Some novel locality results for the blob code spanning tree representation

Authors:
Tim Paulden;David K. Smith
Affiliations:
University of Exeter;University of Exeter
Venue:
Proceedings of the 9th annual conference on Genetic and evolutionary computation
Year:
2007

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Abstract

The Blob Code is a bijective tree code that represents each tree on n labelled vertices as a string of n-2 vertex labels. In recent years, several researchers have deployed the Blob Code as a GA representation, and have reported promising results across a range of tree-based optimization problems. In this paper, we exploit a recently discovered linear-time decoding algorithm for the Blob Code to develop some novel locality results, extending previous work by Julstrom. Let Δ be the random variable representing the number of tree edges that are changed by a random single-element string mutation. Under the Blob Code, we demonstrate that pessimal mutations (i.e., mutations for which Δ=n-1) can arise for any n 4. However, as n grows, the probability of perfect mutation P(Δ=1) approaches one, following a power-law relationship, and E(Δ) approaches two. These results show that the locality of the Blob Code is high, but not as high as that of Dandelion-like codes. We also show that the choice of mutation position places restrictions on the range of Δ, and therefore influences the distribution of Δ. In particular, mutating the kth element of a Blob string alters at most n-k tree edges.