Simple Multi-Party Set Reconciliation

As users migrate information to cloud storage, many distributed cloud-based services use multiple loosely consistent replicas of user information to avoid the high overhead of more tightly coupled synchronization. Periodically, the information must be synchronized, or reconciled. One can place this problem in the theoretical framework of {\em set reconciliation}: two parties $A_1$ and $A_2$ each hold a set of keys, named $S_1$ and $S_2$ respectively, and the goal is for both parties to obtain $S_1 \cup S_2$. Typically, set reconciliation is interesting algorithmically when sets are large but the set difference $|S_1-S_2|+|S_2-S_1|$ is small. In this setting the focus is on accomplishing reconciliation efficiently in terms of communication; ideally, the communication should depend on the size of the set difference, and not on the size of the sets. In this paper, we extend recent approaches using Invertible Bloom Lookup Tables (IBLTs) for set reconciliation to the multi-party setting. In this setting there are three or more parties $A_1,A_2,\ldots,A_n$ holding sets of keys $S_1,S_2,\ldots,S_n$ respectively, and the goal is for all parties to obtain $\cup_i S_i$. This could of course be done by pairwise reconciliations, but we seek more effective methods. Our methodology uses network coding techniques in conjunction with IBLTs, allowing efficiency in network utilization along with efficiency obtained by passing messages of size $O(|\cup_i S_i - \cap_i S_i|)$. Further, our approach can function even if the number of parties is not exactly known in advance, and in many cases can be used to determine which parties contain keys not in the joint union. By connecting reconciliation with network coding, we can allow for substantially more efficient reconciliation methods that apply to a number of natural distributed computing problems.