Communication Complexity of Permutation-Invariant Functions

Motivated by the quest for a broader understanding of communication complexity of simple functions, we introduce the class of "permutation-invariant" functions. A partial function $f:\{0,1\}^n \times \{0,1\}^n\to \{0,1,?\}$ is permutation-invariant if for every bijection $\pi:\{1,\ldots,n\} \to \{1,\ldots,n\}$ and every $\mathbf{x}, \mathbf{y} \in \{0,1\}^n$, it is the case that $f(\mathbf{x}, \mathbf{y}) = f(\mathbf{x}^{\pi}, \mathbf{y}^{\pi})$. Most of the commonly studied functions in communication complexity are permutation-invariant. For such functions, we present a simple complexity measure (computable in time polynomial in $n$ given an implicit description of $f$) that describes their communication complexity up to polynomial factors and up to an additive error that is logarithmic in the input size. This gives a coarse taxonomy of the communication complexity of simple functions. Our work highlights the role of the well-known lower bounds of functions such as 'Set-Disjointness' and 'Indexing', while complementing them with the relatively lesser-known upper bounds for 'Gap-Inner-Product' (from the sketching literature) and 'Sparse-Gap-Inner-Product' (from the recent work of Canonne et al. [ITCS 2015]). We also present consequences to the study of communication complexity with imperfectly shared randomness where we show that for total permutation-invariant functions, imperfectly shared randomness results in only a polynomial blow-up in communication complexity after an additive $O(\log \log n)$ overhead

Communication with Contextual Uncertainty

We introduce a simple model illustrating the role of context in communication and the challenge posed by uncertainty of knowledge of context. We consider a variant of distributional communication complexity where Alice gets some information $x$ and Bob gets $y$, where $(x,y)$ is drawn from a known distribution, and Bob wishes to compute some function $g(x,y)$ (with high probability over $(x,y)$). In our variant, Alice does not know $g$, but only knows some function $f$ which is an approximation of $g$. Thus, the function being computed forms the context for the communication, and knowing it imperfectly models (mild) uncertainty in this context. A naive solution would be for Alice and Bob to first agree on some common function $h$ that is close to both $f$ and $g$ and then use a protocol for $h$ to compute $h(x,y)$. We show that any such agreement leads to a large overhead in communication ruling out such a universal solution. In contrast, we show that if $g$ has a one-way communication protocol with complexity $k$ in the standard setting, then it has a communication protocol with complexity $O(k \cdot (1+I))$ in the uncertain setting, where $I$ denotes the mutual information between $x$ and $y$. In the particular case where the input distribution is a product distribution, the protocol in the uncertain setting only incurs a constant factor blow-up in communication and error. Furthermore, we show that the dependence on the mutual information $I$ is required. Namely, we construct a class of functions along with a non-product distribution over $(x,y)$ for which the communication complexity is a single bit in the standard setting but at least $\Omega(\sqrt{n})$ bits in the uncertain setting.Comment: 20 pages + 1 title pag

The Power of Shared Randomness in Uncertain Communication

In a recent work (Ghazi et al., SODA 2016), the authors with Komargodski and Kothari initiated the study of communication with contextual uncertainty, a setup aiming to understand how efficient communication is possible when the communicating parties imperfectly share a huge context. In this setting, Alice is given a function f and an input string x, and Bob is given a function g and an input string y. The pair (x,y) comes from a known distribution mu and f and g are guaranteed to be close under this distribution. Alice and Bob wish to compute g(x,y) with high probability. The lack of agreement between Alice and Bob on the function that is being computed captures the uncertainty in the context. The previous work showed that any problem with one-way communication complexity k in the standard model (i.e., without uncertainty, in other words, under the promise that f=g) has public-coin communication at most O(k(1+I)) bits in the uncertain case, where I is the mutual information between x and y. Moreover, a lower bound of Omega(sqrt{I}) bits on the public-coin uncertain communication was also shown. However, an important question that was left open is related to the power that public randomness brings to uncertain communication. Can Alice and Bob achieve efficient communication amid uncertainty without using public randomness? And how powerful are public-coin protocols in overcoming uncertainty? Motivated by these two questions: - We prove the first separation between private-coin uncertain communication and public-coin uncertain communication. Namely, we exhibit a function class for which the communication in the standard model and the public-coin uncertain communication are O(1) while the private-coin uncertain communication is a growing function of n (the length of the inputs). This lower bound (proved with respect to the uniform distribution) is in sharp contrast with the case of public-coin uncertain communication which was shown by the previous work to be within a constant factor from the certain communication. This lower bound also implies the first separation between public-coin uncertain communication and deterministic uncertain communication. Interestingly, we also show that if Alice and Bob imperfectly share a sequence of random bits (a setup weaker than public randomness), then achieving a constant blow-up in communication is still possible. - We improve the lower-bound of the previous work on public-coin uncertain communication. Namely, we exhibit a function class and a distribution (with mutual information I approx n) for which the one-way certain communication is k bits but the one-way public-coin uncertain communication is at least Omega(sqrt{k}*sqrt{I}) bits. Our proofs introduce new problems in the standard communication complexity model and prove lower bounds for these problems. Both the problems and the lower bound techniques may be of general interest

Communication over an Arbitrarily Varying Channel under a State-Myopic Encoder

We study the problem of communication over a discrete arbitrarily varying channel (AVC) when a noisy version of the state is known non-causally at the encoder. The state is chosen by an adversary which knows the coding scheme. A state-myopic encoder observes this state non-causally, though imperfectly, through a noisy discrete memoryless channel (DMC). We first characterize the capacity of this state-dependent channel when the encoder-decoder share randomness unknown to the adversary, i.e., the randomized coding capacity. Next, we show that when only the encoder is allowed to randomize, the capacity remains unchanged when positive. Interesting and well-known special cases of the state-myopic encoder model are also presented.Comment: 16 page

Guessing a password over a wireless channel (on the effect of noise non-uniformity)

A string is sent over a noisy channel that erases some of its characters. Knowing the statistical properties of the string's source and which characters were erased, a listener that is equipped with an ability to test the veracity of a string, one string at a time, wishes to fill in the missing pieces. Here we characterize the influence of the stochastic properties of both the string's source and the noise on the channel on the distribution of the number of attempts required to identify the string, its guesswork. In particular, we establish that the average noise on the channel is not a determining factor for the average guesswork and illustrate simple settings where one recipient with, on average, a better channel than another recipient, has higher average guesswork. These results stand in contrast to those for the capacity of wiretap channels and suggest the use of techniques such as friendly jamming with pseudo-random sequences to exploit this guesswork behavior.Comment: Asilomar Conference on Signals, Systems & Computers, 201