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 and position. 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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References
Baker, B.S., Coffman Jr., E.G., Rivest, R.L.: Orthogonal packings in two dimensions. SIAM Journal on Computing 9(4), 846–855 (1980)
Kenyon, C., Rémila, E.: Approximate strip packing. In: Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci., pp. 31–36 (1996)
Csirik, J., Woeginger, G.J.: Shelf algorithms for on-line strip packing. Information Processing Letters 63, 171–175 (1997)
Seiden: On the online bin packing problem. JACM: Journal of the ACM 49 (2002)
van Vliet, A.: An improved lower bound for online bin packing algorithms. Information Processing Letters 43, 277–284 (1992)
Coppersmith, D., Raghavan, P.: Multidimensional online bin packing: Algorithms and worst case analysis. orl 8, 17–20 (1989)
Epstein, L., van Stee, R.: Optimal online bounded space multidimensional packing. In: Proc. 15th Symp. on Discrete Algorithms (SODA), pp. 214–223. ACM/SIAM (2004)
Breukelaar, R., Demaine, E.D., Hohenberger, S., Hoogeboom, H.J., Kosters, W.A., Liben-Nowell, D.: Tetris is hard, even to approximate. Internat. J. Comput. Geom. Appl. 14, 41–68 (2004)
Azar, Y., Epstein, L.: On two dimensional packing. J. Algor. 25, 290–310 (1997)
Coffman Jr., E., Downey, P.J., Winkler, P.: Packing rectangles in a strip. Acta Inform. 38, 673–693 (2002)
de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications. Springer, Berlin (2000)
Fekete, S.P., Schepers, J.: New classes of lower bounds for the bin packing problem. Math. Progr. 91, 11–31 (2001)
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Fekete, S.P., Kamphans, T., Schweer, N. (2009). Online Square Packing. In: Dehne, F., Gavrilova, M., Sack, JR., Tóth , C.D. (eds) Algorithms and Data Structures. WADS 2009. Lecture Notes in Computer Science, vol 5664. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03367-4_27
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DOI: https://doi.org/10.1007/978-3-642-03367-4_27
Publisher Name: Springer, Berlin, Heidelberg
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