Approximation Algorithms for Inscribing or Circumscribing an Axially Symmetric Polygon to a Convex Polygon

Abstract

Given a convex polygon P with n vertices, we present algorithms to determine approximations of the largest axially symmetric convex polygon S contained in P, and the smallest such polygon S′ that contains P. More precisely, for any ε> 0, we can find an axially symmetric convex polygon Q ⊂ P with area |Q|>(1–ε)|S| in time O(n+1/ε 3/2), and we can find an axially symmetric convex polygon Q′ containing P with area |Q′|<(1+ε)|S′| in time O(n+(1/ε 2)log(1/ε)). If the vertices of P are given in a sorted array, we can obtain the same results in time O((1/ε √ )logn+1/ε 3/2 ) O((1/ε)log⁡n+1/ε3/2) and O((1/ε)log n+(1/ε 2)log(1/ε)) respectively.