Yield-Aware Analog IC Design and Optimization in Nanometer-scale Technologies by António Manuel Lourenço Canelas & Jorge Manuel Correia Guilherme & Nuno Cavaco Gomes Horta

Author:António Manuel Lourenço Canelas & Jorge Manuel Correia Guilherme & Nuno Cavaco Gomes Horta
Language: eng
Format: epub
ISBN: 9783030415365
Publisher: Springer International Publishing

Finding the partitions that result in the optimum clusters definition may seem an easy problem to solve since the number of data points is finite. However, in clustering algorithms, like k-means, it is possible to demonstrate that this is an NP-hard problem [6] and exhaustive search for the optimum solutions is not possible, since the number of non-empty partitions for n data points into K disjoint groups is a Stirling number of the second kind [7], which is given by (4.2).

(4.2)

where

According to [8], using (4.2) for n = 25 data points and a number of clusters K = 4, there are approximately 4.69 × 1013 different disjoint partitions. The impressive number of possible partitions is increased if the problem must also explore all possible number of clusters combinations, in order to find the optimum number of clusters to group n data points (4.3):

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