3 research outputs found
A Tight Lower Bound on the Sub-Packetization Level of Optimal-Access MSR and MDS Codes
The first focus of the present paper, is on lower bounds on the
sub-packetization level of an MSR code that is capable of carrying out
repair in help-by-transfer fashion (also called optimal-access property). We
prove here a lower bound on which is shown to be tight for the case
by comparing with recent code constructions in the literature.
We also extend our results to an MDS code over the vector alphabet.
Our objective even here, is on lower bounds on the sub-packetization level
of an MDS code that can carry out repair of any node in a subset of
nodes, where each node is repaired (linear repair) by
help-by-transfer with minimum repair bandwidth. We prove a lower bound on
for the case of . This bound holds for any and
is shown to be tight, again by comparing with recent code constructions in the
literature. Also provided, are bounds for the case .
We study the form of a vector MDS code having the property that we can repair
failed nodes belonging to a fixed set of nodes with minimum repair
bandwidth and in optimal-access fashion, and which achieve our lower bound on
sub-packetization level . It turns out interestingly, that such a code
must necessarily have a coupled-layer structure, similar to that of the Ye-Barg
code.Comment: Revised for ISIT 2018 submissio
Relaxed Models for Adversarial Streaming: The Advice Model and the Bounded Interruptions Model
Streaming algorithms are typically analyzed in the oblivious setting, where
we assume that the input stream is fixed in advance. Recently, there is a
growing interest in designing adversarially robust streaming algorithms that
must maintain utility even when the input stream is chosen adaptively and
adversarially as the execution progresses. While several fascinating results
are known for the adversarial setting, in general, it comes at a very high cost
in terms of the required space. Motivated by this, in this work we set out to
explore intermediate models that allow us to interpolate between the oblivious
and the adversarial models. Specifically, we put forward the following two
models:
(1) *The advice model*, in which the streaming algorithm may occasionally ask
for one bit of advice.
(2) *The bounded interruptions model*, in which we assume that the adversary
is only partially adaptive.
We present both positive and negative results for each of these two models.
In particular, we present generic reductions from each of these models to the
oblivious model. This allows us to design robust algorithms with significantly
improved space complexity compared to what is known in the plain adversarial
model