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arXiv:1109.2563 · Conference on Innovations in Theoretical Computer Science

The garden-hose model

Harry Buhrman, Serge Fehr, Christian Schaffner, Florian Speelman

Published Jan 3, 2013• 27 citations
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Abstract

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 -> {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 ∈ {0,1}n and, similarly, Bob connects his ends of the pipes in a way that is determined solely by his input y ∈ {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).

DOI
10.1145/2422436.2422455

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In this chapter we consider several, more advanced, topics related to the two-party communication model. Direct Sum The direct-sum problem is the following: Alice gets two inputs x f ∈ X f and x g ∈ X g . Bob gets two inputs y f ∈ Y f and y g ∈ Y g . They wish to compute both f ( x f , y f ) and g ( x g , y g ). The obvious solution would be for Alice and Bob to use the best protocol for f to compute the first value, f ( x f , y f ) and the best protocol for g to compute the second value, g ( x g , y g ). We stress that the two subproblems are totally independent. Thus one would tend to conjecture that nothing better than the obvious solution can be done: Alice and Bob cannot “save” any communication over the obvious protocol. As we shall see, in some cases and for some measures of complexity, this intuition is wrong. Denote by D ( f , g ) the (deterministic) communication complexity of this computation. Similarly, we define all other complexity measures such as R ( f , g ), N ( f , g ), and so forth. We also use the notation D ( f ℓ ) as the (deterministic) communication complexity of computing f and ℓ instances; that is, computing f ( x 1 , y 1 ), f ( x 2 , y 2 ), …, f ( x ℓ , y ℓ Open Problem 4.1: Can D ( f , g ) be smaller than D ( f ) + D ( g )? How much smaller can it be? How much smaller can D ( f ℓ ) be compared to ℓ · D ( f )? In some cases we are not interested in computing both f and g but rather some function of the two.

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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 -> {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 ∈ {0,1}n and, similarly, Bob connects his ends of the pipes in a way that is determined solely by his input y ∈ {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 study the communication complexity of computing functions F: {0,1}ⁿ × {0,1}ⁿ → {0,1} in the memoryless communication model. Here, Alice is given x ∈ {0,1}ⁿ, Bob is given y ∈ {0,1}ⁿ and their goal is to compute F(x,y) subject to the following constraint: at every round, Alice receives a message from Bob and her reply to Bob solely depends on the message received and her input x (in particular, her reply is independent of the information from the previous rounds); the same applies to Bob. The cost of computing F in this model is the maximum number of bits exchanged in any round between Alice and Bob (on the worst case input x,y). In this paper, we also consider variants of our memoryless model wherein one party is allowed to have memory, the parties are allowed to communicate quantum bits, only one player is allowed to send messages. We show that some of these different variants of our memoryless communication model capture the garden-hose model of computation by Buhrman et al. (ITCS'13), space-bounded communication complexity by Brody et al. (ITCS'13) and the overlay communication complexity by Papakonstantinou et al. (CCC'14). Thus the memoryless communication complexity model provides a unified framework to study all these space-bounded communication complexity models. We establish the following main results: (1) We show that the memoryless communication complexity of F equals the logarithm of the size of the smallest bipartite branching program computing F (up to a factor 2); (2) We show that memoryless communication complexity equals garden-hose model of computation; (3) We exhibit various exponential separations between these memoryless communication models. We end with an intriguing open question: can we find an explicit function F and universal constant c > 1 for which the memoryless communication complexity is at least c log n? Note that c ≥ 2+ε would imply a Ω(n^{2+ε}) lower bound for general formula size, improving upon the best lower bound by [Nečiporuk, 1966].
We study the communication complexity of computing functions F:{0,1}n×{0,1}n→{0,1}F:\{0,1\}^n\times \{0,1\}^n \rightarrow \{0,1\}F:{0,1}n×{0,1}n→{0,1} in the memoryless communication model. Here, Alice is given x∈{0,1}nx\in \{0,1\}^nx∈{0,1}n, Bob is given y∈{0,1}ny\in \{0,1\}^ny∈{0,1}n and their goal is to compute F(x,y) subject to the following constraint: at every round, Alice receives a message from Bob and her reply to Bob solely depends on the message received and her input x; the same applies to Bob. The cost of computing F in this model is the maximum number of bits exchanged in any round between Alice and Bob (on the worst case input x,y). In this paper, we also consider variants of our memoryless model wherein one party is allowed to have memory, the parties are allowed to communicate quantum bits, only one player is allowed to send messages. We show that our memoryless communication model capture the garden-hose model of computation by Buhrman et al. (ITCS'13), space bounded communication complexity by Brody et al. (ITCS'13) and the overlay communication complexity by Papakonstantinou et al. (CCC'14). Thus the memoryless communication complexity model provides a unified framework to study space-bounded communication models. We establish the following: (1) We show that the memoryless communication complexity of F equals the logarithm of the size of the smallest bipartite branching program computing F (up to a factor 2); (2) We show that memoryless communication complexity equals garden-hose complexity; (3) We exhibit various exponential separations between these memoryless communication models. We end with an intriguing open question: can we find an explicit function F and universal constant c>1 for which the memoryless communication complexity is at least clog⁡nc \log nclogn? Note that c≥2+εc\geq 2+\varepsilonc≥2+ε would imply a Ω(n2+ε)Ω(n^{2+\varepsilon})Ω(n2+ε) lower bound for general formula size, improving upon the best lower bound by Nečiporuk in 1966.
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)Θ(n)Θ(n) bounds for Inner Product mod 2 and Disjointness), as well as an O(n⋅log⁡3n)O(n\cdot \log^3 n)O(n⋅log3n) 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)GH(f)GH(f) of the order Ω(n2+ε)Ω(n^{2+ε})Ω(n2+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.
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.
In this chapter we consider several, more advanced, topics related to the two-party communication model. Direct Sum The direct-sum problem is the following: Alice gets two inputs x f ∈ X f and x g ∈ X g . Bob gets two inputs y f ∈ Y f and y g ∈ Y g . They wish to compute both f ( x f , y f ) and g ( x g , y g ). The obvious solution would be for Alice and Bob to use the best protocol for f to compute the first value, f ( x f , y f ) and the best protocol for g to compute the second value, g ( x g , y g ). We stress that the two subproblems are totally independent. Thus one would tend to conjecture that nothing better than the obvious solution can be done: Alice and Bob cannot “save” any communication over the obvious protocol. As we shall see, in some cases and for some measures of complexity, this intuition is wrong. Denote by D ( f , g ) the (deterministic) communication complexity of this computation. Similarly, we define all other complexity measures such as R ( f , g ), N ( f , g ), and so forth. We also use the notation D ( f ℓ ) as the (deterministic) communication complexity of computing f and ℓ instances; that is, computing f ( x 1 , y 1 ), f ( x 2 , y 2 ), …, f ( x ℓ , y ℓ Open Problem 4.1: Can D ( f , g ) be smaller than D ( f ) + D ( g )? How much smaller can it be? How much smaller can D ( f ℓ ) be compared to ℓ · D ( f )? In some cases we are not interested in computing both f and g but rather some function of the two.