Time and Space Lower Bounds for Nonblocking Implementations

Abstract

We show the following time and space complexity lower bounds. Let $\cal{I}$ be any randomized nonblocking n-process implementation of any object in set A from any combination of objects in set B, where A = {increment, fetch&add, modulo k counter (for any $k \ge 2n$), LL/SC bit, k-valued compare&swap (for any $k \ge n$), single-writer snapshot}, and B = {resettable consensus} $\cup$ {historyless objects such as registers and swap registers}. The space complexity of $\cal{I}$ is at least n-1. Moreover, if $\cal{I}$ is deterministic, both its time and space complexity are at least n-1. These lower bounds hold even if objects used in the implementation are of unbounded size. This improves on some of the $\Omega(\sqrt{n})$ space complexity lower bounds of Fich, Herlihy, and Shavit [i Proceedings of the 12th Annual ACM Symposium on Principles of Distributed Computing, Ithaca, NY, 1993, pp. 241--249; J. Assoc. Comput. Mach., 45 (1998), pp. 843--862]. It also shows the near optimality of some known wait-free implementations in terms of space complexity.