Locality-sensitive hashing (LSH) has found widespread use as a fundamental primitive, particularly to accelerate nearest neighbor search. An LSH scheme for a similarity function $S:\mathcal{X} \times \mathcal{X} \to [0,1]$ is a distribution over hash functions on $\mathcal{X}$ with the property that the probability of collision of any two elements $x,y\in \mathcal{X}$ is exactly equal to $S(x,y)$. However, not all similarity functions admit exact LSH schemes. The notion of LSH distortion measures how multiplicatively close a similarity function is to having an LSH scheme. In this work, we study the LSH distortion of the Ulam and Cayley similarities, which are popular similarity measures on permutations of $n$ elements. We show that the Ulam similarity admits a sublinear LSH distortion of $O(n / \sqrt{\log n})$; we also prove a lower bound of $Ω(n^{0.12})$ on the best LSH distortion achievable. On the other hand, we show that the LSH distortion of the Cayley similarity is $Θ(n)$.

翻译：局部敏感哈希（LSH）作为一种基础性工具已得到广泛应用，特别在加速最近邻搜索中。对于相似函数$S:\mathcal{X} \times \mathcal{X} \to [0,1]$，LSH方案定义了$\mathcal{X}$上哈希函数的一个分布，其性质是任意两个元素$x,y\in \mathcal{X}$的碰撞概率恰好等于$S(x,y)$。然而，并非所有相似函数都存在精确LSH方案。LSH失真度量用于衡量某个相似函数与"存在LSH方案"这一性质之间的乘法逼近程度。本文研究了Ulam相似性和Cayley相似性的LSH失真——这两种相似性是针对$n$个元素排列的常用度量。我们证明Ulam相似性可实现$O(n / \sqrt{\log n})$的次线性LSH失真，同时证明了其可达的最佳LSH失真下界为$\Omega(n^{0.12})$。另一方面，我们证明Cayley相似性的LSH失真为$\Theta(n)$。