New Bounds for 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 Θ(n) bounds for Inner Product mod 2 and Disjointness), as well as an O(n • 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 Ω(n 2+ ) 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. * This work is funded by the Singapore Ministry of Education (partly through the Academic Research Fund Tier 3 MOE2012-T3-1-009) and by the Singapore National Research Foundation.1 It should be mentioned that even though Alice and Bob choose to not communicate in any other way, their intentions are not hostile and neither will deviate from a previously agreed upon protocol.