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Online Square Packing with Gravity

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Abstract

We analyze the problem of packing squares in an online fashion: Given a semi-infinite strip of width 1 and an unknown sequence of squares of side length in [0,1] that arrive from above, one at a time. The objective is to pack these items as they arrive, minimizing the resulting height. Just like in the classical game of Tetris, each square must be moved along a collision-free path to its final destination. In addition, we account for gravity in both motion (squares must never move up) and position (any final destination must be supported from below). A similar problem has been considered before; the best previous result is by Azar and Epstein, who gave a 4-competitive algorithm in a setting without gravity (i.e., with the possibility of letting squares “hang in the air”) based on ideas of shelf packing: Squares are assigned to different horizontal levels, allowing an analysis that is reminiscent of some bin-packing arguments. We apply a geometric analysis to establish a competitive factor of 3.5 for the bottom-left heuristic and present a \(\frac{34}{13} \approx 2.6154\)-competitive algorithm.

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Notes

  1. That is, we take the square with the topmost bottom line. If there is more than one, we take the leftmost of these squares.

  2. The charge to the bottom of \(\tilde{A}_{1}\) can be reduced to \(\frac{3}{4}\) by considering the larger one of the rectangles, R 1 and the one induced by Q, Q′, and P, as well as the triangle below the larger rectangle formed by \(D_{l}^{h}\) and \(D_{r}^{h}\). However, this does not lead to a better competitive ratio, because these costs are already dominated by the cost for holes of Type T.

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Authors and Affiliations

  1. Department of Computer Science, Algorithms Group, Braunschweig University of Technology, Mühlenpfordtstrasse 23, 38106, Braunschweig, Germany

    Sándor P. Fekete, Tom Kamphans & Nils Schweer

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  1. Sándor P. Fekete
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  2. Tom Kamphans
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  3. Nils Schweer
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Correspondence to Sándor P. Fekete.

Additional information

Tom Kamhans was supported by DFG grant FE 407/8-3, project “ReCoNodes”. A preliminary extended abstract summarizing the results of this paper appeared in [15].

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Fekete, S.P., Kamphans, T. & Schweer, N. Online Square Packing with Gravity. Algorithmica 68, 1019–1044 (2014). https://doi.org/10.1007/s00453-012-9713-8

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