Range-efficient consistent sampling and locality-sensitive hashing for polygons

Locality-sensitive hashing (LSH) is a fundamental technique for similarity search and similarity
estimation in high-dimensional spaces. The basic idea is that similar objects should produce hash
collisions with probability significantly larger than objects with low similarity. We consider LSH
for objects that can be represented as point sets in either one or two dimensions. To make the
point sets finite size we consider the subset of points on a grid. Directly applying LSH (e.g. minwise hashing) to these point sets would require time proportional to the number of points. We
seek to achieve time that is much lower than direct approaches.

Technically, we introduce new primitives for range-efficient consistent sampling (of independent
interest), and show how to turn such samples into LSH values. Another application of our
technique is a data structure for quickly estimating the size of the intersection or union of a set
of preprocessed polygons. Curiously, our consistent sampling method uses transformation to a
geometric problem.