In the rendezvous problem, two computing entities (called agents) located at different vertices in a graph have to meet at the same vertex. In this paper, we consider the synchronous neighborhood rendezvous problem, where the agents are initially located at two adjacent vertices. While this problem can be trivially solved in O(Δ) rounds (Δ is the maximum degree of the graph), it is highly challenging to reveal whether that problem can be solved in o(Δ) rounds, even assuming the rich computational capability of agents. The only known result is that the time complexity of O($O(sqrt{n})$) 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 $ ilde{O}(sqrt{nDelta/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 δ is the minimum degree of the graph. The second algorithm assumes that the largest vertex ID is O(n), and achieves $ ilde{O}left( rac{n}{sqrt{delta}} ight)$-round time complexity without using whiteboards. These algorithms attain o(Δ)-round complexity in the case of $delta = {omega}(sqrt{n} log n)$ and δ=ω(n2/3log4/3n) respectively. We also prove that four unconventional assumptions of our algorithm, bounded minimum degree, accessibility to neighborhood IDs, initial distance one, and randomization are all inherently necessary for attaining fast rendezvous. That is, one can obtain the Ω(n)-round lower bound if either one of them is removed.

A passively mobile system is an abstract notion of mobile ad-hoc networks. It is a collection of agents with computing devices. Agents move in a region, but the algorithm cannot control their physical behavior (i.e., how they move). The population protocol model is one of the promising models in which the computation proceeds by the pairwise communication between two agents. The communicating agents update their states by a specified transition function (algorithm). In this paper, we consider a general form of the aggregation problem with a base station. The base station is a special agent having the computational power more powerful than others. In the aggregation problem, the base station has to sum up for inputs distributed to other agents. We propose an algorithm that solves the aggregation problem in sub-linear parallel time using a relatively small number of states per agent. More precisely, our algorithm solves the aggregation problem with input domain X in O(√n log2 n) parallel time and O(|X|2) states per agent (except for the base station) with high probability.