New Bounds for the Garden-Hose Model

TL;DR

This paper advances understanding of the garden-hose model by establishing new bounds, including linear lower bounds for specific functions and an efficient simulation of logical formulas, highlighting the model's complexity and limitations.

Contribution

It provides improved lower bounds based on non-deterministic communication complexity and introduces an efficient simulation of logical formulas within the garden-hose model.

Findings

Linear  bounds for Inner Product mod 2 and Disjointness

An $O(n \, ext{log}^3 n)$ upper bound for Distributed Majority

Simulation of AND, OR, XOR formulas in the garden-hose model

Abstract

We show new results about the garden-hose model. Our main results include improved lower bounds based on non-deterministic communication complexity (leading to the previously unknown $\Theta(n)$ bounds for Inner Product mod 2 and Disjointness), as well as an $O(n\cdot \log^3 n)$ upper bound for the Distributed Majority function (previously conjectured to have quadratic complexity). We show an efficient simulation of formulae made of AND, OR, XOR gates in the garden-hose model, which implies that lower bounds on the garden-hose complexity $GH(f)$ of the order $\Omega(n^{2+\epsilon})$ will be hard to obtain for explicit functions. Furthermore we study a time-bounded variant of the model, in which even modest savings in time can lead to exponential lower bounds on the size of garden-hose protocols.