Stuffed IBLTs: 最优线性多重集草图
Stuffed IBLTs: Optimal Linear Multiset Sketches
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中文总结 AI 辅助
本文提出Stuffed IBLT线性草图，在容量和多重性限制下实现高概率精确恢复，空间接近信息论最优，常数时间更新，线性时间解码，改进现有权衡。
中文摘要 AI 辅助
线性草图是将向量 $v$ 随机线性映射到低维草图向量的方法，旨在保留关于 $v$ 的相关信息。我们考虑针对 $u \in N$ 的向量 $v \in Z^u$ 的草图，设计用于从草图中精确恢复 $v$。具体而言，我们的 Stuffed IBLT 是一种线性草图，配置容量 $n \in N$ 和多重性限制 $L \in N$，当 $||v||_0 \leq n$ 且 $||v||_\infty \leq L$ 时，能以高概率恢复 $v$。该草图可在对 $v$ 进行无限制更新的情况下高效维护，即 $v$ 在两次解码请求之间不受任何约束。这使得该草图适用于流式算法和解决（多重）集对账问题。对于任意正常数 $c$、$\epsilon$，且当 $n$ 足够大以及 $u \geq n^{1+\Omega(1)}$ 时，Stuffed IBLT 的空间使用量在信息论最优值的 $1+\epsilon$ 因子范围内，同时允许常数时间的更新，解码时间为 $O(n)$，失败概率为 $n^{-c}$。这改进了所有先前具有类似功能的构造（包括可逆布隆查找表（IBLT））在空间/时间/错误概率上的权衡。Stuffed IBLT 的性能基本上是我们所能期望的最佳性能，除了对 $c$ 和 $\epsilon$ 的依赖。我们明确给出了对这些参数的依赖，并进一步证明了一个下界，表明在基于剥离（peeling）的方法类别中，对 $c$ 的依赖是最优的。我们的改进来自于 Walzer 的空间耦合技术（SODA '21）、Houen、Pagh 和 Walzer 的纯度启发式方法（SOSA '23）以及 backyarding 技术（Belazzougui、Kucherov 和 Walzer，ESA '24；Fleischhacker、Green Larsen、Obremski 和 Simkin，ICALP '24）的巧妙结合，从而消除了过去方法的瓶颈。
英文摘要
A \emph{linear sketch} is a randomized linear mapping of a vector $v$ to a lower dimensional sketch vector, designed to preserve relevant information about $v$. We consider sketches of vectors $v \in Z^u$ (for $u \in N$), designed for exact recovery of $v$ from its sketch. Concretely, our \emph{Stuffed IBLT} is a linear sketch configured with a capacity $n \in N$ and a multiplicity limit $L \in N$ and will recover $v$ with high probability whenever $||v||_0 \leq n$ and $||v||_\infty \leq L$. The sketch can be maintained efficiently under unrestricted updates to $v$, i.e., $v$ is not subject to any constraints in between decoding requests. This makes the sketch useful for streaming algorithms and for solving the (multi)set reconciliation problem. For any positive constants $c$, $ε$, and for large enough $n$ and $u \geq n^{1+Ω(1)}$, the space usage of a Stuffed IBLT is within a factor $1+ε$ from the information-theoretic optimum while allowing updates in constant time, and decoding in time $O(n)$ with failure probability $n^{-c}$. This improves the space/time/error probability trade-off over all prior constructions with similar functionality, including the Invertible Bloom Lookup Table (IBLT). The performance of the Stuffed IBLT is essentially the best we could hope for, up to the dependence on $c$ and $ε$. We make the dependence on these parameters explicit, and further show a lower bound demonstrating that the dependence on $c$ is optimal within the class of peeling-based approaches. Our improvement comes from a careful combination of Walzer's spatial coupling technique (SODA '21), the purity heuristic of Houen, Pagh, and Walzer (SOSA '23), and backyarding (Belazzougui, Kucherov, and Walzer, ESA '24; Fleischhacker, Green Larsen, Obremski, and Simkin, ICALP '24), allowing us to eliminate bottlenecks of past approaches.
发表机构
- Max Planck Institute for Informatics（马克斯·普朗克计算机科学研究所）
- University of Copenhagen（哥本哈根大学）
- Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）
机构由 AI 辅助整理，请以论文原文为准。
