Fast Computation of the Modality of Polygons

Abstract The modality of a vertex of a polygon is the number of minima or maxima in the distance function from the vertex under consideration to each of the other vertices taken in order around the polygon. Modality was defined by Avis, Toussaint, and Bhattacharya ( Comput. Math Appl. , 8 (1982), 153–156) and has been further studied by Toussaint and Bhattacharya (Technical Report, No. SOCS-81.3, School of Computer Science, McGill University, January 1981). These authors have defined modality for polygons, both convex and nonconvex, but have not given an algorithm to compute the modality of a polygon, other than the obvious algorithm derived from the definition, which is quadratic in the number of vertices of the polygon. Our paper gives a modality-determination algorithm for convex polygons whose running time is linear in the sum of the number of vertices and the total modality of the polygon. As a special case, we can test if a convex polygon is unimodal (a term introduced by Avis et al. ) in time O ( n ) for a polygon with n vertices. This latter result is shown to be optimal to within a constant factor. We then extend our technique to nonconvex polygons and derive a modality-determination algorithm which is better than the obvious algorithm, but probably not optimal. We conclude by posing some open problems and indicating a connection between modality determination and a range query problem which has received some attention in the Computational Geometry literature.