Counting networks

The counting problem requires n asynchronous processors to assign themselves successive values. A solution is linearizable if the order of the values assigned re ects the real-time order in which theywere requested. Linearizable counting lies at the heart of concurrent timestamp generation, as well as concurrent implementations of shared counters, FIFO bu ers, and similar data structures. We consider solutions to the linearizable counting problem in a multiprocessor architecture in which processors communicate by applying read-modify-write operations to a shared memory. Linearizable counting algorithms can be judged by three criteria: the memory contention produced, whether processors are required to wait for one another, and how longittakes a processor to choose a value (the latency). A solution is ideal if it has low contention, low latency, and it eschews waiting. The conventional software solution, where processes synchronize at a single variable, avoids waiting and has low latency, but has high contention. In this paper we give two newcounting network constructions, one with low latency and low contention, but that requires processors to wait for one another, and one with low contention and no waiting, but that has high latency. Finally,weprove that these trade-o s are inescapable: an ideal linearizable counting algorithm is impossible. Since ideal non-linearizable counting algorithms exist, these results establish a substantial complexity gap between linearizable and non-linearizable counting