Linearizable counting networks

Abstract

The counting problem requires n asynchronous processes to assign themselves successive values. A solution is linearizable if the order of the values assigned reflects the real-time order in which they were requested. Linearizable counting lies at the heart of concurrent time-stamp generation, as well as concurrent implementations of shared counters, FIFO buffers, and similar data structures. We consider solutions to the linearizable counting problem in a multiprocessor architecture in which processes 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 processes are required to wait for one another, and how long it takes a process 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 new constructions based on counting networks, one with low latency and low contention, but that requires processes to wait for one another, and one with low contention and no waiting, but that has high latency. Finally, we prove that these trade-offs 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.