Asynchronous Deterministic Rendezvous for Multiple Mobile Agents in Trees

This paper reveals the relation between the time complexity and the space complexity for the ren- dezvous problem with k agents in asynchronous tree networks with n nodes. The rendezvous prob- lem requires k ≥ 2 agents to meet at a single node. We propose three algorithms for the ren- dezvous problem. All the algorithms guarantee that all the agents meet at a single node if the tree is not symmetric. Otherwise, all the agents gather at two neighboring nodes. Firstly, we present the simple asymptotically time-optimal algorithm, which has time complexity O(n) and space com- plexity O(n logn) per agent. The second algo- rithm achieves better space complexity O(n) at the cost of time complexity and its time complexity is O(n logn). Lastly, we present the asymptoti- cally space-optimal one. This algorithm has space complexity O(logn) and time complexity O(∆n8) where ∆ is the maximum degree of the tree.