2013
DOI: 10.1016/j.camwa.2012.09.008
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Idempotent and tropical mathematics; complexity of algorithms and interval analysis

Abstract: A very brief introduction to tropical and idempotent mathematics is presented. Tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the Planck constant tends to zero taking imaginary values. In the framework of idempotent mathematics usually constructions and algorithms are more simple with respect to their traditional analogs. We especially examine algorithms of tropical/idempotent mathematics generated by a collection of basic semiring (or semifield) operation… Show more

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Cited by 5 publications
(4 citation statements)
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References 53 publications
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“…In [19] is pointed out that many algorithms for solving the stationary matrix Bellman equations have complexity 3 ( ) O n . However, NP-hard problems exist.…”
Section: Resultsmentioning
confidence: 99%
“…In [19] is pointed out that many algorithms for solving the stationary matrix Bellman equations have complexity 3 ( ) O n . However, NP-hard problems exist.…”
Section: Resultsmentioning
confidence: 99%
“…In [43], this approach was applied to modeling cell cycle, which is a central process in cancer biology. More conceptually, introducing qualitative and extremely simplified kinetic system descriptions (such as logical equations) might be seen as "dequantization" of continuous kinetics, by analogy with how this procedure justifies the application of tropical algebras in real-life problems [44]. One can speculate that tropical algebras should be well suited for qualitative solutions of complex models in biology because the robust cell fate decisions are made based on comparing the orders of magnitudes of biomolecule concentrations rather than on very precise values [45].…”
Section: Probabilistic and Continuous Flavors Of Logical Modelingmentioning
confidence: 99%
“…Jerrum and Snir [15] have shown that their exponential lower bound also holds in the tropical semiring (R, +, min) (see, e.g., [19,Section 8.5] and references therein). Since our algorithms extend straightforwardly into the tropical setting, we conclude that the circuit complexity of the minimum cost arborescence problem drops from exponential to polynomial as one passes from the tropical semiring to the tropical semifield (R, +, −, min).…”
mentioning
confidence: 99%