edge (i.e the edge whose end points are in S 1 ), it turns away from the new S 1 - point. Whenever it hits an S 2 -edge, it turns away from the S 2 -point. Two diagrams can be merged in O(n) time, which implies an O(n log n)-time algorithm if the divide step is carried out on a balanced way

4 Figure 3. Subdivision of P into S 1 and S 2 and merging their Voronoi diagrams V(S 1 ) and V(S 2 ) Figure 4. Final Voronoi diagram V(P) 3 Applications of Voronoi diagrams Voronoi diagrams have a surprising variety of uses, e.g. nearest neighbour search, facility location, path planning, etc. They can also be used to model trading areas of different cities, to guide robots, and even to describe and simulate the growth of crystals. The dual of the Voronoi diagram, Delaunay triangulation, can be used for an effective finding Euclidean minimum spanning tree. The report [DRYSDALE, 1993] briefly summarises applications from archaeology to zoology. Now we characterise some of these applications in more detail. 3.1 Point location For a given point set P in the plane we want to construct a data structure which, for a given query point x, finds the point of P nearest to x as quickly as possible. This problem arises directly in some practical situations or, more significantly, as a subroutine in more complicated problems. We need to determine the region of the Voronoi diagram of P containing x

5 3.2 Robot motion planning Consider a disc-shaped robot in the plane. It should pass among a set P of point obstacles, getting from a given start position to a given target position and touching none of the obstacles [MATOUŠEK, 2000]. It is easy way to see that if such a passage is possible at all, the robot always walks along the edges of the Voronoi diagram of P, which define the possible channels that maximise the distance to the obstacles, except for the initial and final segments of the tour. This allows us to reduce the robot motion problem to a graph search problem: we define a subgraph of the Voronoi diagram consisting of the edges that are passable for the robot. Instead of spheres of influence points, we can consider the spheres of influence of other objects, such as disjoint polygons. This is what we get if we have a circular robot moving amidst polygonal obstacles. The case when points are situated on a rectilinear plane with rectangular obstacles is studied in [GUHA and SUZUKI, 1997]. 4 Applications of Delaunay triangulation A frequent problem in practice is the problem of constructing a minimum spanning tree in the Euclidean plane (EMST). This problem can be easily solved by well-known Jarník s (Prims s), Kruskal s or Borůvka s polynomial algorithms for the Minimum Spanning Tree Problem in graphs (MSTG) when we construct from the given set of points a complete graph whose edges are represented by straight lines between each pair of points and their weights correspond to Euclidean distances of these points. All these algorithms have the same asymptotic running time O(( V + E ) log V ) for a graph G=(V,E) [MOUNT, 1999], but unfortunately the time complexity of the complete graph construction is higher, it equals to O( V 2 ) and therefore the total running time of the algorithm is O( V 2 ). Another approach for solving the EMST is based on the concept of the Delaunay triangulation. We refer to the fact that when searching for a current edge of minimal weight in minimum spanning tree algorithms, it suffices to look over the edges of the Delaunay triangulation. It results from the following assertions [HWANG et al., 1992], [IVANOV and TUZHILIN, 1994]. Lemma 4.1: Let P = P P be an arbitrary partition of a finite set P of points of the plane, and assume that e is the shortest segment joining the sets P and P. Then e is an edge of the Delaunay triangulation for the set P. Proof. Suppose that e = AB is not an edge of the Delaunay triangulation D(P) for the set P and let A P and B P. Then the perpendicular to the segment AB at its middle point C does not contain an edge of the Voronoi diagram, and the boundary of the Voronoi region V(A) intersects the segment AB at some point X lying between the point C and the point A. Let h be the edge of the V(A) intersecting AB. Then h lies on the perpendicular to the segment AA at its middle point, where A is some point from P and A lies inside the circle of diameter AB centeres at C. If A belongs to the set P, then e is not a shortest segment joining P and P, since A B < AB. Analogously, A cannot belong to P, since A A < AB. Theorem 4.2: Let P be an arbitrary finite set of points of the plane, and assume that T is some minimum spanning tree spanning P. Then the edges of T are edges of the Delaunay triangulation D(P) for the set P. Proof. Each step of Jarník s algorithm for constructing an MST spanning P consists of constructing a tree T spanning T(P) P, and adding an edge e that joins a vertex from T(P) with one from P T(P) such that the length of e is the least possible among the lengths of the edges of this type

6 Using these results, we can construct the minimum spanning tree in the Euclidean plane as follows: Firstly, we construct a Voronoi diagram from the given set of n points (in O(n log n) time), then the corresponding Delaunay triangulation (in O(n) time) and finally the EMST (in O(n) time [CHERITON and TARJAN, 1976]). Therefore, the Delaunay based approach is more efficient that this one based on constructing the complete graph consuming O(n 2 ) time. Figure 5. Voronoi diagram, Delaunay triangulation and Euclidean mimimum spanning tree Let us note that there are algorithms for constructing the Delaunay triangulation directly without precomputation of the Voronoi diagram [FORTUNE, 1997], [FORTUNE, 1992]. Delaunay triangulation can also be used for approximate solving the Euclidean Steiner Minimal Tree Problem, for more detail see [BEASLEY and GOFFINET, 1994]. 5 Conclusion The purpose of this paper was to introduce a fundamental geometric structure of computational geometry Voronoi diagram, and summarise basic algorithms for computing - 6 -

7 Voronoi diagrams in the plane and show typical examples of their applications. Voronoi diagrams are an aid of collision-free motions in robotics. In the scene of point obstacles, when moving on the edges of the Voronoi diagram the robot always keeps as far as possible from the neighbourhood obstacles. The Delaunay based approach presented in Section 4 is a very important application of solving the Euclidean minimum spanning tree (EMST), because the EMST is a good approximation of the Euclidean Steiner minimum tree (Steiner ratio in the worst case equals to 1.15). There are many other applications of Voronoi diagrams that were not presented here, e.g. in cartography (piece together satellite photography into large mosaic maps), geography (analyzing patterns of urban settlements), crystallography (Voronoi diagram models crystal growth where each crystal has a different growth rate), biology (model and analyse plant competition), etc. More general algorithms, for sets of points with the Euclidean metric in E d are presented in [FORTUNE, 1992]. Algorithms for constructing higher-dimensional Delaunay triangulations are described in [SHEWCHUK, 2000]. Acknowledgement: This paper was supported by the research project CEZ: J22/98: Non-Traditional Study Methods of Complex and Uncertain Systems 6 References AMATO, N. and RAMOS, E.A.: On Computing Voronoi Diagrams by Divide-Prune-and- Conquer. In Proceedings of the 12th Annual Symposium on Computational Geometry SCG 95. Philadelphia, 1995, pp AURENHAMMER, F.: Voronoi Diagrams A Survey of a Fundamental Geometric Data Structure. ACM Computing Surveys, Vol. 23, No. 3, 1991, pp BEASLEY, J.E. and GOFFINET, F.: A Delaunay Triangulation-Based Heuristic for the Euclidean Steiner Problem. Networks, Vol. 24, 1994, pp CHERITON, D. and TARJAN, R.E.: Finding Minimum Spanning Trees. SIAM Journal on Computing, Vol. 5, No. 4, 1976, pp CHEW, L.P. and FORTUNE, S.: Sorting Helps for Voronoi Diagrams. Algorithmica, Vol. 18, 1997, pp CHOSET, H.: Incremental Construction of the Generalized Voronoi Diagram, the Generalized Voronoi Graph, and the Hierarchical Generalized Voronoi Graph. In Proceedings of the First CGC Workshop on Computational Geometry. Baltimore (Maryland), 1996, 10 pp. De BERG, M., van KREVELD, M., OVERMARS, M. and SCHWARZKOPF, O: Computational Geometry: Algorithms and Applications. Springer-Verlag, Berlin, 2000 (2nd rev. ed.). ISBN DEHNE, F. and KLEIN, R.: The Big Sweep : On the Power of the Wavefront Approach to Voronoi Diagrams. Algorithmica, Vol. 17, 1997, pp DRYSDALE, S.: Voronoi Diagrams: Applications from Archaeology to Zoology. Technical Report, Regional Geometry Institute, Smith College, FORTUNE, S.: Voronoi Diagrams and Delaunay Triangulations. In GOODMAN, J.E. and O ROURKE, J. (eds.): Handbook of Discrete and Computational Geometry. CRC Press, New York, 1997, pp FORTUNE, S.: Voronoi Diagrams and Delaunay Triangulations. In DU, D.A. and HWANG, F.K. (eds.): Euclidean Geometry and Computers. World Scientific Publishing, Singapore, 1992, pp HWANG, F.K., RICHARDS, D.S. and WINTER, P.: The Steiner Tree Problem. North- Holland, Amsterdam, ISBN X. GUHA, S. and SUZUKI, I.: Proximity Problems for Points on a Rectilinear Plane with Rectangular Obstacles. Algorithmica, Vol. 17, 1997, pp

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