Schema disruption in tree-structured chromosomes

Abstract

We study if and when the inequality dp(H) ≤ rel∆(H) holds for schemas H in chromosomes that are structured as trees. The disruption probability dp(H) is the probability that a random cut of a tree limb will separate two fixed nodes of H. The relative diameter rel∆(H) is the ratio (max distance between two fixed nodes in H) / (max distance between two tree nodes), and measures how close together are the fixed nodes of H. Inequality dp(H) ≤ rel∆(H) is of significance in proving Schema Theorems for non-linear chromosomes, and so bears upon the success we can expect from genetic algorithms. For linear chromosomes, dp(H) = rel∆(H). Our results include the following. There is no constant c such that dp(H) ≤ c · rel∆(H) holds for arbitrary schemas and trees. This is illustrated in trees with eccentric, stringy shapes. Matters improve for dense, ball-like trees, explained herein. Inequality dp(H) ≤ rel∆(H) always holds in such trees, except for certain atypically large schemas. Thus, the more compact are our tree-structured chromosomes, the better we can expect our genetic algorithms to work.

A tree-based genetic algorithm for building rectilinear Steiner arborescences

“…For traffic flow problems in a city, to structure a solution as a grid or graph may be most natural. Theoretical issues with these ideas have appeared in the literature; to cite but one author, there is Greene [5], [6].…”