Dynamic Voronoi Diagrams in Motion Planning

Given a set of n points in the Euclidean plane each of which is continuously moving along a given trajectory. At each instant of time, these points define a Voronoi diagram which also changes continuously, except for certain critical instances so-called topological events. In [/to 90], an efficient method is presented of maintaining the Voronoi diagram over time. Recently Guibas, Mitchell and Roos [GuMiKo 91] improved the trivial quartic upper bound on the number of topological events by almost a linear factor to the nearly cubic upper bound of O(n ~ A,(n)) topological events, where A~(n) is the maximum length of an (n, 8)-Davenport-Schinzel sequence and s is a constant depending on the motion of the sites. Each topological event uses only O(log n) time (which is worst-case optimal). Now in this work, we present a new algorithm for planning the motion of a disc in a dynamic scene of moving sites which is based on the corresponding sequence of Voronoi diagrams. Thereby we make use of the well-known fact, that locally the Voronoi edges are the safest paths in the dynamic scene. We present a quite simple approach combining local and global strategies for planning a feasible path through the dynamic scene. One basic advantage of our algorithm is that only the topological structure of the dynamic Voronoi diagram is required for the computation. Additionally, our goal oriented approach provides that we can maintain an existing feasible path over time. This guarantees that we reach the goal if there is a feasible path in the dynamic scene at all. Finally our approach can easily be extended to general convex objects. 1 I n t r o d u c t i o n The Voronoi diagram is one of the most fundamental data structures in computational geometry. In its most general form, the Voronoi diagram VD(S) of a set S of n objects in a space • is a subdivision of this space into maximal regions, so that all points within a given region have the same nearest neighbor in S with regard to a general distance measure d. Shamos and Hoey [ShHo 75] introduced the Voronoi diagram for a finite set of points in the Euclidean plane IE 2 into the field of computational geometry, providing the first efficient algorithm for its computation. Since then, Voronoi diagrams in many variations have appeared throughout the algorithmic literature; see, for example, [ChEd 87], [Ya 87], [Ro 89] and [Au 90]. One problem of recent interest has been of allowing the set of objects S to vary continuously over t ime along given trajectories. This "dynamic" version has been studied by [ImSuIm 89], [Au~nro 90] and [Ro 90]. ~ work was supported by the Deutsche Forschungsgemeinschaft (DFG) under contract (No 88/10 1) and (No 88/10 2).