Bounds on the costs of register implementations

Abstract

A fundamental aspect of any concurrent system is how processes communicate with each other. Ultimately, all communication involves concurrent reads and writes of shared memory cells, or registers. The stronger the guarantees provided by a register, the more useful it is to the user, but the harder it may be to implement in practice. Thus it is of interest to determine which types of registers can implement which other types of registers. The types of registers studied in this paper are safe vs. regular, 1-reader vs. n-readers, and binary vs. k-ary. Algorithms for various implementations have been previously developed. These have, for the most part, concentrated on the relative computability between different types of registers. In contrast, this paper studies the relative complexity of such algorithms, by considering the costs incurred when implementing one type of register (the logical register) with registers of another type (physical registers). The cost measures considered are the number of physical registers and the number of reads and writes on the physical registers required to implement the logical register. Bounds on the number of physical operations can be easily converted to provide time bounds for the logical operations. Tight bounds are obtained on the cost measures in many cases, and interesting trade-offs between the cost measures are identified. The lower bounds are shown using information-theoretic techniques. Two new algorithms are presented that improve on the costs of previously known algorithms: the hypercube algorithm implements a k-ary safe register out of binary safe registers, requiring only one physical write per logical write; and the tree algorithm implements a k-ary regular register out of binary regular registers, requiring only log k physical operations per logical operation. Both algorithms use novel combinatorial techniques.