The paper studies problems of reduction (decomposition) of OLAP-hypercube multidimensional data models. When decomposing large hyper-cubes of multidimensional data into sub-cube components the goal is to increase the computational performance of analytical OLAP systems, which is related to decreasing computational complexity of reduction methods for solving OLAP-data analysis problems with respect to the computational complexity of non-reduction methods, applied to data directly all over the hypercube. The paper formalizes the concepts of reduction and non-reduction methods and gives a definition of the upper bound for the change in the computational complexity of reduction methods in the decomposition of the problem of analyzing multidimensional OLAP-data in comparison with non-reduction methods in the class of exponential degree of computational complexity.The exact values of the upper bound for changing computational complexity are obtained for the hypercube decomposition into two sub-cubes on sets consisting of an even and an odd number of sub-cube structures, and its main properties are given, which are used to determine the decomposition efficiency. A formula for the efficiency of decomposition into two sub-cube structures for reduction of OLAP data analysis problems is obtained, and it is shown that with an increase in the dimension “n” of the lattice specifying the number of sub-cubes in the hypercube data structure, the efficiency of such a decomposition obeys an exponential law with an exponent “n/2”, regardless of the parity “n”. The examples show the possibility to use the values (found) of the upper bound for the change in computational complexity to establish the effectiveness criteria for reduction methods and the expediency of decomposition in specific cases.The paper results can be used in processing and analysis of information arrays of hypercube structures of analytical OLAP systems belonging to the Big-Data or super-large computer systems of multidimensional data.
The paper investigates the problems of reduction (decomposition) of multidimensional data models in the form of hypercube OLAP structures. OLAP data processing does not allow changes in the dimension of space. With the increase in data volumes, the productivity of computing cubic structures decreases. Methods for reducing large data cubes to sub-cubes with smaller volumes can solve the problem of reducing computing performance.The reduction problems are considered for cases when the cube lattice has already determined criteria aggregation, and the cube decomposition into smaller cubes is needed to reduce the computation time of the full lattice when dynamically changing data in the cube.The objective of the paper is to find conditions for reducing the computational complexity of solving data analysis problems by reduction methods, to obtain exact quantitative boundaries for reducing the complexity of decomposition methods from the class of polynomial degrees of complexity, to establish the nature of the dependence of computational performance on the structural properties of a hypercube, and to determine the quantitative boundaries of computational performance for solving decomposition problems of data aggregation .The study of the computational complexity of decomposition methods for the analysis of multidimensional hyper-cubes of polynomial-logarithmic and polynomial degrees of complexity is carried out. An exact upper limit is found for reducing the complexity of decomposition methods for analyzing the initial OLAP - data hypercube with respect to non-decomposition ones and based on them criteria are proved for the effective application of reduction methods for analyzing hypercube structures in comparison with traditional non-reduction methods.Examples of decomposition methods of cube structures are presented, both reducing and increasing computational complexity in comparison with calculations using the full model.The results obtained can be used in processing and analysis of information arrays of hypercube structures of analytical OLAP-systems belonging to the BigData class, or ultra-large computer multidimensional data systems.
The article investigates the problems of reduction (decomposition) of multidimensional data models in terms of hypercube OLAP-structures. Describes the case when a data structure is defined by the array that slices and dices the hypercube into the odd number of subcubes, and this set of subcube structures becomes decomposed. Defines an exact upper bound for increasing a computational performance of methods to analyze OLAP-data on subcubes, which determines the decomposition approach efficiency in comparison with the OLAP-data analysis on a complete unreduced hypercube. A compared efficiency of the hypercube decomposition into two subcubes on the sets consisting of the even and odd number of subcube structures has shown that with considerable data partitioning for methods of a polynomial complexity degree the decomposition efficiency essentially is independent on this factor and rises with increasing complexity degree of methods applied.When using the mathematical methods to study decomposition (reduction) of large hyper-cubes of multidimensional data of analytical OLAP systems into subcube components, there is a need to find conditions for minimising the computational complexity of methods to solve the problems of the OLAP hyper-cube analysis during data decomposition in comparison with using these methods for analyzing large amounts of information that is accumulated directly in the hyper-cubes of multidimensional OLAP-data to establish the criteria for decreasing or increasing computational performance when applying methods on the subcube components (reduction methods) as compared to applying these methods on a hypercube (non-reduction or traditional methods), depending on one or another degree of complexity of complex methods.The article provides an accurate quantitative estimate of decreasing computational complexity of reduction methods for analyzing OLAP-cubes as compared to the non-reduction methods in the case when said methods have the polynomial complexity and the original hypercube array of data comprises the odd number of subcubes.
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