Voronoi Diagram of Polygonal Chains under the Discrete Fr´echet Distance Sergey Bereg1, Kevin Buchin2,MaikeBuchin2, Marina Gavrilova3, and Binhai Zhu4 1 Department of Computer Science, University of Texas at Dallas, Richardson, TX 75083, USA
[email protected] 2 Department of Information and Computing Sciences, Universiteit Utrecht, The Netherlands {buchin,maike}@cs.uu.nl 3 Department of Computer Science, University of Calgary, Calgary, Alberta T2N 1N4, Canada
[email protected] 4 Department of Computer Science, Montana State University, Bozeman, MT 59717-3880, USA
[email protected] Abstract. Polygonal chains are fundamental objects in many applica- tions like pattern recognition and protein structure alignment. A well- known measure to characterize the similarity of two polygonal chains is the (continuous/discrete) Fr´echet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in d-dimension under the discrete Fr´echet distance. Given a set C of n polygonal chains in d-dimension, each with at most k vertices, we prove fundamental proper- ties of such a Voronoi diagram VDF (C). Our main results are summarized as follows. dk+ – The combinatorial complexity of VDF (C)isatmostO(n ). dk – The combinatorial complexity of VDF (C)isatleastΩ(n )fordi- mension d =1, 2; and Ω(nd(k−1)+2) for dimension d>2. 1 Introduction The Fr´echet distance was first defined by Maurice Fr´echet in 1906 [8]. While known as a famous distance measure in the field of mathematics (more specifi- cally, abstract spaces), it was first applied in measuring the similarity of polygo- nal curves by Alt and Godau in 1992 [2].