Real Analysis Exchange

Computational Complexity of Fractal Sets

Kamo Hiroyasu, Takeuti Izumi, and Kawamura Kiko

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Abstract

In studies on fractal geometry, it is important to determine whether the classification by means of computational complexity is independent of the classification by means of fractal dimension. In this paper, we show that each self-similar set defined by polynomial time computable functions is polynomial time computable, if the self-similar set satisfies a polynomial time open set condition. This fact provides us examples of sets whose computational complexity are polynomial time computable, and which have non integer Hausdorff dimension. We also construct a set with computational complexity NP-complete and with an integer Hausdorff dimension. These two examples establish the independence of computational complexity and Hausdorff dimension.

Article information

Source
Real Anal. Exchange, Volume 26, Number 2 (2000), 773-794.

Dates
First available in Project Euclid: 27 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.rae/1214571367

Mathematical Reviews number (MathSciNet)
MR1844393

Zentralblatt MATH identifier
1009.68059

Subjects
Primary: 03F60: Constructive and recursive analysis [See also 03B30, 03D45, 03D78, 26E40, 46S30, 47S30] 28A80: Fractals [See also 37Fxx] 03D15: Complexity of computation (including implicit computational complexity) [See also 68Q15, 68Q17] 68Q25: Analysis of algorithms and problem complexity [See also 68W40]

Keywords
Time complexity Fractals Self-similar sets

Citation

Hiroyasu, Kamo; Kiko, Kawamura; Izumi, Takeuti. Computational Complexity of Fractal Sets. Real Anal. Exchange 26 (2000), no. 2, 773--794. https://projecteuclid.org/euclid.rae/1214571367


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