The Space Complexity of Long-lived and One-Shot Timestamp Implementations

TL;DR

This paper analyzes the space complexity of implementing long-lived and one-shot timestamp objects in distributed systems, establishing tight bounds and revealing a gap in resource requirements between the two types.

Contribution

It improves lower bounds on register requirements for long-lived timestamps and introduces tight bounds for one-shot timestamps, highlighting a fundamental complexity gap.

Findings

Long-lived timestamp systems require at least n/6 registers.

One-shot timestamp systems require at least sqrt{n} registers.

A gap exists in space complexity between one-shot and long-lived timestamp implementations.

Abstract

This paper is concerned with the problem of implementing an unbounded timestamp object from multi-writer atomic registers, in an asynchronous distributed system of n processors with distinct identifiers where timestamps are taken from an arbitrary universe. Ellen, Fatourou and Ruppert (2008) showed that sqrt{n}/2-O(1) registers are required for any obstruction-free implementation of long-lived timestamp systems from atomic registers (meaning processors can repeatedly get timestamps). We improve this existing lower bound in two ways. First we establish a lower bound of n/6 - O(1) registers for the obstruction-free long-lived timestamp problem. Previous such linear lower bounds were only known for constrained versions of the timestamp problem. This bound is asymptotically tight; Ellen, Fatourou and Ruppert (2008) constructed a wait-free algorithm that uses n-1 registers. Second we show that sqrt{n} - O(1) registers are required for any obstruction-free implementation of one-shot timestamp systems(meaning each processor can get a timestamp at most once). We show that this bound is also asymptotically tight by providing a wait-free one-shot timestamp system that uses fewer than 2 sqrt{n} registers, thus establishing a space complexity gap between one-shot and long-lived timestamp systems.