195 research outputs found
Communication-Rounds Tradeoffs for Common Randomness and Secret Key Generation
We study the role of interaction in the Common Randomness Generation (CRG)
and Secret Key Generation (SKG) problems. In the CRG problem, two players,
Alice and Bob, respectively get samples and
with the pairs , , being drawn independently
from some known probability distribution . They wish to communicate so as
to agree on bits of randomness. The SKG problem is the restriction of the
CRG problem to the case where the key is required to be close to random even to
an eavesdropper who can listen to their communication (but does not have access
to the inputs of Alice and Bob). In this work, we study the relationship
between the amount of communication and the number of rounds of interaction in
both the CRG and the SKG problems. Specifically, we construct a family of
distributions , parametrized by integers , and ,
such that for every there exists a constant for which CRG
(respectively SKG) is feasible when with
rounds of communication, each consisting of bits, but when
restricted to rounds of interaction, the total communication must
exceed bits. Prior to our work no separations were
known for .Comment: 41 pages, 3 figure
Information Value of Two-Prover Games
We introduce a generalization of the standard framework for studying the difficulty of two-prover games. Specifically, we study the model where Alice and Bob are allowed to communicate (with information constraints) - in contrast to the usual two-prover game where they are not allowed to communicate after receiving their respective input. We study the trade-off between the information cost of the protocol and the achieved value of the game after the protocol.
In particular, we show the connection of this trade-off and the amortized behavior of the game (i.e. repeated value of the game).
We show that if one can win the game with at least (1 - epsilon)-probability by communicating at most epsilon bits of information,
then one can win n copies with probability at least 2^{-O(epsilon n)}. This gives an intuitive explanation why Raz\u27s counter-example to strong parallel repetition [Raz2008] (the odd cycle game) is a counter-example to strong parallel repetition - one can win the odd-cycle game on a cycle of length by communicating O(m^{-2})-bits where m is the number of vertices.
Conversely, for projection games, we show that if one can win n copies with probability larger than (1-epsilon)^n,
then one can win one copy with at least (1 - O(epsilon))-probability by communicating O(epsilon) bits of information.
By showing the equivalence between information value and amortized value, we give an alternative direction for further works in studying amortized behavior of the two-prover games.
The main technical tool is the "Chi-Squared Lemma" which bounds the information cost of the protocol in terms of Chi-Squared distance,
instead of usual divergence. This avoids the square loss from using Pinsker\u27s Inequality
Near-Quadratic Lower Bounds for Two-Pass Graph Streaming Algorithms
We prove that any two-pass graph streaming algorithm for the -
reachability problem in -vertex directed graphs requires near-quadratic
space of bits. As a corollary, we also obtain near-quadratic space
lower bounds for several other fundamental problems including maximum bipartite
matching and (approximate) shortest path in undirected graphs.
Our results collectively imply that a wide range of graph problems admit
essentially no non-trivial streaming algorithm even when two passes over the
input is allowed. Prior to our work, such impossibility results were only known
for single-pass streaming algorithms, and the best two-pass lower bounds only
ruled out space algorithms, leaving open a large gap between
(trivial) upper bounds and lower bounds
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