Reducing Compare-and-Swap to Consensus Number One Primitives

The consensus number of an object is the maximum number of processes among which binary consensus can be solved using any number of instances of the object and read-write registers. Herlihy [6] showed in his seminal work that if an object has a consensus number of n, then there is a universal construction for a wait-free and linearizable implementation of any non-trivial concurrent object or data structure that is shared among n processes. Thus, a synchronization object such as compare-and-swap with an infinite consensus number and the corresponding instruction can be viewed as "strong". On the other hand, a synchronization object such as fetch-and-add with consensus number two and the corresponding fetch-and-add instruction can be viewed as "weak". Ellen et al. [2] observed recently that an object supporting two weak instructions can also achieve infinite consensus number like an object that supports one strong instruction. Using Herlihy's universal construction, this implies that ignoring concerns about efficiency, one can design any concurrent data structure or algorithm using only weak instructions. However, is it possible that a combination of weak instructions is really powerful enough to efficiently replace a strong instruction, like compare-and-swap, without incurring a large overhead in time or space? In this paper, we answer this question by giving an O(1) time wait-free and linearizable implementation of a compare-and-swap register shared among n processes using read-write registers and O(1) registers that support two synchronization primitives half-max and max-write, each having consensus number one. Thus, any algorithm that solves some arbitrary synchronization problem using read-write and compare-and-swap registers can be transformed into an algorithm that has the same asymptotic time complexity and only uses consensus number one instructions.