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Algorithmic Folding Complexity

par Cardinal, Jean ;Demaine, Erik D. ;Demaine, Martin L.;Imahori, Shinji S.;Ito, Tsuyoshi;Kiyomi, Masashi;Langerman, Stefan ;Uehara, Ryuhei;Uno, Takeaki
Référence Graphs and combinatorics, 27, 3, page (341-351)
Publication Publié, 2011-05

Article révisé par les pairs

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Résumé :

How do we most quickly fold a paper strip (modeled as a line) to obtain a desired mountain-valley pattern of equidistant creases (viewed as a binary string)? Define the folding complexity of a mountain-valley string as the minimum number of simple folds required to construct it. We first show that the folding complexity of a length-n uniform string (all mountains or all valleys), and hence of a length-n pleat (alternating mountain/valley), is O(lg2n). We also show that a lower bound of the complexity of the problems is Ω(lg2n/lg lg n). Next we show that almost all mountain-valley patterns require Ω(n/lg n) folds, which means that the uniform and pleat foldings are relatively easy problems. We also give a general algorithm for folding an arbitrary sequence of length n in O(n/lg n) folds, meeting the lower bound up to a constant factor. © 2011 Springer.