12 research outputs found
Access vs. Bandwidth in Codes for Storage
Maximum distance separable (MDS) codes are widely used in storage systems to
protect against disk (node) failures. A node is said to have capacity over
some field , if it can store that amount of symbols of the field.
An MDS code uses nodes of capacity to store information
nodes. The MDS property guarantees the resiliency to any node failures.
An \emph{optimal bandwidth} (resp. \emph{optimal access}) MDS code communicates
(resp. accesses) the minimum amount of data during the repair process of a
single failed node. It was shown that this amount equals a fraction of
of data stored in each node. In previous optimal bandwidth
constructions, scaled polynomially with in codes with asymptotic rate
. Moreover, in constructions with a constant number of parities, i.e. rate
approaches 1, is scaled exponentially w.r.t. . In this paper, we focus
on the later case of constant number of parities , and ask the following
question: Given the capacity of a node what is the largest number of
information disks in an optimal bandwidth (resp. access) MDS
code. We give an upper bound for the general case, and two tight bounds in the
special cases of two important families of codes. Moreover, the bounds show
that in some cases optimal-bandwidth code has larger than optimal-access
code, and therefore these two measures are not equivalent.Comment: This paper was presented in part at the IEEE International Symposium
on Information Theory (ISIT 2012). submitted to IEEE transactions on
information theor
An Alternate Construction of an Access-Optimal Regenerating Code with Optimal Sub-Packetization Level
Given the scale of today's distributed storage systems, the failure of an
individual node is a common phenomenon. Various metrics have been proposed to
measure the efficacy of the repair of a failed node, such as the amount of data
download needed to repair (also known as the repair bandwidth), the amount of
data accessed at the helper nodes, and the number of helper nodes contacted.
Clearly, the amount of data accessed can never be smaller than the repair
bandwidth. In the case of a help-by-transfer code, the amount of data accessed
is equal to the repair bandwidth. It follows that a help-by-transfer code
possessing optimal repair bandwidth is access optimal. The focus of the present
paper is on help-by-transfer codes that employ minimum possible bandwidth to
repair the systematic nodes and are thus access optimal for the repair of a
systematic node.
The zigzag construction by Tamo et al. in which both systematic and parity
nodes are repaired is access optimal. But the sub-packetization level required
is where is the number of parities and is the number of
systematic nodes. To date, the best known achievable sub-packetization level
for access-optimal codes is in a MISER-code-based construction by
Cadambe et al. in which only the systematic nodes are repaired and where the
location of symbols transmitted by a helper node depends only on the failed
node and is the same for all helper nodes. Under this set-up, it turns out that
this sub-packetization level cannot be improved upon. In the present paper, we
present an alternate construction under the same setup, of an access-optimal
code repairing systematic nodes, that is inspired by the zigzag code
construction and that also achieves a sub-packetization level of .Comment: To appear in National Conference on Communications 201
Fundamental Limits on Communication for Oblivious Updates in Storage Networks
In distributed storage systems, storage nodes intermittently go offline for
numerous reasons. On coming back online, nodes need to update their contents to
reflect any modifications to the data in the interim. In this paper, we
consider a setting where no information regarding modified data needs to be
logged in the system. In such a setting, a 'stale' node needs to update its
contents by downloading data from already updated nodes, while neither the
stale node nor the updated nodes have any knowledge as to which data symbols
are modified and what their value is. We investigate the fundamental limits on
the amount of communication necessary for such an "oblivious" update process.
We first present a generic lower bound on the amount of communication that is
necessary under any storage code with a linear encoding (while allowing
non-linear update protocols). This lower bound is derived under a set of
extremely weak conditions, giving all updated nodes access to the entire
modified data and the stale node access to the entire stale data as side
information. We then present codes and update algorithms that are optimal in
that they meet this lower bound. Next, we present a lower bound for an
important subclass of codes, that of linear Maximum-Distance-Separable (MDS)
codes. We then present an MDS code construction and an associated update
algorithm that meets this lower bound. These results thus establish the
capacity of oblivious updates in terms of the communication requirements under
these settings.Comment: IEEE Global Communications Conference (GLOBECOM) 201
Long MDS Codes for Optimal Repair Bandwidth
MDS codes are erasure-correcting codes that can
correct the maximum number of erasures given the number of
redundancy or parity symbols. If an MDS code has r parities
and no more than r erasures occur, then by transmitting all
the remaining data in the code one can recover the original
information. However, it was shown that in order to recover a
single symbol erasure, only a fraction of 1/r of the information
needs to be transmitted. This fraction is called the repair
bandwidth (fraction). Explicit code constructions were given in
previous works. If we view each symbol in the code as a vector
or a column, then the code forms a 2D array and such codes
are especially widely used in storage systems. In this paper, we
ask the following question: given the length of the column l, can
we construct high-rate MDS array codes with optimal repair
bandwidth of 1/r, whose code length is as long as possible? In
this paper, we give code constructions such that the code length
is (r + 1)log_r l
The MDS Queue: Analysing the Latency Performance of Erasure Codes
In order to scale economically, data centers are increasingly evolving their
data storage methods from the use of simple data replication to the use of more
powerful erasure codes, which provide the same level of reliability as
replication but at a significantly lower storage cost. In particular, it is
well known that Maximum-Distance-Separable (MDS) codes, such as Reed-Solomon
codes, provide the maximum storage efficiency. While the use of codes for
providing improved reliability in archival storage systems, where the data is
less frequently accessed (or so-called "cold data"), is well understood, the
role of codes in the storage of more frequently accessed and active "hot data",
where latency is the key metric, is less clear.
In this paper, we study data storage systems based on MDS codes through the
lens of queueing theory, and term this the "MDS queue." We analytically
characterize the (average) latency performance of MDS queues, for which we
present insightful scheduling policies that form upper and lower bounds to
performance, and are observed to be quite tight. Extensive simulations are also
provided and used to validate our theoretical analysis. We also employ the
framework of the MDS queue to analyse different methods of performing so-called
degraded reads (reading of partial data) in distributed data storage
Data Secrecy in Distributed Storage Systems under Exact Repair
The problem of securing data against eavesdropping in distributed storage
systems is studied. The focus is on systems that use linear codes and implement
exact repair to recover from node failures.The maximum file size that can be
stored securely is determined for systems in which all the available nodes help
in repair (i.e., repair degree , where is the total number of nodes)
and for any number of compromised nodes. Similar results in the literature are
restricted to the case of at most two compromised nodes. Moreover, new explicit
upper bounds are given on the maximum secure file size for systems with
. The key ingredients for the contribution of this paper are new results
on subspace intersection for the data downloaded during repair. The new bounds
imply the interesting fact that the maximum data that can be stored securely
decreases exponentially with the number of compromised nodes.Comment: Submitted to Netcod 201
Optimal Locally Repairable and Secure Codes for Distributed Storage Systems
This paper aims to go beyond resilience into the study of security and
local-repairability for distributed storage systems (DSS). Security and
local-repairability are both important as features of an efficient storage
system, and this paper aims to understand the trade-offs between resilience,
security, and local-repairability in these systems. In particular, this paper
first investigates security in the presence of colluding eavesdroppers, where
eavesdroppers are assumed to work together in decoding stored information.
Second, the paper focuses on coding schemes that enable optimal local repairs.
It further brings these two concepts together, to develop locally repairable
coding schemes for DSS that are secure against eavesdroppers.
The main results of this paper include: a. An improved bound on the secrecy
capacity for minimum storage regenerating codes, b. secure coding schemes that
achieve the bound for some special cases, c. a new bound on minimum distance
for locally repairable codes, d. code construction for locally repairable codes
that attain the minimum distance bound, and e. repair-bandwidth-efficient
locally repairable codes with and without security constraints.Comment: Submitted to IEEE Transactions on Information Theor