Elsevier

Journal of Complexity

Volume 29, Issues 3–4, June–August 2013, Pages 248-262
Journal of Complexity

On the complexity of computing the Hausdorff distance

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Abstract

We study the computational complexity of the Hausdorff distance of two curves on the two-dimensional plane, in the context of the Turing machine-based complexity theory of real functions. It is proved that the Hausdorff distance of any two polynomial-time computable curves is a left-Σ2P real number. Conversely, for any tally set A in Σ2P, there exist two polynomial-time computable curves such that set A is computable in polynomial time relative to the Hausdorff distance of these two curves. More generally, we show that, for any set A in Σ2P, there exist two polynomial-time computable curves such that set A is computable in polynomial time relative to the Hausdorff distances of subcurves of these two curves.

Keywords

Hausdorff distance
Computational complexity
Polynomial-time
Two-dimensional plane
Curves
Turing machine

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