基于对称区组设计的无条带纠删码的副本阈值

中文摘要 AI 辅助

本文研究了基于对称平衡不完全区组设计（SBIBD）的无条带纠删编码中的一个基本问题：每个对象究竟需要多少个副本才能保证从任意至多$p$个节点故障中恢复？我们细化了广义SBIBD $(v,k,\lambda)$ 构造的已知充分恢复保证。对于$p=1$，一个副本是必要且充分的；对于$p\ge2$，$q=\lambda(p-1)+2$个副本（$q\le k$）可保证恢复。对于$p=2$，恢复只需一轮，一般情况下至多两轮。为了研究该数量的紧致性，我们针对固定的零对角线SBIBD代表矩阵$A$定义了通用副本阈值$q^\ast(A,p)$。我们确定了$p=1$和$p=2$时的$q^\ast(A,p)$。对于$p\ge3$，我们识别出成对分离集和私有目标置换作为决定充分数量是否紧致或可进一步减少的结构。这些条件给出了除$\lambda>1$且$A$不包含大小为$p$的成对分离集之外的所有情况下的精确阈值。我们改进了剩余情况的界，并将其精确阈值留作开放问题。最后，我们给出了成对分离集的充分条件，并展示了仿射重标记如何实现私有目标置换。

英文摘要

This paper investigates a fundamental question in stripeless erasure coding based on symmetric balanced incomplete block designs (SBIBDs): how many replicas per object are precisely required to guarantee recovery from any set of at most $p$ node failures? We refine the known sufficient recovery guarantee for the generalized SBIBD $(v,k,λ)$ construction. One replica is necessary and sufficient for $p=1$, and $q=λ(p-1)+2$ replicas ($q\le k$) guarantee recovery for $p\ge2$. Recovery takes one round for $p=2$ and at most two rounds in general. To study the tightness of this count, we define the universal replica threshold $q^\ast(A,p)$ for a fixed zero-diagonal SBIBD representative~$A$. We determine $q^\ast(A,p)$ for $p=1$ and $p=2$. For $p\ge3$, we identify pairwise separated sets and private target permutations as the structures determining whether the sufficient count is tight or can be further reduced. These conditions give exact thresholds for all but the case where $λ>1$ and $A$ contains no pairwise separated set of size~$p$. We improve the bounds for this remaining case and leave its exact threshold open. Finally, we give sufficient conditions for pairwise separated sets and show how affinity relabeling can realize private target permutations.

发表机构

• IEIT SYSTEMS Co., Ltd.（IEIT系统有限公司）

机构由 AI 辅助整理，请以论文原文为准。