АНАЛИЗ ВОЗМОЖНОСТЕЙ СОВРЕМЕННЫХ ВЫЧИСЛИТЕЛЬНЫХ ТЕХНОЛОГИЙ ДЛЯ РАСЧЕТА ТОЧНЫХ ПРИБЛИЖЕНИЙ РАСПРЕДЕЛЕНИЙ ВЕРОЯТНОСТЕЙ ЗНАЧЕНИЙ СТАТИСТИК

Abstract

In the paper we consider the solution of a computationally expensive problem such as calculationof statistics probability distribution with the help of modern computer technologies. To reducecomputational complexity and to provide a sufficient level of criteria efficiency not less thanthe specified threshold, we suggest to use Δ-exact approximations. To calculate exact approximations,we use the method of second order, based on solution of a system of linear equations. Owingto this method, it is possible to calculate exact approximations for the maximum values of sampleparameters for available computational resource. The most laborious part of the method of secondorder is the procedure of sequential detection of the vectors of possible solutions and test if thevectors belong to the set of solutions. The system solution set membership test for the vectors ofpossible solutions is data independent, so the algorithm can be data-parallelized. We give the algorithmcomplexity equation for calculation of exact approximations of statistics probability distributions.Using this equation, we calculated the complexity of modern practical problems for thesamples with the parameters (N, n) of the alphabet power and the sample size: (256,1280),(128,640), (128, 320), and (192,3200) for the accuracy of calculations =10-5. The computationalcomplexity is 9.68·1022-1.60·1052 operations, and its average value is about 4.55·1025 operations,the number of tested vectors is 6.50·1023-1.39·1050, and the number of solutions is 4.67·1012-5.60·1025, respectively. The total solution time for clock-round duration of calculations cannotexceed 30 days or 2.592·106 sec. For the obtained complexity evaluation, we analysed abilities ofmodern cluster computer systems based on general-purpose processors, graphic accelerators, andFPGA-based reconfigurable computer systems. For each technology, we determined the number ofcomputational nodes needed for calculation of exact approximations with the specified parametersduring the specified time. We proved that it is impossible to obtain a solution for the required parametersof exact approximations of statistics probability with the help of the reviewed moderncomputer technologies. In conclusion, we claim that it is necessary to analyse the abilities of advancedcomputer technologies based of quantum and photonic computers, and also hybrid computersystems for calculation of exact approximations of statistics probability distributions withthe specified parameters during reasonable time.