  Locality-sensitive hashing (LSH) is an important tool for managing
high-dimensional noisy or uncertain data, for example in connection with data
cleaning (similarity join) and noise-robust search (similarity search).
However, for a number of problems the LSH framework is not known to yield good
solutions, and instead ad hoc solutions have been designed for particular
similarity and distance measures. For example, this is true for
output-sensitive similarity search/join, and for indexes supporting annulus
queries that aim to report a point close to a certain given distance from the
query point.
  In this paper we initiate the study of distance-sensitive hashing (DSH), a
generalization of LSH that seeks a family of hash functions such that the
probability of two points having the same hash value is a given function of the
distance between them. More precisely, given a distance space $(X,
\text{dist})$ and a "collision probability function" (CPF) $f\colon
\mathbb{R}\rightarrow [0,1]$ we seek a distribution over pairs of functions
$(h,g)$ such that for every pair of points $x, y \in X$ the collision
probability is $\Pr[h(x)=g(y)] = f(\text{dist}(x,y))$. Locality-sensitive
hashing is the study of how fast a CPF can decrease as the distance grows. For
many spaces, $f$ can be made exponentially decreasing even if we restrict
attention to the symmetric case where $g=h$. We show that the asymmetry
achieved by having a pair of functions makes it possible to achieve CPFs that
are, for example, increasing or unimodal, and show how this leads to principled
solutions to problems not addressed by the LSH framework. This includes a novel
application to privacy-preserving distance estimation. We believe that the DSH
framework will find further applications in high-dimensional data management.
