We have introduced the problem of reconciling sets of sets, and developed a number of protocols to solve it. Along the way, we also developed a new set difference estimator, improving on the state of the art. There are a number of open problems remaining in this area. None of our protocols are obvious tight, so any of the bounds could possibly be improved, but we are also lacking non-trivial lower bounds. There is also the question of the minimum number of rounds of communication required to efficiently solve SSRU. Theorem 2.3.8 is our only protocol for SSRU using ˜O(d) total communication and a constant number of rounds of communication, however it uses 4 rounds while conceivably 2 rounds may suffice as they do in set reconciliation. There is also work to be done investigating how these protocols can best be optimized for use in practice.
Chapter 3
Graph Reconciliation
3.1
Introduction
In this chapter, we introduce a general model for reconciling unlabeled graphs, as well as specific models for reconciling random unlabeled graphs, and unlabeled directed forests. In the general model, Alice and Bob each have an unlabeled graph, GA = (VA, EA) and GB =
(VB, EB) respectively, where |VA| =|VB| =n, and only d edges need to be changed (added
or deleted) in EA to make GA isomorphic to GB. We assume our word size w = Ω(logn).
Alice and Bob wish to communicate so that they both end up with the same graph. We note that if GA and GB were labeled graphs, then the problem would be equivalent to set
reconciliation on their sets of labeled edges.
There are various ways to formalize what we want the final graph to be. The simplest, which we focus on, is the one way version of the problem, where we want Bob to end up with a graph isomorphic toGA. Alternatively, we might wish for them to end up with something
corresponding to the union of the two graphs, but this is not always well defined. Figure 3.1 gives an example of two graphs where adding one edge to each will yield isomorphic graphs, but there are multiple distinct ways to do so which yield graphs that are not isomorphic.
+
=
or
Figure 3.1: Depending on how we rotate these two graphs before merging them, we get different (non-isomorphic) results. There is no way to add an edge to only one graph and get isomorphic results, but here we show two different ways to add one edge to each graph and get isomorphic results.
Consequently, we focus on one-way reconciliation, and simply note that our techniques can be generally be extended to most natural two-way versions.
This problem has applications to any setting where we wish to combine large graph based datasets in a communication efficient manner, but we do not possess common labelings of the vertices. For example, these datasets could be maps or social networks.
Most of this chapter is devoted to random graph reconciliation, in which GA and GB
are both close (in edge edits) to a random graph. Our protocols for this problem combine techniques for random graph isomorphism with set of sets reconciliation. Specifically, several graph isomorphism and graph watermarking techniques for random graphs develop what we call here signatures for the vertices, where a signature corresponds to a set related to the neighborhood of the vertex [BK79, CP08]. We adapt this methodology for reconciliation, reconciling a perturbed random graph by first reconciling the set of signatures of the vertices, which can themselves be represented as sets that have undergone a small number of total changes.
We also develop a protocol for graph reconciliation when GA and GB are forests. This
3.1.1
Related Work
Graph reconciliation is closely related to the problem of graph isomorphism [Bab16, BK79, CP08], which can be thought of as reconciling with no errors, and graph watermarking [EGL+16], which involves determining if one graph is a perturbation of another. We incor- porate approaches from these lines of work to construct reconciliation protocols for families of random graphs.
Graph reconciliation is also closely related to the problem of graph matching, also known as graph alignment or social network de-anonymization [KHG15, KYG15, KL14, YG13]. The underlying problem is essentially the same, but there are not two communicating agents; one simply seeks an algorithm that aligns input graphs. In the graph reconciliation setting, where the goal is to use small communication, the number of differences between the graphs is gen- erally assumed to be very small, which is a setting largely unexplored in the graph matching setting. This assumption allows us to reconcile random graphs even with adversarial edits and no additional information, while graph matching assumes random edits and assumes knowledge of some initial set of matching vertices.
Reconciling forests as a communication problem does not appear to have been studied previously. Related work, again without communication, includes computing the edit dis- tance between trees [Bil05]. Tree edit distance considers a different set of graph updates than we consider (vertex updates instead of edge updates).