Computationally efficient sup-t transitive closure for sparse fuzzy binary relations

The property of transitivity is one of the most important for fuzzy binary relations, especially in the cases when they are used for the representation of real-life similarity or ordering information. As far as the algorithmic part of the actual calculation of the transitive closure of such relations is concerned, works in the literature mainly focus on crisp symmetric relations, paying little attention to the case of general fuzzy binary relations. Most works that deal with the algorithmic part of the transitive closure of fuzzy relations focus only on the case of max–min transitivity, disregarding other types of transitivity. In this paper, after formalizing the notion of sparseness and providing a representation model for sparse relations that displays both computational and storage merits, we propose an algorithm for the incremental update of fuzzy sup-t transitive relations. The incremental transitive update (ITU) algorithm achieves the re-establishment of transitivity when an already transitive relation is only locally disturbed. Based on this algorithm, we propose an extension to handle the sup-t transitive closure of any fuzzy binary relation, through a novel incremental transitive closure (ITC) algorithm. The ITU and ITC algorithms can be applied on any fuzzy binary relation and t-norm; properties such as reflexivity, symmetricity and idempotency are not a requirement. Under the specified assumptions for the average sparse relation, both of the proposed algorithms have considerably smaller computational complexity than the conventional approach; this is established both theoretically and verified via appropriate computing experiments.