Curves that must be retraced

Authors:
Xiaoyang Gu;Jack H. Lutz;Elvira Mayordomo
Affiliations:
LinkedIn Corporation, 2029 Stierlin Court, Mountain View, CA 94043, USA;Department of Computer Science, Iowa State University, Ames, IA 50011, USA;Departamento de Informática e Ingeniería de Sistemas, Universidad de Zaragoza, 50018 Zaragoza, Spain
Venue:
Information and Computation
Year:
2011

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Abstract

We exhibit a polynomial time computable plane curve @C that has finite length, does not intersect itself, and is smooth except at one endpoint, but has the following property. For every computable parametrization f of @C and every positive integer m, there is some positive-length subcurve of @C that f retraces at least m times. In contrast, every computable curve of finite length that does not intersect itself has a constant-speed (hence non-retracing) parametrization that is computable relative to the halting problem.