A Fuzzy Relation Approach to Single Linkage

Abstract

Single linkage is equivalent to sub-dominant ultrametric. Many algorithms are available for constructing these two objects. But all of then, except one which was proposed by Gondran, are very tedious because of the lack of algebraic structure. Gondran used a special algebraic system as theoretical bases. But it seems rather artificial. In this paper, we propose a more formal approach based on fuzzy relation. The main result presented here is the equivalence between sub-dominant ultrametric and the min-max transitive closure of a symmetric and anti reflexive fuzzy relation. This property enables us to construct an easy and efficient algorithm. At the end of this paper we will find its relationship with Gondran's approach.

Keywords

• single linkage
• sub-dominant ultametric and Gondran’s algebraic system