The Minimal Dependency Relation for Causal Event Ordering in Distributed Computing

Abstract

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 ensuring the necessary and sufficient dependency constraints. In this 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.

An Efficient Causal Group Communication Protocol for Free Scale Peer-to-Peer Networks

“…We notice that in our protocol, as for the minimal causal algorithm in [14], the likelihood that the worst case will occur approaches zero as the number of participants in the group grows. This is because the likelihood that k concurrent messages occur decreases inversely proportional to the size of the communication group.…”

“…The Immediate Dependency Relation [14] is the propagation threshold of the control information, regarding the messages sent in the causal past which must be transmitted to ensure a causal delivery. IDR is denoted as "↓" and its formal definition is as follows.…”

“…Causal information that includes the messages immediately preceding a given message is sufficient to ensure a causal delivery of such message [14].…”

“…As the proposed protocol depends on the Immediate Dependency Relation protocol [14], the size of the control information of message m depends on the number of concurrent messages that form an IDR with m.…”

“…The first category includes the protocols designed for plain peer-to-peer networks. We can classify [12,14,24,25] in this category. One disadvantage of these protocols is that the size of the control information in the worst case scenario grows with the number of peers in the network.…”