A discrete distribution p, over [n], is a k-histogram if its probability distribution function can be represented as a piece-wise constant function with k pieces. Such a function is represented by a list of k intervals and k corresponding values. We consider the following problem: given a collection of samples from a distribution p, find a k-histogram that (approximately) minimizes the \ell_2 distance to the distribution p. We give time and sample efficient algorithms for this problem.

We further provide algorithms that distinguish distributions that have the property of being a k-histogram from distributions that are \epsilon-far from any k-histogram in the \ell_1 distance and \ell_2 distance respectively.

A discrete distribution p, over [n], is a k-histogram if its probability distribution function can be
represented as a piece-wise constant function with k pieces. Such a function
is
represented by a list of k intervals and k corresponding values. We consider
the following problem: given a collection of samples from a distribution p,
find a k-histogram that (approximately) minimizes the \ell_2 distance to the
distribution p.
We give time and sample efficient algorithms for this problem.

We further provide algorithms that distinguish distributions that have the
property of being a k-histogram from distributions that are \eps-far from
any
k-histogram in the \ell_1 distance and \ell_2 distance respectively.