We investigate two fundamental problems in mobile computing: exploration and rendezvous, with two distinct mobile agents in an unknown graph. The agents may communicate by reading and writing information on whiteboards that are located at all nodes. They both move along one adjacent edge at every time-step. In the exploration problem, the agents start from the same arbitrary node and must traverse all the edges. We present an algorithm achieving collective exploration in $m$ time-steps, where $m$ is the number of edges of the graph. This improves over the guarantee of depth-first search, which requires $2m$ time-steps. In the rendezvous problem, the agents start from different nodes of the graph and must meet as fast as possible. We present an algorithm guaranteeing rendezvous in at most $\frac{3}{2}m$ time-steps. This improves over the so-called `wait for Mommy' algorithm which is based on depth-first search and which also requires $2m$ time-steps. Importantly, all our guarantees are derived from a more general asynchronous setting in which the speeds of the agents are controlled by an adversary at all times. Our guarantees generalize to weighted graphs, when replacing the number of edges $m$ with the sum of all edge lengths. We show that our guarantees are met with matching lower-bounds in the asynchronous setting.

翻译：我们研究了移动计算中的两个基本问题：探索与交会，涉及两个不同的移动代理在一个未知图中。代理可以通过读取和写入位于所有节点上的白板信息进行通信。它们每个时间步都沿着一条相邻边移动。在探索问题中，代理从同一个任意节点出发，必须遍历所有边。我们提出了一种算法，可在$m$个时间步内实现集体探索，其中$m$为图的边数。这改进了深度优先搜索所需的$2m$个时间步保证。在交会问题中，代理从图的不同节点出发，必须尽快相遇。我们提出了一种算法，保证至多在$\frac{3}{2}m$个时间步内实现交会。这改进了基于深度优先搜索的所谓“等待妈妈”算法，后者同样需要$2m$个时间步。重要的是，我们所有的保证都源于一个更一般的异步设置，其中代理的速度始终由对手控制。我们的保证可推广到加权图，此时需将边数$m$替换为所有边长之和。我们证明了在异步设置中，这些保证与匹配的下界相吻合。