Efficient Algorithms for Approximating Polygonal Chains

An error criterion for measuring the quality of an approximation is defined and a simple randomized algorithm is presented, with expected running time O(n4/3 + δ) , to solve the problem of approximating a polygonal chain C by another Polygonal Chain C' whose vertices are constrained to be a subset of the set of vertices of C .

Abstract

We consider the problem of approximating a 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' . Based on a framework introduced by Imai and Iri [25], we define an error criterion for measuring the quality of an approximation. We consider two problems. (1) Given a polygonal chain C and a parameter ɛ \geq 0 , compute an approximation of C , among all approximations whose error is at most ɛ , that has the smallest number of vertices. We present an O(n 4/3 + δ ) -time algorithm to solve this problem, for any δ > 0; the constant of proportionality in the running time depends on δ . (2) Given a polygonal chain C and an integer k , compute an approximation of C with at most k vertices whose error is the smallest among all approximations with at most k vertices. We present a simple randomized algorithm, with expected running time O(n 4/3 + δ ) , to solve this problem.