On the performance effects of unbiased module encapsulation

A recent theoretical investigation of modular representations shows that certain modularizations can introduce a distance bias into a landscape. This was a static analysis, and empirical investigations were used to connect formal results to performance. Here we replace this experimentation with an introductory runtime analysis of performance. We study a base-line, unbiased modularization that makes use of a complete module set (CMS), with special focus on strings that grow logarithmically with the problem size. We learn that even unbiased modularizations can have profound effects on problem performance. Our (1+1) CMS-EA optimizes a generalized OneMax problem in Ω(n²) time, provably worse than a (1+1) EA. More generally, our (1+1) CMS-EA optimizes a particular class of concatenated functions in O(2^lm k n) time, where l_m is the length of module strings and k is the number of module positions, when the modularization is aligned with the problem separability. We compare our results to known results for traditional EAs, and develop new intuition about modular encapsulation. We observe that search in the CMS-EA is essentially conducted at two levels (intra- and extra-module) and use this observation to construct a module trap, requiring super-polynomial time for our CMS-EA and O(n ln n) for the analogous EA.