Locality Sensitive Hashing (LSH) is an effective method of indexing a set of items to support efficient nearest neighbors queries in high-dimensional spaces. The basic idea of LSH is that similar items should produce hash collisions with higher probability than dissimilar items. We study LSH for (not necessarily convex) polygons, and use it to give efficient data structures for similar shape retrieval. Arkin et al. represent polygons by their "turning function" - a function which follows the angle between the polygon's tangent and the $ x $-axis while traversing the perimeter of the polygon. They define the distance between polygons to be variations of the $ L_p $ (for $p=1,2$) distance between their turning functions. This metric is invariant under translation, rotation and scaling (and the selection of the initial point on the perimeter) and therefore models well the intuitive notion of shape resemblance. We develop and analyze LSH near neighbor data structures for several variations of the $ L_p $ distance for functions (for $p=1,2$). By applying our schemes to the turning functions of a collection of polygons we obtain efficient near neighbor LSH-based structures for polygons. To tune our structures to turning functions of polygons, we prove some new properties of these turning functions that may be of independent interest. As part of our analysis, we address the following problem which is of independent interest. Find the vertical translation of a function $ f $ that is closest in $ L_1 $ distance to a function $ g $. We prove tight bounds on the approximation guarantee obtained by the translation which is equal to the difference between the averages of $ g $ and $ f $.

中文翻译：

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有效相似多边形检索的局部敏感哈希

局部敏感哈希（LSH）是索引一组项目的有效方法，以支持高维空间中有效的最近邻居查询。LSH的基本思想是，相似项比非相似项产生散列冲突的可能性更高。我们研究（不一定是凸的）多边形的LSH，并使用它为相似的形状检索提供有效的数据结构。Arkin等。表示多边形的“车削函数”，该函数遵循多边形的切线和$ x $轴之间的角度，同时遍历多边形的周长。它们将多边形之间的距离定义为其转弯函数之间的$ L_p $（对于$ p = 1,2 $）距离的变化。该指标在翻译后是不变的，旋转和缩放（以及在周界上选择初始点），因此可以很好地模拟形状相似的直观概念。我们针对函数的$ L_p $距离的几种变化（对于$ p = 1,2 $），开发并分析了LSH近邻数据结构。通过将我们的方案应用于多边形集合的车削函数，我们可以获得有效的基于近邻LSH的多边形结构。为了使我们的结构适应多边形的车削功能，我们证明了这些车削功能的一些新特性，这些特性可能是独立引起关注的。作为我们分析的一部分，我们解决了独立关注的以下问题。求出距函数L $$最近的函数L $ f $的垂直平移。