Revisiting Locality in Binary-Integer Representations

TL;DR

This paper analyzes the locality properties of nonredundant bitstring-to-integer representations in genetic algorithms, deriving bounds and examining their impact on algorithm performance, especially comparing standard binary and Gray encodings.

Contribution

It introduces tailored locality metrics for these representations, derives bounds and expected values, and evaluates their predictive power on GEA performance, challenging previous assumptions about Gray codes.

Findings

Standard binary has no worse locality than Gray codes.

Locality does not explain the superior performance of Gray codes.

Weak locality representations aid exploration, strong ones aid exploitation.

Abstract

Mutation and recombination operators play a key role in determining the speed and quality of Genetic and Evolutionary Algorithms (GEAs). Prior work has analyzed the effects of these operators on genotypic variation, often using locality metrics that measure the sensitivity and stability of genotype-phenotype representations to these operators. In this paper, we focus on an important subset of representations, namely nonredundant bitstring-to-integer representations, and analyze them through the lens of Rothlauf's widely used locality metrics. We first define locality metrics equivalent to Rothlauf's that are tailored to our domain: the \textit{point locality} for single-bit mutation and \textit{general locality} for recombination. With these definitions, we derive tight bounds and a closed form expected value for point locality. For general locality we show that it is asymptotically equivalent across all representations and operators. We also recreate three established GEA experiments to understand the predictive power of point locality on GEA performance, focusing on two popular and often juxtaposed representations: standard binary and binary reflected Gray. We show that standard binary has provably no worse locality than any Gray encoding, including binary reflected Gray. We discuss this result in the context of previous studies that found binary reflected Gray to outperform standard binary, and we argue that locality cannot be the explanation for strong performance. Finally, we provide empirical evidence that weak point locality representations can be beneficial to performance in the exploration phase of the GEA, while strong point locality representations are more beneficial in the exploitation phase.