Quasi-Kautz Digraphs for Peer-to-Peer Networks

We consider deterministic feasibility and time complexity of two fundamental tasks in distributed computing: consensus and mutual exclusion. Processes have different labels and communicate through a multiple access channel. The adversary wakes up some processes in possibly different rounds. In any round, every awake process either listens or transmits. The message of a process i is heard by all other awake processes, if i is the only process to transmit in a given round. If more than one process transmits simultaneously, there is a collision and no message is heard. We consider three characteristics that may or may not exist in the channel: collision detection (listening processes can distinguish collision from silence), the availability of a global clock showing the round number, and the knowledge of the number n of all processes. If none of the above three characteristics is available in the channel, we prove that consensus and mutual exclusion are infeasible; if at least one of them is available, both tasks are feasible, and we study their time complexity. Collision detection is shown to cause an exponential gap in complexity: if it is available, both tasks can be performed in time logarithmic in n, which is optimal, and without collision detection both tasks require linear time. We then investigate both consensus and mutual exclusion in the absence of collision detection, but under alternative presence of the two other features. With global clock, we give an algorithm whose time complexity linearly depends on n and on the wake-up time, and an algorithm whose complexity does not depend on the wake-up time and differs from the linear lower bound only by a factor O(\log^2 n). If n is known, we also show an algorithm whose complexity differs from the linear lower bound only by a factor O(\log^2 n).