2 research outputs found
Bandwidth Cost of Code Conversions in Distributed Storage: Fundamental Limits and Optimal Constructions
Erasure codes have become an integral part of distributed storage systems as
a tool for providing data reliability and durability under the constant threat
of device failures. In such systems, an code over a finite field
encodes message symbols into codeword symbols from
which are then stored on different nodes in the system.
Recent work has shown that significant savings in storage space can be obtained
by tuning and to variations in device failure rates. Such a tuning
necessitates code conversion: the process of converting already encoded data
under an initial code to its equivalent under a final
code. The default approach to conversion is to reencode data, which places
significant burden on system resources. Convertible codes are a recently
proposed class of codes for enabling resource-efficient conversions. Existing
work on convertible codes has focused on minimizing access cost, i.e., the
number of code symbols accessed during conversion. Bandwidth, which corresponds
to the amount of data read and transferred, is another important resource to
optimize.
In this paper, we initiate the study on the fundamental limits on bandwidth
used during code conversion and present constructions for bandwidth-optimal
convertible codes. First, we model the code conversion problem using network
information flow graphs with variable capacity edges. Second, focusing on MDS
codes and an important parameter regime called the merge regime, we derive
tight lower bounds on the bandwidth cost of conversion. The derived bounds show
that bandwidth cost can be significantly reduced even in regimes where access
cost cannot be reduced as compared to the default approach. Third, we present a
new construction for MDS convertible codes which matches the proposed lower
bound and is thus bandwidth-optimal during conversion
Constrained Linear Representability of Polymatroids and Algorithms for Computing Achievability Proofs in Network Coding
The constrained linear representability problem (CLRP) for polymatroids
determines whether there exists a polymatroid that is linear over a specified
field while satisfying a collection of constraints on the rank function. Using
a computer to test whether a certain rate vector is achievable with vector
linear network codes for a multi-source network coding instance and whether
there exists a multi-linear secret sharing scheme achieving a specified
information ratio for a given secret sharing instance are shown to be special
cases of CLRP. Methods for solving CLRP built from group theoretic techniques
for combinatorial generation are developed and described. These techniques form
the core of an information theoretic achievability prover, an implementation
accompanies the article, and several computational experiments with interesting
instances of network coding and secret sharing demonstrating the utility of the
method are provided.Comment: submitted to IEEE Transactions on Information Theory, (this version:
corrected figure 9