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From Cleveland (1993): Bin packing is a computer problem that has challenged mathematicians working on the foundations of theoretical computer science. Suppose a large number of files of different sizes are to be written on floppies. No file can be split between two floppies, but we want to waste as little space as possible. Unfortunately, any algorithm that guarantees the minimum possible empty space takes an enormous amount of computation time unless the number of files is quite small. Fortunately, there are heuristic algorithms that run fast and do an extremely good job of packing, even though they do not guarantee the minimum of empty space. One is first fit decreasing. The files are packed from largest to smallest. For each file, the first floppy is tried; if it has sufficient empty space, the file is written, and if not, the second floppy is tried. If the second file has sufficient space, the file is written and if not, the third floppy is tried. The algorithm proceeds in this way until a floppy with space, possibly a completely empty one, is found. To supplement the theory of bin packing with empirical results, mathematicians and computer scientists have run simulations, computer experiments in which bins are packed with randomly generated weights. For one data set from one experiment, the weights were randomly selected from the interval 0 to 0.8 and packed in bins of size one. The number of weights, n, for each simulation run took one of 11 values: 125,250,500, and so forth by factors of 2 up to 128000. There were 25 runs for each of the 11 different numbers of weights, which makes 25 x 11 = 275 runs in all. For each run of the experiment, the performance of the algorithm was measured by the total amount of empty space in the bins that were used. We will study log empty space to enhance our understanding of multiplicative effects.

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A data frame with 275 rows and 2 variables:

- empty.space
total amount of empty space in the bins that were used

- number.weights
number of weights

Cleveland W. S. (1993). “Visualizing Data”. Hobart Press.

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