Abstract
The unbounded-error communication complexity of a Boolean function $F$ is the limit of the $\epsilon$-error randomized complexity of $F$ as $\epsilon\to1/2.$ Communication complexity with weakly unbounded error is defined similarly but with an additive penalty term that depends on $1/2-\epsilon$. Explicit functions are known whose two-party communication complexity with unbounded error is logarithmic compared to their complexity with weakly unbounded error. Chattopadhyay and Mande (ECCC TR16-095, Theory of Computing 2018) recently generalized this exponential separation to the number-on-the-forehead multiparty model. We show how to derive such an exponential separation from known two-party work, achieving a quantitative improvement along the way. We present several proofs here, some as short as half a page. In more detail, we construct a $k$-party communication problem $F\colon(\{0,1\}^{n})^{k}\to\{0,1\}$ that has complexity $O(\log n)$ with unbounded error and $\Omega(\sqrt n\,/\,4^{k})$ with weakly unbounded error, reproducing the bounds of Chattopadhyay and Mande. In addition, we prove a quadratically stronger separation of $O(\log n)$ versus $\Omega(n\,/\,4^k)$ using a nonconstructive argument. A preliminary version of this paper appeared in ECCC, Report TR16-138, 2016.
Highlights
The number-on-the-forehead model, due to Chandra et al [9], is the most powerful model of multiparty communication
An input (x1, x2, . . . , xk) is distributed among the k players by giving the i-th player the arguments x1, . . . , xi−1, xi+1, . . . , xk but not xi. This arrangement can be visualized as having the k players seated in a circle with xi written on the i-th player’s forehead, whence the name of the model
We refer to PP(F) as the communication complexity of F with weakly unbounded error
Summary
The number-on-the-forehead model, due to Chandra et al [9], is the most powerful model of multiparty communication. The players’ objective is to compute F on any given input with minimal communication To this end, each player privately holds an unbounded supply of uniformly random bits which he can use in deciding what message to send at any given point in the protocol. The ε-error randomized communication complexity Rε (F) of a given function F is the least cost of a protocol that computes F with probability of error at most ε on every input. There are two standard ways to define the complexity of F in this setting, both inspired by probabilistic polynomial time for Turing machines: UPP(F) = inf Rε (F) The former quantity, introduced by Paturi and Simon [21], is called the communication complexity of F with unbounded error, in reference to the fact that the error probability can be arbitrarily close to 1/2.
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