Upper and Lower Bounds for Deterministic Approximate Objects

TL;DR

This paper investigates the step complexity of approximate shared objects, specifically max registers and counters, allowing for multiplicative errors, and establishes bounds for their wait-free implementations.

Contribution

It introduces and analyzes the complexity bounds for $k$-multiplicative-accurate max registers and counters, a novel relaxation of traditional shared objects.

Findings

Provided upper bounds for implementation complexity.

Established lower bounds for wait-free implementations.

Analyzed the impact of multiplicative error on complexity.

Abstract

Relaxing the sequential specification of shared objects has been proposed as a promising approach to obtain implementations with better complexity. In this paper, we study the step complexity of relaxed variants of two common shared objects: max registers and counters. In particular, we consider the $k$-multiplicative-accurate max register and the $k$-multiplicative-accurate counter, where read operations are allowed to err by a multiplicative factor of $k$ (for some $k \in \mathbb{N}$). More accurately, reads are allowed to return an approximate value $x$ of the maximum value $v$ previously written to the max register, or of the number $v$ of increments previously applied to the counter, respectively, such that $v/k \leq x \leq v \cdot k$. We provide upper and lower bounds on the complexity of implementing these objects in a wait-free manner in the shared memory model.