# Constant Space and Non-Constant Time in Distributed Computing

## TL;DR

This paper explores the relationship between time and space complexity in distributed algorithms, demonstrating that some problems can be solved with constant space but require non-constant time, thus establishing a new complexity class.

## Contribution

It introduces a new complexity class in distributed computing by analyzing problems solvable with constant space but non-constant time, filling a gap in existing theory.

## Key findings

- Existence of non-trivial graph problems solvable with constant space but requiring non-constant time
- Constant communication rounds correspond to constant states visited, but not vice versa
- Raises questions about other possible time-space complexity combinations in distributed algorithms

## Abstract

While the relationship of time and space is an established topic in traditional centralised complexity theory, this is not the case in distributed computing. We aim to remedy this by studying the time and space complexity of algorithms in a weak message-passing model of distributed computing. While a constant number of communication rounds implies a constant number of states visited during the execution, the other direction is not clear at all. We consider several graph families and show that indeed, there exist non-trivial graph problems that are solvable by constant-space algorithms but that require a non-constant running time. This provides us with a new complexity class for distributed computing and raises interesting questions about the existence of further combinations of time and space complexity.