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Zhifang Zhang

Publications and source records attributed to Zhifang Zhang.

At least 37 records · Page 2Linked to original sources
arXiv2022

An Efficient Piggybacking Design Framework with Sub-packetization $l\le r$ for All-Node Repair

Piggybacking design has been widely applied in distributed storage systems since it can greatly reduce the repair bandwidth with small sub-packetization. Compared with other existing erasure codes, piggybacking is more convenient to operate and the I/O cost is lower. In this paper, we propose a new efficient design which can further reduce the repair bandwidth with the sub-packetization $l\le r$ where $r = n-k$. Generally, we let $l\le 8$. Compared with other analogous designs, our design has lower $l$ and the value of $l$ is more flexible. Moreover, our design can repair all nodes with small repair bandwidth. Therefore our piggybacking design is more feasible.

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arXiv2022

Rack-Aware Regenerating Codes with Multiple Erasure Tolerance

In a modern distributed storage system, storage nodes are organized in racks, and the cross-rack communication dominates the system bandwidth. In We study the rack-aware storage system where all storage nodes are organized in racks and within each rack the nodes can communicate freely without taxing the system bandwidth. Rack-aware regenerating codes (RRCs) were proposed for minimizing the repair bandwidth for single erasures. In the initial setting of RRCs, the repair of a single node requires the participation of all the remaining nodes in the rack containing the failed node as well as a large number of helper racks containing no failures. Consequently, the repair may be infeasible in front of multiple node failures. In this work, a relaxed repair model that can tolerate multiple node failures by simultaneously reducing the intra-rack connections and cross-rack connections is proposed. A tradeoff between the storage and repair bandwidth under the relaxed repair model is derived, and parameters of the two extreme points on the tradeoff curve are characterized for the minimum storage and minimum bandwidth respectively. Moreover, two codes corresponding to the extreme points are explicitly constructed over the fields of size comparable to the code length and with the lowest sub-packetization. Finally, for the convenience of practical use, systematic encoding processes for the two codes are also established.

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arXiv2021

Explicit Construction of Minimum Bandwidth Rack-Aware Regenerating Codes

In large data centers, storage nodes are organized in racks, and the cross-rack transmission dominates the bandwidth cost. For the repair of single node failures, codes achieving the tradeoff between the storage redundancy and cross-rack repair bandwidth are called rack-aware regenerating codes (RRCs). In this work, we give the first explicit construction of RRCs with the minimum repair bandwidth (i.e., the cross-rack bandwidth equals the storage size per node). Our construction applies to all admissible parameters and has the lowest sub-packetization level. Moreover, the underlying finite fields are of size comparable to the number of storage nodes, which makes our codes more implementable in practice. Finally, for the convenience of practical use, we also establish a transformation to convert our codes into systematic codes.

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arXiv2021

Rack-Aware Regenerating Codes with Fewer Helper Racks

We consider the rack-aware storage system where \(n\) nodes are organized in \(\bar{n}\) racks each containing \(u\) nodes, and any \(k\) nodes can retrieve the stored file. Moreover, any single node erasure can be recovered by downloading data from \(\bar{d}\) helper racks as well as the remaining \(u\!-\!1\) nodes in the same rack. Previous work mostly focuses on minimizing the cross-rack repair bandwidth under the condition \(\bar{d}\geq \bar{k}\), where \(\bar{k}=\lfloor\frac{k}{u}\rfloor\). However, \(\bar{d}\geq \bar{k}\) is not an intrinsic condition for the rack-aware storage model. In this paper, we establish a tradeoff between the storage overhead and cross-rack repair bandwidth for the particularly interesting case \(\bar{d}\!<\!\bar{k}\). Furthermore, we present explicit constructions of codes with parameters lying on the tradeoff curve respectively at the minimum storage point and minimum bandwidth point. The codes are scalar or have sub-packetization \(\bar{d}\), and operate over finite fields of size comparable to \(n\). Regarding \(\bar{d}\) as the repair degree, these codes combine the advantage of regenerating codes in minimizing the repair bandwidth and that of locally repairable codes in reducing the repair degree. Moreover, they also abandon the restriction of MBR codes having storage overhead no less than \(2\times\) and that of high-rate MSR codes having exponential sub-packetization level.

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arXiv2019

A Capacity-Achieving $T$-PIR Scheme Based On MDS Array Codes

Suppose a database containing $M$ records is replicated in each of $N$ servers, and a user wants to privately retrieve one record by accessing the servers such that identity of the retrieved record is secret against any up to $T$ servers. A scheme designed for this purpose is called a $T$-private information retrieval ($T$-PIR) scheme. In this paper we focus on the field size of $T$-PIR schemes. We design a generalcapacity-achieving $T$-PIR scheme whose queries are generated by using some {\rm MDS } array codes. It only requires field size $q\geq\sqrt[\ell]{N}$, where $\ell=\min\{t^{M-2},(n-t)^{M-2}\}$, $~t=T/{\rm gcd}(N,T)$,$~n=N/{\rm gcd}(N,T)$ and has the optimal sub-packetization $Nn^{M-2}$. Comparing with existing capacity-achieving $T$-PIR schemes, our scheme has the following advantage, that is, its field size monotonically decreases as the number of records $M$ grows. In particular, the binary field is sufficient for building a capacity-achieving T-PIR scheme as long as $M\geq 2+\lceil\log_μ\log_2N\rceil$, where $μ=\min\{t,n-t\}>1$.

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arXiv2019

On Sub-Packetization and Access Number of Capacity-Achieving PIR Schemes for MDS Coded Non-Colluding Servers

Consider the problem of private information retrieval (PIR) over a distributed storage system where $M$ records are stored across $N$ servers by using an $[N,K]$ MDS code. For simplicity, this problem is usually referred as the coded PIR problem. In 2016, Banawan and Ulukus designed the first capacity-achieving coded PIR scheme with sub-packetization $KN^{M}$ and access number $MKN^{M}$, where capacity characterizes the minimal download size for retrieving per unit of data, and sub-packetization and access number are two metrics closely related to implementation complexity. In this paper, we focus on minimizing the sub-packetization and the access number for linear capacity-achieving coded PIR schemes. We first determine the lower bounds on sub-packetization and access number, which are $Kn^{M-1}$ and $MKn^{M-1}$, respectively, in the nontrivial cases (i.e. $N\!>\!K\!\geq\!1$ and $M\!>\!1$), where $n\!=\!N/{\rm gcd}(N,K)$. We then design a general linear capacity-achieving coded PIR scheme to simultaneously attain these two bounds, implying tightness of both bounds.

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arXiv2019

An Improved Cooperative Repair Scheme for Reed-Solomon Codes

Dau et al. recently extend Guruswami and Wootters' scheme (STOC'2016) to cooperatively repair two or three erasures in Reed-Solomon (RS) codes. However, their scheme restricts to either the case that the characteristic of $F$ divides the extension degree $[F\!:\!B]$ or some special failure patterns, where $F$ is the base field of the RS code and $B$ is the subfield of the repair symbols. In this paper, we derive an improved cooperative repair scheme that removes all these restrictions. That is, our scheme applies to any characteristic of $F$ and can repair all failure patterns of two or three erasures.

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arXiv2018

Scalar MSCR Codes via the Product Matrix Construction

An $(n,k,d)$ cooperative regenerating code provides the optimal-bandwidth repair for any $t~(t\!>\!1)$ node failures in a cooperative way. In particular, an MSCR (minimum storage cooperative regenerating) code retains the same storage overhead as an $(n,k)$ MDS code. Suppose each node stores $α$ symbols which indicates the sub-packetization level of the code. A scalar MSCR code attains the minimum sub-packetization, i.e., $α=d-k+t$. By now, all existing constructions of scalar MSCR codes restrict to very special parameters, eg. $d=k$ or $k=2$, etc. In a recent work, Ye and Barg construct MSCR codes for all $n,k,d,t$, however, their construction needs $α\approx{\rm exp}(n^t)$ which is almost infeasible in practice. In this paper, we give an explicit construction of scalar MSCR codes for all $d\geq \max\{2k-1-t,k\}$, which covers all possible parameters except the case of $k\leq d\leq 2k-2-t$ when $k<2k-1-t$. Moreover, as a complementary result, for $k<d<2k-2-t$ we prove the nonexistence of linear scalar MSCR codes that have invariant repair spaces. Our construction and most of the previous scalar MSCR codes all have invariant repair spaces and this property is appealing in practice because of convenient repair. As a result, this work presents an almost full description of linear scalar MSCR codes.

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arXiv2018

Building Capacity-Achieving PIR Schemes with Optimal Sub-Packetization over Small Fields

Suppose a database containing $M$ records is replicated across $N$ servers, and a user wants to privately retrieve one record by accessing the servers such that identity of the retrieved record is secret against any up to $T$ servers. A scheme designed for this purpose is called a $T$-private information retrieval ($T$-PIR) scheme. Three indexes are concerned for PIR schemes: (1)rate, indicating the amount of retrieved information per unit of downloaded data. The highest achievable rate is characterized by the capacity; (2) sub-packetization, reflexing the implementation complexity for linear schemes; (3) field size. We consider linear schemes over a finite field. In this paper, a general $T$-PIR scheme simultaneously attaining the optimality of almost all of the three indexes is presented. Specifically, we design a linear capacity-achieving $T$-PIR scheme with sub-packetization $\!dn^{M-1}\!$ over a finite field $\mathbb{F}_q$, $q\geq N$. The sub-packetization $\!dn^{M-1}\!$, where $\!d\!=\!{\rm gcd}(N,T)\!$ and $\!n\!=\!N/d$, has been proved to be optimal in our previous work. The field size of all existing capacity-achieving $T$-PIR schemes must be larger than $Nt^{M-2}$ where $t=T/d$, while our scheme reduces the field size by an exponential factor.

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arXiv2017

Capacity-Achieving PIR Schemes with Optimal Sub-Packetization

Suppose a database containing $M$ records is replicated across $N$ servers, and a user wants to privately retrieve one record by accessing the servers such that identity of the retrieved record is secret against any up to $T$ servers. A scheme designed for this purpose is called a private information retrieval (PIR) scheme. In practice, capacity-achieving and small sub-packetization are both desired for PIR schemes, because the former implies the highest download rate and the latter usually means simple realization. For general values of $N,T,M$, the only known capacity-achieving PIR scheme was designed by Sun and Jafar in 2016 with sub-packetization $N^M$. In this paper, we design a linear capacity-achieving PIR scheme with much smaller sub-packetization $dn^{M-1}$, where $d={\rm gcd}(N,T)$ and $n=N/d$. Furthermore, we prove that for any linear capacity-achieving PIR scheme it must have sub-packetization no less than $dn^{M-1}$, implying our scheme has the optimal sub-packetization. Moreover, comparing with Sun and Jafar's scheme, our scheme reduces the field size by a factor of $\frac{1}{Nd^{M-2}}$.

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arXiv2017

Private Multi-File Retrieval From Distributed Databases

Suppose there are $N$ distributed databases each storing a full set of $M$ independent files. A user wants to retrieve $r$ out of the $M$ files without revealing the identity of the $r$ files. When $r=1$ it is the classic problem of private information retrieval (PIR). In this paper we study the problem of private multi-file retrieval (PMFR) which covers the case of general $r$. We first prove an upper bound on the capacity of PMFR schemes which indicates the minimum possible download size per unit of retrieved files. Then we design a general PMFR scheme which happens to attain the upper bound when $r\geq\frac{M}{2}$, thus achieving the optimal communication cost. As $r$ goes down we show the trivial approach of executing $r$ independent PIR instances achieves the near optimal communication cost. Comparing with the capacity-achieving PIR schemes, our PMFR scheme reduces the number of subpackages needed for each file from $N^M$ to $N^2$, which implies a great reduction of implementation complexity.

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arXiv2017

Bounds and Constructions for Linear Locally Repairable Codes over Binary Fields

For binary $[n,k,d]$ linear locally repairable codes (LRCs), two new upper bounds on $k$ are derived. The first one applies to LRCs with disjoint local repair groups, for general values of $n,d$ and locality $r$, containing some previously known bounds as special cases. The second one is based on solving an optimization problem and applies to LRCs with arbitrary structure of local repair groups. Particularly, an explicit bound is derived from the second bound when $d\geq 5$. A specific comparison shows this explicit bound outperforms the Cadambe-Mazumdar bound for $5\leq d\leq 8$ and large values of $n$. Moreover, a construction of binary linear LRCs with $d\geq6$ attaining our second bound is provided.

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arXiv2015

Achieving Arbitrary Locality and Availability in Binary Codes

The $i$th coordinate of an $(n,k)$ code is said to have locality $r$ and availability $t$ if there exist $t$ disjoint groups, each containing at most $r$ other coordinates that can together recover the value of the $i$th coordinate. This property is particularly useful for codes for distributed storage systems because it permits local repair and parallel accesses of hot data. In this paper, for any positive integers $r$ and $t$, we construct a binary linear code of length $\binom{r+t}{t}$ which has locality $r$ and availability $t$ for all coordinates. The information rate of this code attains $\frac{r}{r+t}$, which is always higher than that of the direct product code, the only known construction that can achieve arbitrary locality and availability.

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arXiv2014

An Integer Programming Based Bound for Locally Repairable Codes

The locally repairable code (LRC) studied in this paper is an $[n,k]$ linear code of which the value at each coordinate can be recovered by a linear combination of at most $r$ other coordinates. The central problem in this work is to determine the largest possible minimum distance for LRCs. First, an integer programming based upper bound is derived for any LRC. Then by solving the programming problem under certain conditions, an explicit upper bound is obtained for LRCs with parameters $n_1>n_2$, where $n_1 = \left\lceil \frac{n}{r+1} \right\rceil$ and $n_2 = n_1 (r+1) - n$. Finally, an explicit construction for LRCs attaining this upper bound is presented over the finite field $\mathbb{F}_{2^m}$, where $m\geq n_1r$. Based on these results, the largest possible minimum distance for all LRCs with $r \le \sqrt{n}-1$ has been definitely determined, which is of great significance in practical use.

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arXiv2014

Repair Locality From a Combinatorial Perspective

Repair locality is a desirable property for erasure codes in distributed storage systems. Recently, different structures of local repair groups have been proposed in the definitions of repair locality. In this paper, the concept of regenerating set is introduced to characterize the local repair groups. A definition of locality $r^{(δ-1)}$ (i.e., locality $r$ with repair tolerance $δ-1$) under the most general structure of regenerating sets is given. All previously studied locality turns out to be special cases of this definition. Furthermore, three representative concepts of locality proposed before are reinvestigated under the framework of regenerating sets, and their respective upper bounds on the minimum distance are reproved in a uniform and brief form. Additionally, a more precise distance bound is derived for the square code which is a class of linear codes with locality $r^{(2)}$ and high information rate, and an explicit code construction attaining the optimal distance bound is obtained.

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arXiv2013

Repair Locality with Multiple Erasure Tolerance

In distributed storage systems, erasure codes with locality $r$ is preferred because a coordinate can be recovered by accessing at most $r$ other coordinates which in turn greatly reduces the disk I/O complexity for small $r$. However, the local repair may be ineffective when some of the $r$ coordinates accessed for recovery are also erased. To overcome this problem, we propose the $(r,δ)_c$-locality providing $δ-1$ local repair options for a coordinate. Consequently, the repair locality $r$ can tolerate $δ-1$ erasures in total. We derive an upper bound on the minimum distance $d$ for any linear $[n,k]$ code with information $(r,δ)_c$-locality. For general parameters, we prove existence of the codes that attain this bound when $n\geq k(r(δ-1)+1)$, implying tightness of this bound. Although the locality $(r,δ)$ defined by Prakash et al provides the same level of locality and local repair tolerance as our definition, codes with $(r,δ)_c$-locality are proved to have more advantage in the minimum distance. In particular, we construct a class of codes with all symbol $(r,δ)_c$-locality where the gain in minimum distance is $Ω(\sqrt{r})$ and the information rate is close to 1.

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arXiv2012

Network Coding Based on Chinese Remainder Theorem

Random linear network code has to sacrifice part of bandwidth to transfer the coding vectors, thus a head of size k log|T| is appended to each packet. We present a distributed random network coding approach based on the Chinese remainder theorem for general multicast networks. It uses a couple of modulus as the head, thus reduces the size of head to O(log k). This makes it more suitable for scenarios where the number of source nodes is large and the bandwidth is limited. We estimate the multicast rate and show it is satisfactory in performance for randomly designed networks.

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