Recently, Harry Buhrman et al introduced a novel communication complexity model, called “garden-hose”. This model sprout from a research on using quantum properties to allow for position based cryptography. SAT is one of the fundamental NP-Complete problem - finding a satisfying assignment to a CNF (conjunctive Normal Form) formula, or proving that none exists. We will not get into the details of the quantum physics, neither we are going to explain the internal work of SAT solver. Instead we will start from the mathematical garden hose model; describe the way we used SAT solver as a tool; give some lower and upper bounds on implementing the equality function; and conclude with open questions.Recently, Harry Buhrman et al introduced a novel communication complexity model, called “garden-hose”. This model sprout from a research on using quantum properties to allow for position based cryptography. SAT is one of the fundamental NP-Complete problem - finding a satisfying assignment to a CNF (conjunctive Normal Form) formula, or proving that none exists. We will not get into the details of the quantum physics, neither we are going to explain the internal work of SAT solver. Instead we will start from the mathematical garden hose model; describe the way we used SAT solver as a tool; give some lower and upper bounds on implementing the equality function; and conclude with open questions.

Information Theory and Applications Workshop (ITA)Feb 1, 20143 citations

Complexity Equals (Almost) Anything

Robert Cobb Myers, Shan-Ming Ruan

arXiv:2403.17475Springer eBooksJan 1, 20253 citations

New Bounds for the Garden-Hose Model

Hartmut Klauck, Supartha Podder

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 $O(n\cdot \log^3 n)$ 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 $Ω(n^{2+ε})$ 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.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 $O(n\cdot \log^3 n)$ 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 $Ω(n^{2+ε})$ 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.

arXiv:1412.4904Leibniz international proceedings in informaticsJan 1, 20146 citations

Relative Complexity

Leonard Susskind

SpringerBriefs in physicsJan 1, 20200 citations

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\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

Ω(n^{2+ε})

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.