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Several algorithms of different domains in distributed systems are designed over the principle of the Happened-Before Relation (HBR). One common aspect among them is that they intend to be efficient in their implementation by identifying and ensurin g the necessary and sufficient dependency constraints. In thi s pursuit, some previous works talk about the use of a transitive reduction of the causality. However, none of these works formally prove in a broad manner that such transitive reduction is the minimal expression of the HBR. In this paper, a formal study of the minimal binary relation (transitive reduction) of the HBR is presented, which is called the Immediate Dependency Relation(IDR). The study shows that since the transitive closure of the HBR is antisymmetric and finite, it implies that the IDR is unique. This is important because it means that all of the works that deal with a minimal expression of the HBR discuss the same minimal binary relation. In addition, an extension to the IDR to identify causal immediate dependencies only among a subset of relevant events is presented. Finally, as case of study, the extension of the IDR is applied to the causal delivery of messages.

The Minimal Dependency Relation for Causal Event Ordering in Distributed Computing