1,456 research outputs found
Delays and the Capacity of Continuous-time Channels
Any physical channel of communication offers two potential reasons why its
capacity (the number of bits it can transmit in a unit of time) might be
unbounded: (1) Infinitely many choices of signal strength at any given instant
of time, and (2) Infinitely many instances of time at which signals may be
sent. However channel noise cancels out the potential unboundedness of the
first aspect, leaving typical channels with only a finite capacity per instant
of time. The latter source of infinity seems less studied. A potential source
of unreliability that might restrict the capacity also from the second aspect
is delay: Signals transmitted by the sender at a given point of time may not be
received with a predictable delay at the receiving end. Here we examine this
source of uncertainty by considering a simple discrete model of delay errors.
In our model the communicating parties get to subdivide time as microscopically
finely as they wish, but still have to cope with communication delays that are
macroscopic and variable. The continuous process becomes the limit of our
process as the time subdivision becomes infinitesimal. We taxonomize this class
of communication channels based on whether the delays and noise are stochastic
or adversarial; and based on how much information each aspect has about the
other when introducing its errors. We analyze the limits of such channels and
reach somewhat surprising conclusions: The capacity of a physical channel is
finitely bounded only if at least one of the two sources of error (signal noise
or delay noise) is adversarial. In particular the capacity is finitely bounded
only if the delay is adversarial, or the noise is adversarial and acts with
knowledge of the stochastic delay. If both error sources are stochastic, or if
the noise is adversarial and independent of the stochastic delay, then the
capacity of the associated physical channel is infinite
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
Property Testing via Set-Theoretic Operations
Given two testable properties and , under
what conditions are the union, intersection or set-difference of these two
properties also testable? We initiate a systematic study of these basic
set-theoretic operations in the context of property testing. As an application,
we give a conceptually different proof that linearity is testable, albeit with
much worse query complexity. Furthermore, for the problem of testing
disjunction of linear functions, which was previously known to be one-sided
testable with a super-polynomial query complexity, we give an improved analysis
and show it has query complexity O(1/\eps^2), where \eps is the distance
parameter.Comment: Appears in ICS 201
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