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by Jean Mairesse, Laurent Vuillon
Comput. Sci
http://www.liafa.jussieu.fr/~vuillon/opti2.ps
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Abstract:
In a heap model, solid blocks, or pieces, pile up according to the Tetris game mechanism. An optimal sequence is an infinite sequence of pieces minimizing the asymptotic growth rate of the heap. In a heap model with two pieces, we prove that there always exists an optimal sequence which is either periodic or Sturmian. We completely characterize the cases where the optimal is periodic and the ones where it is Sturmian. The proof is constructive, providing an explicit optimal sequence. We also consider the model where the successive pieces are choosen at random, independently and with some given probabilities. We study the expected growth rate of the heap. For a model with two pieces, the rate is either computed explicitly or given as an infinite series. We show an application for a system of two processes sharing a resource, and we prove that a greedy schedule is not always optimal.
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