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## On one problem of Koffman-Shor connected with strip packing.

#### Abstract

In 1993 Coffman and Shor proposed on-line strip packing algorithm with expected unpacked area (waste area) of order O(n^{2/3}) in a standard probabilistic model, where n is the number of rectangles. In the standard probabilistic model the width and height of each rectangle are independent random variables with uniform distribution in [0,1]. It is wellknown that optimal packing has expected waste of order O(n^{1/2}) and there exists off-line polynomial algorithm that provides this bound.. During more than 10 years the question concerning the possibility to obtain similar upper bound in the class of on-line strip packing algorithms was open. It was also known that in the class of popular so-called shelf algorithms the bound O(n^{2/3}) cannot be improved. In this paper we propose new strip packing algorithm which differs from all known on-line strip packing algorithms. We analyze our strip packing algorithm experimentally and show that the upper bound of expected unpacked area is of order O(n^{1/2}) which is optimal up to constant factor. Our experimnets show that this constant factor is close to 1.5-1.6. Our experiments were done with up to 2000000000 random rectangles. The only restriction is the following: we must know the number n of rectangles in advance. In a popular terminology concerning on-line algorithms this means that our algorithm is closed-end on-line algorithm. The paper proposes a new online strip packing algorithm that provides a quality of packing significantly higher than well-known algorithm of Koffman-Shor.

#### Keywords

#### Edition

Proceedings of the Institute for System Programming, vol. 22, 2012, pp. 456-462.

ISSN 2220-6426 (Online), ISSN 2079-8156 (Print).

DOI: 10.15514/ISPRAS-2012-22-24

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