改进的圆形$k$ -不匹配草图
Improved Circular $k$-Mismatch Sketches
摘要：
相同长度的两个字符串$S_1$和$S_2$之间的移位距离$\mathsf{sh}(S_1,S_2)$被定义为$S_1$和$S_2$的任何旋转（循环移位）之间的最小汉明距离。我们研究的问题，素描的移位距离，这是以下的通信复杂性问题：字符串$S_1$和$S_2$的长度$n$给两个相同的球员（编码器），谁独立计算草图（摘要）$\mathtt{sk}(S_1)$和$\mathtt{sk}(S_2)$，分别，以便在收到两个草图，第三个球员（解码器）能够计算（或近似）$\mathsf{sh}(S_1,S_2)$具有高概率。本文主要集中在更一般的$k$ -不匹配版本的问题，其中允许解码器宣布失败，如果$\mathsf{sh}(S_1,S_2)>k$，其中$k$是一个参数已知的所有各方。Andoni等人（STOC'13）介绍了精确的圆形$k$ -大小为$\widetilde{O}(k+D(n))$的失配草图，其中$D(n)$是$n$的约数。Andoni等人也证明了他们的草图尺寸在线性同态草图类中是最优的。我们绕过这个下限设计一个（非线性）精确的圆形$k$ -大小$\widetilde{O}(k)$的不匹配草图，这个大小匹配的通信复杂度下限。我们还设计了$(1\pm \varepsilon)$ -近似圆形$k$ -尺寸为$\widetilde{O}(\min(\varepsilon^{-2}\sqrt{k}, \varepsilon^{-1.5}\sqrt{n}))$的失配草图，其改进了Crouch和McGregor的$\widetilde{O}(\varepsilon^{-2}\sqrt{n})$尺寸的草图（近似'11）。
英文摘要：
The shift distance $\mathsf{sh}(S_1,S_2)$ between two strings $S_1$ and $S_2$ of the same length is defined as the minimum Hamming distance between $S_1$ and any rotation (cyclic shift) of $S_2$. We study the problem of sketching the shift distance, which is the following communication complexity problem: Strings $S_1$ and $S_2$ of length $n$ are given to two identical players (encoders), who independently compute sketches (summaries) $\mathtt{sk}(S_1)$ and $\mathtt{sk}(S_2)$, respectively, so that upon receiving the two sketches, a third player (decoder) is able to compute (or approximate) $\mathsf{sh}(S_1,S_2)$ with high probability. This paper primarily focuses on the more general $k$-mismatch version of the problem, where the decoder is allowed to declare a failure if $\mathsf{sh}(S_1,S_2)>k$, where $k$ is a parameter known to all parties. Andoni et al. (STOC'13) introduced exact circular $k$-mismatch sketches of size $\widetilde{O}(k+D(n))$, where $D(n)$ is the number of divisors of $n$. Andoni et al. also showed that their sketch size is optimal in the class of linear homomorphic sketches. We circumvent this lower bound by designing a (non-linear) exact circular $k$-mismatch sketch of size $\widetilde{O}(k)$; this size matches communication-complexity lower bounds. We also design $(1\pm \varepsilon)$-approximate circular $k$-mismatch sketch of size $\widetilde{O}(\min(\varepsilon^{-2}\sqrt{k}, \varepsilon^{-1.5}\sqrt{n}))$, which improves upon an $\widetilde{O}(\varepsilon^{-2}\sqrt{n})$-size sketch of Crouch and McGregor (APPROX'11).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
