Operation-Valency and the Cost of Coordination

This paper introduces operation-valency, a generalization of the valency proof technique originated by Fischer, Lynch, and Paterson. By focusing on critical events that influence the return values of individual operations rather then on critical events that influence a protocol's single return value, the new technique allows us to derive a collection of realistic lower bounds for lock-free implementations of concurrent objects such as linearizable queues, stacks, sets, hash tables, shared counters, approximate agreement, and more. By realistic we mean that they follow the real-world model introduced by Dwork, Herlihy, and Waarts, counting both memory-references and memory-stalls due to contention, and that they allow the combined use of read, write, and read-modify-write operations available on current machines. By using the operation-valency technique, we derive an Ω(√n) non-cached shared memory accesses lower bound on the worst-case time complexity of lock-free implementations of objects in Influence(n), a wide class of concurrent objects including all of those mentioned above, in which an individual operation can be influenced by all others. We also prove the existence of a fundamental relationship between the space complexity, latency, contention, and "influence level" of any lock-free object implementation. Our results are broad in that they hold for implementations combining read/write memory and any collection of read-modify-write operations, and in that they apply even if shared memory words have unbounded size.