arXiv subjects

M. Nikhil Krishnan

Publications and source records attributed to M. Nikhil Krishnan.

17 recordsLinked to original sources

arXiv2025

On MDS Convertible Codes in the Merge Regime

In large-scale distributed storage systems, erasure coding is employed to ensure reliability against disk failures. Recent work by Kadekodi et al. demonstrates that adapting code parameters to varying disk failure rates can lead to significant storage savings without compromising reliability. Such adaptations, known as \emph{code conversions}, motivate the design of \emph{convertible codes}, which enable efficient transformations between codes of different parameters. In this work, we study the setting in which $λ$ codewords of an initial $[n^I = k^I + r^I,\, k^I]$ MDS code are merged into a single codeword of a final $[n^F = λk^I + r^F,\, k^F = λk^I]$ MDS code. We begin by presenting three constructions that achieve optimal \emph{access cost}, defined as the total number of disks accessed during the conversion process. The first two constructions apply when $λ\leq r^I$ and impose specific divisibility conditions on $r^I$ and the field size $q$. These schemes minimize both the per-symbol and the overall access cost. The third construction, which builds on a prior scheme by Kong, achieves minimal access cost while supporting arbitrary parameter regimes. All three constructions require field sizes that are linear in the final code length, and notably, the third construction achieves a field size that matches the lower bound implied by the MDS conjecture in almost all cases. In addition, we propose a construction that optimizes the \emph{bandwidth cost}, defined as the total number of symbols transmitted during conversion. This scheme is a refinement of Maturana and Rashmi's bandwidth-optimal construction based on the piggybacking framework, and achieves reduced sub-packetization.

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arXiv2023

Hierarchical Coded Gradient Aggregation Based on Layered MDS Codes

The growing privacy concerns and the communication costs associated with transmitting raw data have resulted in techniques like federated learning, where the machine learning models are trained at the edge nodes, and the parameter updates are shared with a central server. Because communications from the edge nodes are often unreliable, a hierarchical setup involving intermediate helper nodes is considered. The communication links between the edges and the helper nodes are error-prone and are modeled as straggling/failing links. To overcome the issue of link failures, coding techniques are proposed. The edge nodes communicate encoded versions of the model updates to the helper nodes, which pass them on to the master after suitable aggregation. The primary work in this area uses repetition codes and Maximum Distance Separable (MDS) codes at the edge nodes to arrive at the Aligned Repetition Coding (ARC) and Aligned MDS Coding (AMC) schemes, respectively. We propose using vector codes, specifically a family of layered MDS codes parameterized by a variable $ν$, at the edge nodes. For the proposed family of codes, suitable aggregation strategies at the helper nodes are also developed. At the extreme values of $ν$, our scheme matches the communication costs incurred by the ARC and AMC schemes, resulting in a graceful transition between these schemes.

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arXiv2023

Sequential Gradient Coding For Straggler Mitigation

In distributed computing, slower nodes (stragglers) usually become a bottleneck. Gradient Coding (GC), introduced by Tandon et al., is an efficient technique that uses principles of error-correcting codes to distribute gradient computation in the presence of stragglers. In this paper, we consider the distributed computation of a sequence of gradients $\{g(1),g(2),\ldots,g(J)\}$, where processing of each gradient $g(t)$ starts in round-$t$ and finishes by round-$(t+T)$. Here $T\geq 0$ denotes a delay parameter. For the GC scheme, coding is only across computing nodes and this results in a solution where $T=0$. On the other hand, having $T>0$ allows for designing schemes which exploit the temporal dimension as well. In this work, we propose two schemes that demonstrate improved performance compared to GC. Our first scheme combines GC with selective repetition of previously unfinished tasks and achieves improved straggler mitigation. In our second scheme, which constitutes our main contribution, we apply GC to a subset of the tasks and repetition for the remainder of the tasks. We then multiplex these two classes of tasks across workers and rounds in an adaptive manner, based on past straggler patterns. Using theoretical analysis, we demonstrate that our second scheme achieves significant reduction in the computational load. In our experiments, we study a practical setting of concurrently training multiple neural networks over an AWS Lambda cluster involving 256 worker nodes, where our framework naturally applies. We demonstrate that the latter scheme can yield a 16\% improvement in runtime over the baseline GC scheme, in the presence of naturally occurring, non-simulated stragglers.

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arXiv2023

Explicit Information-Debt-Optimal Streaming Codes With Small Memory

For a convolutional code in the presence of a symbol erasure channel, the information debt $I(t)$ at time $t$ provides a measure of the number of additional code symbols required to recover all message symbols up to time $t$. Information-debt-optimal streaming ($i$DOS) codes are convolutional codes which allow for the recovery of all message symbols up to $t$ whenever $I(t)$ turns zero under the following conditions; (i) information debt can be non-zero for at most $τ$ consecutive time slots and (ii) information debt never increases beyond a particular threshold. The existence of periodically-time-varying $i$DOS codes are known for all parameters. In this paper, we address the problem of constructing explicit, time-invariant $i$DOS codes. We present an explicit time-invariant construction of $i$DOS codes for the unit memory ($m=1$) case. It is also shown that a construction method for convolutional codes due to Almeida et al. leads to explicit time-invariant $i$DOS codes for all parameters. However, this general construction requires a larger field size than the first construction for the $m=1$ case.

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arXiv2022

Adaptive relaying for streaming erasure codes in a three node relay network

This paper investigates adaptive streaming codes over a three-node relayed network. In this setting, a source node transmits a sequence of message packets to a destination through a relay. The source-to-relay and relay-to-destination links are unreliable and introduce at most $N_1$ and $N_2$ packet erasures, respectively. The destination node must recover each message packet within a strict delay constraint $T$. The paper presents achievable streaming codes for all feasible parameters $\{N_1, N_2, T\}$ that exploit the fact that the relay naturally observes the erasure pattern occurring in the link from source to relay, thus it can adapt its relaying strategy based on these observations. In a recent work, Fong et al. provide streaming codes featuring channel-state-independent relaying strategies. The codes proposed in this paper achieve rates higher than the ones proposed by Fong et al. whenever $N_2 > N_1$, and achieve the same rate when $N_2 = N_1$. The paper also presents an upper bound on the achievable rate that takes into account erasures in both links in order to bound the rate in the second link. The upper bound is shown to be tighter than a trivial bound that considers only the erasures in the second link.

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arXiv2021

Explicit Rate-Optimal Streaming Codes with Smaller Field Size

Streaming codes are a class of packet-level erasure codes that ensure packet recovery over a sliding window channel which allows either a burst erasure of size $b$ or $a$ random erasures within any window of size $(τ+1)$ time units, under a strict decoding-delay constraint $τ$. The field size over which streaming codes are constructed is an important factor determining the complexity of implementation. The best known explicit rate-optimal streaming code requires a field size of $q^2$ where $q \ge τ+b-a$ is a prime power. In this work, we present an explicit rate-optimal streaming code, for all possible $\{a,b,τ\}$ parameters, over a field of size $q^2$ for prime power $q \ge τ$. This is the smallest-known field size of a general explicit rate-optimal construction that covers all $\{a,b,τ\}$ parameter sets. We achieve this by modifying the non-explicit code construction due to Krishnan et al. to make it explicit, without change in field size.

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arXiv2020

Codes for Distributed Storage

This chapter deals with the topic of designing reliable and efficient codes for the storage and retrieval of large quantities of data over storage devices that are prone to failure. For long, the traditional objective has been one of ensuring reliability against data loss while minimizing storage overhead. More recently, a third concern has surfaced, namely of the need to efficiently recover from the failure of a single storage unit, corresponding to recovery from the erasure of a single code symbol. We explain here, how coding theory has evolved to tackle this fresh challenge.

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arXiv2020

Staggered Diagonal Embedding Based Linear Field Size Streaming Codes

An $(a,b,τ)$ streaming code is a packet-level erasure code that can recover under a strict delay constraint of $τ$ time units, from either a burst of $b$ erasures or else of $a$ random erasures, occurring within a sliding window of time duration $w$. While rate-optimal constructions of such streaming codes are available for all parameters $\{a,b,τ,w\}$ in the literature, they require in most instances, a quadratic, $O(τ^2)$ field size. In this work, we make further progress towards field size reduction and present rate-optimal $O(τ)$ field size streaming codes for two regimes: (i) $gcd(b,τ+1-a)\ge a$ (ii) $τ+1 \ge a+b$ and $b \mod \ a \in \{0,a-1\}$.

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arXiv2019

Low Field-size, Rate-Optimal Streaming Codes for Channels With Burst and Random Erasures

In this paper, we design erasure-correcting codes for channels with burst and random erasures, when a strict decoding delay constraint is in place. We consider the sliding-window-based packet erasure model proposed by Badr et al., where any time-window of width $w$ contains either up to $a$ random erasures or an erasure burst of length at most $b$. One needs to recover any erased packet, where erasures are as per the channel model, with a strict decoding delay deadline of $τ$ time slots. Presently existing rate-optimal constructions in the literature require, in general, a field-size which grows exponential in $τ$, for a constant $\frac{a}τ$. In this work, we present a new rate-optimal code construction covering all channel and delay parameters, which requires an $O(τ^2)$ field-size. As a special case, when $(b-a)=1$, we have a field-size linear in $τ$. We also present three other constructions having linear field-size, under certain constraints on channel and decoding delay parameters. As a corollary, we obtain low field-size, rate-optimal convolutional codes for any given column distance and column span. Simulations indicate that the newly proposed streaming code constructions offer lower packet-loss probabilities compared to existing schemes, for selected instances of Gilbert-Elliott and Fritchman channels.

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arXiv2018

Erasure Coding for Distributed Storage: An Overview

In a distributed storage system, code symbols are dispersed across space in nodes or storage units as opposed to time. In settings such as that of a large data center, an important consideration is the efficient repair of a failed node. Efficient repair calls for erasure codes that in the face of node failure, are efficient in terms of minimizing the amount of repair data transferred over the network, the amount of data accessed at a helper node as well as the number of helper nodes contacted. Coding theory has evolved to handle these challenges by introducing two new classes of erasure codes, namely regenerating codes and locally recoverable codes as well as by coming up with novel ways to repair the ubiquitous Reed-Solomon code. This survey provides an overview of the efforts in this direction that have taken place over the past decade.

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arXiv2018

Codes with Combined Locality and Regeneration Having Optimal Rate, $d_{\text{min}}$ and Linear Field Size

In this paper, we study vector codes with all-symbol locality, where the local code is either a Minimum Bandwidth Regenerating (MBR) code or a Minimum Storage Regenerating (MSR) code. In the first part, we present vector codes with all-symbol MBR locality, for all parameters, that have both optimal minimum-distance and optimal rate. These codes combine ideas from two popular codes in the distributed storage literature, Product-Matrix codes and Tamo-Barg codes. In the second part which deals with codes having all-symbol MSR locality, we follow a Pairwise Coupling Transform-based approach to arrive at optimal minimum-distance and optimal rate, for a range of parameters. All the code constructions presented in this paper have a low field-size that grows linearly with the code-length $n$.

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arXiv2018

Rate-Optimal Streaming Codes for Channels with Burst and Isolated Erasures

Recovery of data packets from packet erasures in a timely manner is critical for many streaming applications. An early paper by Martinian and Sundberg introduced a framework for streaming codes and designed rate-optimal codes that permit delay-constrained recovery from an erasure burst of length up to $B$. A recent work by Badr et al. extended this result and introduced a sliding-window channel model $\mathcal{C}(N,B,W)$. Under this model, in a sliding-window of width $W$, one of the following erasure patterns are possible (i) a burst of length at most $B$ or (ii) at most $N$ (possibly non-contiguous) arbitrary erasures. Badr et al. obtained a rate upper bound for streaming codes that can recover with a time delay $T$, from any erasure patterns permissible under the $\mathcal{C}(N,B,W)$ model. However, constructions matching the bound were absent, except for a few parameter sets. In this paper, we present an explicit family of codes that achieves the rate upper bound for all feasible parameters $N$, $B$, $W$ and $T$.

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arXiv2017

A Study on the Impact of Locality in the Decoding of Binary Cyclic Codes

In this paper, we study the impact of locality on the decoding of binary cyclic codes under two approaches, namely ordered statistics decoding (OSD) and trellis decoding. Given a binary cyclic code having locality or availability, we suitably modify the OSD to obtain gains in terms of the Signal-To-Noise ratio, for a given reliability and essentially the same level of decoder complexity. With regard to trellis decoding, we show that careful introduction of locality results in the creation of cyclic subcodes having lower maximum state complexity. We also present a simple upper-bounding technique on the state complexity profile, based on the zeros of the code. Finally, it is shown how the decoding speed can be significantly increased in the presence of locality, in the moderate-to-high SNR regime, by making use of a quick-look decoder that often returns the ML codeword.

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arXiv2016

Outer Bounds on the Storage-Repair Bandwidth Tradeoff of Exact-Repair Regenerating Codes

In this paper, three outer bounds on the normalized storage-repair bandwidth (S-RB) tradeoff of regenerating codes having parameter set $\{(n,k,d),(α,β)\}$ under the exact-repair (ER) setting are presented. The first outer bound is applicable for every parameter set $(n,k,d)$ and in conjunction with a code construction known as {\em improved layered codes}, it characterizes the normalized ER tradeoff for the case $(n,k=3,d=n-1)$. It establishes a non-vanishing gap between the ER and functional-repair (FR) tradeoffs for every $(n,k,d)$. The second bound is an improvement upon an existing bound due to Mohajer et al. and is tighter than the first bound, in a regime away from the Minimum Storage Regeneraing (MSR) point. The third bound is for the case of $k=d$, under the linear setting. This outer bound matches with the achievable region of {\em layered codes} thereby characterizing the normalized ER tradeoff of linear ER codes when $k=d=n-1$.

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arXiv2016

On MBR codes with replication

An early paper by Rashmi et. al. presented the construction of an $(n,k,d=n-1)$ MBR regenerating code featuring the inherent double replication of all code symbols and repair-by-transfer (RBT), both of which are important in practice. We first show that no MBR code can contain even a single code symbol that is replicated more than twice. We then go on to present two new families of MBR codes which feature double replication of all systematic message symbols. The codes also possess a set of $d$ nodes whose contents include the message symbols and which can be repaired through help-by-transfer (HBT). As a corollary, we obtain systematic RBT codes for the case $d=(n-1)$ that possess inherent double replication of all code symbols and having a field size of $O(n)$ in comparison with the general, $O(n^2)$ field size requirement of the earlier construction by Rashmi et. al. For the cases $(k=d=n-2)$ or $(k+1=d=n-2)$, the field size can be reduced to $q=2$, and hence the codes can be binary. We also give a necessary and sufficient condition for the existence of MBR codes having double replication of all code symbols and also suggest techniques which will enable an arbitrary MBR code to be converted to one with double replication of all code symbols.

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arXiv2015

The Storage-Repair-Bandwidth Trade-off of Exact Repair Linear Regenerating Codes for the Case $d = k = n-1$

In this paper, we consider the setting of exact repair linear regenerating codes. Under this setting, we derive a new outer bound on the storage-repair-bandwidth trade-off for the case when $d = k = n -1$, where $(n, k, d)$ are parameters of the regenerating code, with their usual meaning. Taken together with the achievability result of Tian et. al. [1], we show that the new outer bound derived here completely characterizes the trade-off for the case of exact repair linear regenerating codes, when $d = k = n -1$. The new outer bound is derived by analyzing the dual code of the linear regenerating code.

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arXiv2014

Evaluation of Codes with Inherent Double Replication for Hadoop

In this paper, we evaluate the efficacy, in a Hadoop setting, of two coding schemes, both possessing an inherent double replication of data. The two coding schemes belong to the class of regenerating and locally regenerating codes respectively, and these two classes are representative of recent advances made in designing codes for the efficient storage of data in a distributed setting. In comparison with triple replication, double replication permits a significant reduction in storage overhead, while delivering good MapReduce performance under moderate work loads. The two coding solutions under evaluation here, add only moderately to the storage overhead of double replication, while simultaneously offering reliability levels similar to that of triple replication. One might expect from the property of inherent data duplication that the performance of these codes in executing a MapReduce job would be comparable to that of double replication. However, a second feature of this class of code comes into play here, namely that under both coding schemes analyzed here, multiple blocks from the same coded stripe are required to be stored on the same node. This concentration of data belonging to a single stripe negatively impacts MapReduce execution times. However, much of this effect can be undone by simply adding a larger number of processors per node. Further improvements are possible if one tailors the Map task scheduler to the codes under consideration. We present both experimental and simulation results that validate these observations.

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