The Garden-Hose Model

We define a new model of communication complexity, called the garden-hose model. Informally, the garden-hose complexity of a function f:{0,1}^n x {0,1}^n to {0,1} is given by the minimal number of water pipes that need to be shared between two parties, Alice and Bob, in order for them to compute the function f as follows: Alice connects her ends of the pipes in a way that is determined solely by her input x \in {0,1}^n and, similarly, Bob connects his ends of the pipes in a way that is determined solely by his input y \in {0,1}^n. Alice turns on the water tap that she also connected to one of the pipes. Then, the water comes out on Alice’s or Bob’s side depending on the function value f(x,y). We prove almost-linear lower bounds on the garden-hose complexity for concrete functions like inner product, majority, and equality, and we show the existence of functions with exponential garden-hose complexity. Furthermore, we show a connection to classical complexity theory by proving that all functions computable in log-space have polynomial garden-hose complexity. We consider a randomized variant of the garden-hose complexity, where Alice and Bob hold pre-shared randomness, and a quantum variant, where Alice and Bob hold pre-shared quantum entanglement, and we show that the randomized garden-hose complexity is within a polynomial factor of the deterministic garden-hose complexity. Examples of (partial) functions are given where the quantum garden-hose complexity is logarithmic in n while the classical garden-hose complexity can be lower bounded by n^c for constant c>0. Finally, we show an interesting connection between the garden-hose model and the (in)security of a certain class of quantum position-verification schemes.

💡 Research Summary

The paper introduces the garden‑hose model, a novel communication‑complexity framework inspired by a simple physical picture: two parties, Alice and Bob, share a set of s water pipes. For each input x∈{0,1}ⁿ (Alice) and y∈{0,1}ⁿ (Bob), they independently connect the ends of the pipes in a one‑to‑one fashion; Alice also attaches a tap to one pipe and turns on the water. The water follows a unique path determined by the combined connections and eventually exits either on Alice’s side (output 0) or Bob’s side (output 1). The minimum number of pipes required to compute a Boolean function f is defined as the garden‑hose complexity GH(f).

The authors first prove that GH(f) is always finite and give a trivial upper bound GH(f) ≤ 2ⁿ+1 by directly mapping inputs to pipe labels and letting Bob pair pipes according to the truth table of f. They then establish an almost‑linear lower bound for functions that are “injective for Alice” or “injective for Bob” (i.e., for any two distinct inputs of one party there exists a counter‑input of the other party that distinguishes the function values). Specifically, they show 4·GH(f)·log GH(f) ≥ n, which yields Ω(n·log n) lower bounds for the inner‑product (IP), equality (EQ), and majority (MAJ) functions. For IP and EQ the bound is tight up to constant factors, as they also present linear‑size constructions (3n+1 pipes for EQ, 4n+1 for IP). MAJ currently admits a quadratic upper bound (≈(n+2)² pipes) and the same Ω(n·log n) lower bound.

A key insight is the connection between GH(f) and deterministic one‑way communication complexity D₁(f): any garden‑hose protocol can be turned into a one‑way protocol where Alice sends a description of her connections using at most GH(f)·log GH(f) bits, so D₁(f) ≤ GH(f)·log GH(f). Conversely, the authors show GH(f) ≤ 2·D(f)+1, where D(f) is the usual deterministic two‑way communication complexity.

The paper further relates GH(f) to space complexity. Every function computable in logarithmic space can be realized with a polynomial‑size garden‑hose protocol, and conversely any function with polynomial GH(f) can be computed by a log‑space machine after a simple pre‑processing step. Hence the class of functions with polynomial garden‑hose complexity coincides with L (up to local preprocessing).

Two extensions are considered. In the randomized garden‑hose model, Alice and Bob share a public random string and are allowed a small error ε; the authors prove that the randomized complexity GH_ε(f) differs from the deterministic GH(f) by at most a polynomial factor. In the quantum variant, the parties share an arbitrary entangled state and may base their wiring decisions on measurement outcomes. They exhibit a partial function for which the quantum garden‑hose complexity GH^Q_ε(f) is logarithmic in n, while the best known classical randomized complexity is n^c for some constant c>0, thereby demonstrating a genuine quantum advantage.

Finally, the authors connect the garden‑hose model to quantum position‑verification (PV) schemes. A PV protocol PV_f is defined by a Boolean function f; an adversarial coalition that tries to break the protocol must teleport a qubit back and forth using EPR pairs. The authors show a one‑to‑one correspondence between attacks on PV_f and garden‑hose strategies, implying that the quantum garden‑hose complexity of f upper‑bounds the number of EPR pairs needed for a successful attack. Consequently, if there exists an f∈P for which any attack requires super‑polynomial EPR resources, then P≠L; conversely, proving lower bounds on GH^Q(f) would yield security guarantees for PV schemes based on classical complexity assumptions.

In summary, the paper defines a new, intuitively appealing model of communication complexity, establishes tight (up to polylogarithmic factors) bounds for several fundamental functions, links the model to log‑space computation and one‑way communication, explores randomized and quantum extensions, and demonstrates relevance to quantum cryptographic protocols. The work opens several avenues for future research, including tighter bounds for majority, explicit exponential‑complexity functions, and deeper investigations of the quantum‑garden‑hose versus entanglement‑cost relationship.