Encoding homotopy of paths in the plane

Abstract

We study the problem of encoding homotopy of simple paths in the plane. We show that the homotopy of a simple path with k edges in the presence of n obstacles can be encoded using O(n log (n + k)) bits. The bound is tight if k = Ω(n1+ε). We present an efficient algorithm for encoding the homotopy of a path. The algorithm can be applied to find homotopic paths among a set of simple paths. We show that the homotopy of a general (not necessary simple) path can be encoded using O(k log n) bits. The bound is tight. The code is based on a homotopic minimum-link path and we present output-sensitive algorithms for computing a path and the code. © Springer-Verlag 2004.