Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 2, pp. 537-549 (2002)

On-line Algorithms for the $q$-adic Covering of the Unit Interval and for Covering a Cube by Cubes

Marek Lassak

Instytut Matematyki i Fizyki ATR, Kaliskiego 7, 85-796 Bydgoszcz, Poland, e-mail:

Abstract: We present efficient algorithms for the on-line $q$-adic covering of the unit interval by sequences of segments. The basic method guarantees covering provided the total length of segments is at least $1+ 2\cdot {1\over q} - {1\over q^3}$. Other algorithms improve this estimate for $q\geq 6$. The unit $d$-dimensional cube can be on-line covered by an arbitrary sequence of cubes whose total volume is at least $2^d+{5\over 3}+{5\over 3}\cdot 2^{-d}$.

Classification (MSC2000): 52C17

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