Approximating monotone polygonal curves using the uniform metric

Abstract

We consider the problem of approximating a monotone polygonal chain C by another polygonal chain C′ whose vertices are constrained to be a subset of the set of vertices of C. The goal is to minimize the number of vertices needed in the approximation C′. We use the uniform metric as the error criterion for the approximation. We consider two problems. (1) Given ε ≥ 0, find an approximation, among all approximations whose error is at most ε, that has the smallest number of vertices. We give an O(n4/3+δ)-time algorithm to solve this problem: throughout this paper, δ > 0 is an arbitrarily small constant, and the constant of proportionality hidden in the big-Oh notation depends on δ. (2) Given an integer k, find an approximation with at most k vertices whose error is the smallest among all approximations with at most k vertices. We give a simple randomized algorithm, with expected running time O(n4/3+δ), to solve this problem. We also present a deterministic version of the algorithm that has the same asymptotic time complexity. Algorithms with close to linear running times are known for the variants of this problem which do not restrict the vertices of the approximation C′ to be a subset of the set of vertices of C. Ours is the first non-trivial instance of a subquadratic-time algorithm for the restricted case.