The Space Complexity of Unbounded Timestamps

Abstract

The timestamp problem captures a fundamental aspect of asynchronous distributed computing. It allows processes to label events throughout the system with timestamps that provide information about the real-time ordering of those events. We consider the space complexity of wait-free implementations of timestamps from shared read-write registers in a system of n processes.

We prove an $\Omega(\sqrt{n})$ lower bound on the number of registers required. If the timestamps are elements of a nowhere dense set, for example the integers, we prove a stronger, and tight, lower bound of n. However, if timestamps are not from a nowhere dense set, this bound can be beaten; we give an algorithm that uses n − 1 (single-writer) registers.

We also consider the special case of anonymous algorithms, where processes do not have unique identifiers. We prove anonymous timestamp algorithms require n registers. We give an algorithm to prove that this lower bound is tight. This is the first anonymous algorithm that uses a finite number of registers. Although this algorithm is wait-free, its step complexity is not bounded. We also present an algorithm that uses O(n ²) registers and has bounded step complexity.