Suppose that Alice and Bob are given each an infinite string, and they want to decide whether their two strings are in a given relation. How much communication do they need? How can communication be even defined and measured for infinite strings? In this article, we propose a formalism for a notion of infinite communication complexity, prove that it satisfies some natural properties and coincid...

We thank Mauricio Karchmer and Avi Wigderson for raising the question and for helpful discussions, and Amos Beimel, Benny Chor, Alon Orlitsky and Steve Ponzio for many interesting comments on earlier versions of this paper. Finally, we thank an anonymous referee for his very helpful comments and criticism. 17 For values of`which are around log n we would like to replace the term p ` log n with ...

where x ∈ {0, 1}ks, y ∈ {0, 1}s, and Γk : {0, 1}ks × {0, 1}s → {0, 1}ks is some function that can “recycle the randomness in x”. In the INW-generator, Γ is the neighbor function of a suitable expander graph or it is a suitable extractor. Nisan’s generator can also be cast in the framework of (1), but the definition is a bit subtle. Here the string x is of length (2k − 1) · s and of the form x =...

In this work we study the direct-sum problem with respect to communication complexity: Consider a relation f deened over f0; 1g n f0; 1g n. Can the communication complexity of simultaneously computing f onìnstances (x 1 ; y 1); : : : ; (x ` ; y `) be smaller than the communication complexity of computing f on thè instances, separately? Let the amortized communication complexity of f be the comm...

Definition 1.1. Formally, a protocol P is a rooted binary tree with each internal node v labelled by either “A” or “B” and two out-edges, one labelled 0 and the other labelled 1. Each leaf has an output label (typically in {0, 1}. There is a function fv associated with node v; if v is labelled A then fv : X → {0, 1} and if v is labelled B then fv : Y → {0, 1}. The bit sent at node v is sent by ...

Can quantum communication be more efficient than its classical counterpart? Holevo’s theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can be communicated but no more. In apparent contradiction, there are distributed computational task...

Quantum entanglement cannot be used to achieve direct communication between remote parties, but it can reduce the communication needed for some problems. Let each of k parties hold some partial input data to some fixed k-variable function f . The communication complexity of f is the minimum number of classical bits required to be broadcasted for every party to know the value of f on their input...

In this section, we introduce a new communication cost called the distributional communication complexity, which will become useful while obtaining lower bounds on randomized communication complexity. Here we consider a probability distribution μ over the set of inputs X × Y. We are interested in the best (least communication) deterministic protocol P that has an error at most ε when the input ...

dots the authors were also able to estimate the number of elongating Pol II complexes along the length of the MS2/lacZ transgene. These calculations suggest that there are approximately 30 elongating Pol II complexes distributed along the length of the transgene at maximal induction. This corresponds to a density of one Pol II complex every 150 basepairs, or one Pol II released from the promote...