# Space Complexity of Fault Tolerant Register Emulations

## TL;DR

This paper explores the space complexity of emulating fault-tolerant registers in cloud storage, providing bounds and separations based on the type of base objects and system parameters.

## Contribution

It generalizes the ABD bound, establishing new lower and upper bounds for resource requirements with various base object types and system configurations.

## Key findings

- Number of registers needed scales with the number of writers.
- Resource requirements decrease as the number of servers increases.
- No separation exists between max-registers and CAS in resource efficiency.

## Abstract

Driven by the rising popularity of cloud storage, the costs associated with implementing reliable storage services from a collection of fault-prone servers have recently become an actively studied question. The well-known ABD result shows that an f-tolerant register can be emulated using a collection of 2f + 1 fault-prone servers each storing a single read-modify-write object type, which is known to be optimal. In this paper we generalize this bound: we investigate the inherent space complexity of emulating reliable multi-writer registers as a fucntion of the type of the base objects exposed by the underlying servers, the number of writers to the emulated register, the number of available servers, and the failure threshold. We establish a sharp separation between registers, and both max-registers (the base object types assumed by ABD) and CAS in terms of the resources (i.e., the number of base objects of the respective types) required to support the emulation; we show that no such separation exists between max-registers and CAS. Our main technical contribution is lower and upper bounds on the resources required in case the underlying base objects are fault-prone read/write registers. We show that the number of required registers is directly proportional to the number of writers and inversely proportional to the number of servers.