  Representing a sparse histogram, or more generally a sparse vector, is a
fundamental task in differential privacy. An ideal solution would use space
close to information-theoretical lower bounds, have an error distribution that
depends optimally on the desired privacy level, and allow fast random access to
entries in the vector. However, existing approaches have only achieved two of
these three goals.
  In this paper we introduce the Approximate Laplace Projection (ALP) mechanism
for approximating k-sparse vectors. This mechanism is shown to simultaneously
have information-theoretically optimal space (up to constant factors), fast
access to vector entries, and error of the same magnitude as the
Laplace-mechanism applied to dense vectors. A key new technique is a unary
representation of small integers, which we show to be robust against
``randomized response'' noise. This representation is combined with hashing, in
the spirit of Bloom filters, to obtain a space-efficient, differentially
private representation.
  Our theoretical performance bounds are complemented by simulations which show
that the constant factors on the main performance parameters are quite small,
suggesting practicality of the technique.
