Fast Neighborhood Rendezvous

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