  In the rendezvous problem, two computing entities (called \emph{agents})
located at different vertices in a graph have to meet at the same vertex. In
this paper, we consider the synchronous \emph{neighborhood rendezvous problem},
where the agents are initially located at two adjacent vertices. While this
problem can be trivially solved in $O(\Delta)$ rounds ($\Delta$ is the maximum
degree of the graph), it is highly challenging to reveal whether that problem
can be solved in $o(\Delta)$ rounds, even assuming the rich computational
capability of agents. The only known result is that the time complexity of
$O(\sqrt{n})$ rounds is achievable if the graph is complete and agents are
probabilistic, asymmetric, and can use whiteboards placed at vertices. Our main
contribution is to clarify the situation (with respect to computational models
and graph classes) admitting such a sublinear-time rendezvous algorithm. More
precisely, we present two algorithms achieving fast rendezvous additionally
assuming bounded minimum degree, unique vertex identifier, accessibility to
neighborhood IDs, and randomization. The first algorithm runs within
$\tilde{O}(\sqrt{n\Delta/\delta} + n/\delta)$ rounds for graphs of the minimum
degree larger than $\sqrt{n}$, where $n$ is the number of vertices in the
graph, and $\delta$ is the minimum degree of the graph. The second algorithm
assumes that the largest vertex ID is $O(n)$, and achieves $\tilde{O}\left(
\frac{n}{\sqrt{\delta}} \right)$-round time complexity without using
whiteboards. These algorithms attain $o(\Delta)$-round complexity in the case
of $\delta = {\omega}(\sqrt{n} \log n)$ and $\delta = \omega(n^{2/3} \log^{4/3}
n)$ respectively.

Distributed, Parallel, and Cluster Computing