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Register a new userFixed point forms of the parallel symmetric sandpile model
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Added by
Tran Thi Thu Huong
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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This paper presents a generalization of the sandpile model, called the parallel symmetric sandpile model, which inherits the rules of the symmetric sandpile model and implements them in parallel. In this new model, at each step the collapsing of the collapsible columns happens at the same time and one collapsible column is able to collapse on the left or on the right but not both. We prove that the set of forms of fixed points of the symmetric sandpile model is the same as the one of that model using parallel update scheme by constructing explicitly the way (in the parallel update scheme) to reach the form of an arbitrary fixed point of the sequential model.
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We study maps on the set of permutations of n generated by the Renyi-Foata map intertwined with other dihedral symmetries (of a permutation considered as a 0-1 matrix). Iterating these maps leads to dynamical systems that in some cases exhibit interesting orbit structures, e.g., every orbit size being a power of two, and homomesic statistics (ones which have the same average over each orbit). In particular, the number of fixed points (aka 1-cycles) of a permutation appears to be homomesic with respect to three of these maps, even in one case where the orbit structures are far from nice. For the most interesting such Foatic action, we give a heap analysis and recursive structure that allows us to prove the fixed-point homomesy and orbit properties, but two other cases remain conjectural.
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We propose a new self-organizing algorithm for fixed-charge network flow problems based on ghost image (GI) processes as proposed in Glover (1994) and adapted to fixed-charge transportation problems in Glover, Amini and Kochenberger (2005). Our self-organizing GI algorithm iteratively modifies an idealized representation of the problem embodied in a parametric ghost image, enabling all steps to be performed with a primal network flow algorithm operating on the parametric GI. Computational tests are carried out on an extensive set of benchmark problems which includes the previous largest set in the literature, comparing our algorithm to the best methods previously proposed for fixed-charge transportation problems, though our algorithm is not specialized to this class. We also provide comparisons for additional more general fixed-charge network flow problems against Cplex 12.8 to demonstrate that the new self-organizing GI algorithm is effective on large problem instances, finding solutions with statistically equivalent objective values at least 700 times faster. The attractive outcomes produced by the current GI/TS implementation provide a significant advance in our ability to solve fixed-cost network problems efficiently and invites its use for larger instances from a variety of application domains.
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