Optimized OR-Sets without Ordering Constraints

Eventual consistency is a relaxation of strong consistency that guarantees that if no new updates are made to a replicated data object, then all replicas will converge. The conflict free replicated datatypes (CRDTs) of Shapiro et al. are data structures whose inherent mathematical structure guarantees eventual consistency. We investigate a fundamental CRDT called Observed-Remove Set (OR-Set) that robustly implements sets with distributed add and delete operations. Existing CRDT implementations of OR-Sets either require maintaining a permanent set of “tombstones” for deleted elements, or imposing strong constraints such as causal order on message delivery. We formalize a concurrent specification for OR-Sets without ordering constraints and propose a generalized implementation of OR-sets without tombstones that provably satisfies strong eventual consistency. We introduce Interval Version Vectors to succinctly keep track of distributed time-stamps in systems that allow out-of-order delivery of messages. The space complexity of our generalized implementation is competitive with respect to earlier solutions with causal ordering. We also formulate k-causal delivery, a generalization of causal delivery, that provides better complexity bounds.