8 research outputs found
Some Lower Bounds in Dynamic Networks with Oblivious Adversaries
This paper considers several closely-related problems in synchronous dynamic networks with oblivious adversaries, and proves novel Omega(d + poly(m)) lower bounds on their time complexity (in rounds). Here d is the dynamic diameter of the dynamic network and m is the total number of nodes. Before this work, the only known lower bounds on these problems under oblivious adversaries were the trivial Omega(d) lower bounds. Our novel lower bounds are hence the first non-trivial lower bounds and also the first lower bounds with a poly(m) term. Our proof relies on a novel reduction from a certain two-party communication complexity problem. Our central proof technique is unique in the sense that we consider the communication complexity with a special leaker. The leaker helps Alice and Bob in the two-party problem, by disclosing to Alice and Bob certain "non-critical" information about the problem instance that they are solving
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 is permutation-invariant if for every bijection
and every , it is the case that . 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 given
an implicit description of ) 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 overhead
New Separations Results for External Information
We obtain new separation results for the two-party external information
complexity of boolean functions. The external information complexity of a
function is the minimum amount of information a two-party protocol
computing must reveal to an outside observer about the input. We obtain the
following results:
1. We prove an exponential separation between external and internal
information complexity, which is the best possible; previously no separation
was known.
2. We prove a near-quadratic separation between amortized zero-error
communication complexity and external information complexity for total
functions, disproving a conjecture of \cite{Bravermansurvey}.
3. We prove a matching upper showing that our separation result is tight
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Complexity Theory
Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness and randomness extraction. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes