Anat Ganor

Exponential Separation of Information and Communication

We show an exponential gap between communication complexity and information complexity, by giving an explicit example for a communication task (relation), with information complexity ≤ O(k), and distributional communication complexity ≥2k. This shows that a communication protocol cannot always be compressed to its internal information. By a result of Braverman [1], our gap is the largest possible. By a result of Braverman and Rao [2], our example shows a gap between communication complexity and amortized communication complexity, implying that a tight direct sum result for distributional communication complexity cannot hold.

Exponential Separation of Information and Communication for Boolean Functions

We show an exponential gap between communication complexity and information complexity for boolean functions, by giving an explicit example of a partial function with information complexity ≤ O(k), and distributional communication complexity ≥ 2k. This shows that a communication protocol for a partial boolean function cannot always be compressed to its internal information. By a result of Braverman [Bra12], our gap is the largest possible. By a result of Braverman and Rao [BR11], our example shows a gap between communication complexity and amortized communication complexity, implying that a tight direct sum result for distributional communication complexity of boolean functions cannot hold, answering a long standing open problem. Our techniques build on [GKR14], that proved a similar result for relations with very long outputs (double exponentially long in k). In addition to the stronger result, the current work gives a simpler proof, benefiting from the short output length of boolean functions. Another (conceptual) contribution of our work is the relative discrepancy method, a new rectangle-based method for proving communication complexity lower bounds for boolean functions, powerful enough to separate information complexity and communication complexity.

Exponential separation of communication and external information

We show an exponential gap between communication complexity and external information complexity, by analyzing a communication task suggested as a candidate by Braverman. Previously, only a separation of communication complexity and internal information complexity was known. More precisely, we obtain an explicit example of a search problem with external information complexity ≤ O(k), with respect to any input distribution, and distributional communication complexity ≥ 2k, with respect to some input distribution. In particular, this shows that a communication protocol cannot always be compressed to its external information. By a result of Braverman, our gap is the largest possible. Moreover, since the upper bound of O(k) on the external information complexity of the problem is obtained with respect to any input distribution, our result implies an exponential gap between communication complexity and information complexity (both internal and external) in the non-distributional setting of Braverman. In this setting, no gap was previously known, even for internal information complexity.

Two Sides of the Coin Problem

In the coin problem, one is given n independent flips of a coin that has bias b > 0 towards either Head or Tail. The goal is to decide which side the coin is biased towards, with high confidence. An optimal strategy for solving the coin problem is to apply the majority function on the n samples. This simple strategy works as long as b > c(1/sqrt n) for some constant c. However, computing majority is an impossible task for several natural computational models, such as bounded width read once branching programs and AC^0 circuits. Brody and Verbin proved that a length n, width w read once branching program cannot solve the coin problem for b < O(1/(log n)^w). This result was tightened by Steinberger to O(1/(log n)^(w-2)). The coin problem in the model of AC^0 circuits was first studied by Shaltiel and Viola, and later by Aaronson who proved that a depth d size s Boolean circuit cannot solve the coin problem for b < O(1/(log s)^(d+2)). This work has two contributions: 1. We strengthen Steinbergers result and show that any Santha-Vazirani source with bias b < O(1/(log n)^(w-2)) fools length n, width w read once branching programs. In other words, the strong independence assumption in the coin problem is completely redundant in the model of read once branching programs, assuming the bias remains small. That is, the exact same result holds for a much more general class of sources. 2. We tighten Aaronsons result and show that a depth d, size s Boolean circuit cannot solve the coin problem for b < O(1/(log s)^(d-1)). Moreover, our proof technique is different and we believe that it is simpler and more natural.

Space Pseudorandom Generators by Communication Complexity Lower Bounds

In 1989, Babai, Nisan and Szegedy [BNS92] gave a construction of a pseudorandom generator for logspace, based on lower bounds for multiparty communication complexity. The seed length of their pseudorandom generator was 2 ( p log n) , because the best lower bounds for multiparty communication complexity are relatively weak. Subsequently, pseudorandom generators for logspace with seed length O(log 2 n) were given by [N92] and [INW94]. In this paper, we show how to use the pseudorandom generator construction of [BNS92] to obtain a third construction of a pseudorandom generator with seed length O(log 2 n), achieving the same parameters as [N92] and [INW94]. We achieve this by concentrating on protocols in a restricted model of multiparty communication complexity that we call the conservative one-way unicast model and is based on the conservative one-way model of [DJS98]. We observe that bounds in the conservative one-way unicast model (rather than the standard Number On the Forehead model) are sufficient for the pseudorandom generator construction of [BNS92] to work. Roughly speaking, in a conservative one-way unicast communication protocol, the players speak in turns, one after the other in a fixed order, and every message is visible only to the next player. Moreover, before the beginning of the protocol, each player only knows the inputs of the players that speak after she does and a certain function of the inputs of the players that speak before she does. We prove a lower bound for the communication complexity of conservative one-way unicast communication protocols that compute a family of functions obtained by compositions of strong extractors. Our final pseudorandom generator construction is related to, but different from the constructions of [N92] and [INW94].