Divisive-agglomerative algorithm and complexity of automatic classification problems

An algorithm of solution of the Automatic Classification (AC for brevity) problem is set forth in the paper. In the AC problem, it is required to find one or several partitions, starting with the given pattern matrix or dissimilarity ∕ similarity matrix. The three-level scheme of the algorithm is suggested. At the internal level, the frequency minimax dichotomy algorithm is described. At the intermediate level, this algorithm is repeatedly used at alternations of divisive and agglomerative stages, which causes the construction of a classifications family. At the external level, several runs of the algorithm of the intermediate level are completed; thereafter among all the constructed classifications families the set of all the different classifications is selected. The latest set is taken as a set of all the solutions of the given AC problem. In many cases, this set of solutions can be significantly contracted (sometimes to one classification). The ratio of cardinality of the set of solutions to cardinality of the set of all the classifications found at the external level is taken as a measure of complexity of the initial AC problem. For classifications of parliament members according to their vote results, the general notion of complexity is interpreted as consistence or rationality of this parliament policy. For “tossing” deputies or ∕ and whole fractions the corresponding clusters become poorly distinguished and partially perplexing that results in relatively high value of complexity of their classifications. By contrast, under consistent policy, deputy’s clusters are clearly distinguished and the complexity level is low enough (i.e. in a given parliament the level of consistency, accordance, rationality is high). The mentioned reasoning was applied to analysis of activity of 2-nd, 3-rd and 4-th RF Duma (Russian parliament,19962007). The classifications based upon one-month votes were constructed for every month. Calculation of an average complexity for every Duma have demonstrated its almost three times decrease in the 3-rd Duma as compared to the 2-nd Duma as well as its subsequent essential increase in the 4-th Duma as compared to the 3-nd Duma. The decrease of the suggested index was the most pronounced in 2002 in the wake of the “political peculiar point” – creation of the party “United Russia” 01.12.2001. In 2002 the complexity was equal to 0.096 that was significantly less when in any other year at the consider 12-years period. The introduced notions allow suggesting new meaningful interpretations of activity of various election bodies, including different country parliaments, international organizations and board of large corporations.