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
$[n, k]$
code over a finite field
$\mathbb {F}_{q}$
encodes
$k$
message symbols from
$\mathbb {F}_{q}$
into
$n$
codeword symbols from
$\mathbb {F}_{q}$
which are then stored on
$n$
different nodes in the system. Recent work has shown that significant savings in storage space can be obtained by tuning
$n$
and
$k$
to variations in device failure rates. Such a tuning necessitates
code conversion
: the process of converting already encoded data under an initial
$[n^{ I}, k^{ I}]$
code to its equivalent under a final
$[n^{ F}, k^{ F}]$
code. The default approach to conversion is to re- encode the data under the new code, 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 the 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 during conversions. In this paper, we study 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 conversion bandwidth. The derived bounds show that conversion bandwidth can be significantly reduced as compared to the default approach even in regions where it has been shown that access cost cannot be reduced. Third, we present a new construction for MDS convertible codes which matches the proposed lower bound and is thus bandwidth-optimal during conversion.