Time/Contention Trade-Offs for Multiprocessor Synchronization

Abstract

We establish trade-offs between time complexity and write- and access-contention for solutions to the mutual exclusion problem. The write-contention ( access-contention ) of a concurrent program is the number of processes that may be simultaneously enabled to write (access by reading and/or writing) the same shared variable. Our notion of time complexity distinguishes between local and remote accesses of shared memory. We show that, for any N -process mutual exclusion algorithm, if write-contention is w , and if at most v remote variables can be accessed by a single atomic operation, then there exists an execution involving only one process in which that process executes Ω (log vw N ) remote operations for entry into its critical section. We further show that, among these operations,[formula]distinct remote variables are accessed. For algorithms with access-contention c , we show that the latter bound can be improved to Ω (log vc N ). The last two of these bounds imply that a trade-off between contention and time complexity exists even if coherent caching techniques are employed. In most shared-memory multiprocessors, an atomic operation may access only a constant number of remote variables. In fact, most commonly-available synchronization primitives (e.g., read, write, test-and-set, load-and-store, compare-and-swap, and fetch-and-add) access only one remote variable. In this case, the first and the last of our bounds are asymptotically tight. Our results have a number of important implications regarding specific concurrent programming problems. For example, the time bounds that we establish apply not only to the mutual exclusion problem, but also to a class of decision problems that includes the leader-election problem. Also, because the execution that establishes these bounds involves only one process, it follows that “fast mutual exclusion” requires arbitrarily high write-contention. Although such conclusions are interesting in their own right, we believe that the most important contribution of our work is to identify a time complexity measure for asynchronous concurrent programs that strikes a balance between being conceptually simple and having a tangible connection to real performance.