Abstract

. Motivated by problems in computational geometry, this paper investigates the complexity of finding a fixed-point of the composition of two or three continuous functions that are defined piecewise. It shows that certain cases require nested binary search taking \Theta(log 2 n) time. Other cases can be solved in logarithmic time by using a pruneand -search technique that may make tentative discards and later revoke or certify them. This work finds application in optimal subroutines that compute approximations to convex polygons, dense packings, and Voronoi vertices for Euclidean and polygonal distance functions. 1 Introduction Several fundamental problems in computational geometry can be expressed as a search for special k-tuple with one element drawn from each of k lists. Examples with k equal to 2 or 3 include common tangents to two convex polygons, special chords in a polygon, and Voronoi vertices (circles or polygons tangent to three given points or polygons). Many of these p...

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