The complexity of object reconciliation, and open problems related to set difference and coding

We explore the connections between the classical problems of set difference and error correction codes, motivated by some recent results on Invertible Bloom Filters (communication-efficient set difference) and Biff Codes (fast error correction coding based on set difference). In particular, we seek to understand how these results generalize to settings where many parties communicate over a network represented by a graph, and the goal is for the parties to reconcile the objects owned by each, for some suitable definition of reconcile. Our general framework encompasses standard problems such as rumor spreading and network coding. We suggest that generalizing to other objects such as sequences with other measures such as edit distance may lead to a theory of reconciling objects over graphs. Such a theory may have practical consequences for modern cloud-based deployments.