  A conditional sampling oracle for a probability distribution D returns
samples from the conditional distribution of D restricted to a specified subset
of the domain. A recent line of work (Chakraborty et al. 2013 and Cannone et
al. 2014) has shown that having access to such a conditional sampling oracle
requires only polylogarithmic or even constant number of samples to solve
distribution testing problems like identity and uniformity. This significantly
improves over the standard sampling model where polynomially many samples are
necessary.
  Inspired by these results, we introduce a computational model based on
conditional sampling to develop sublinear algorithms with exponentially faster
runtimes compared to standard sublinear algorithms. We focus on geometric
optimization problems over points in high dimensional Euclidean space. Access
to these points is provided via a conditional sampling oracle that takes as
input a succinct representation of a subset of the domain and outputs a
uniformly random point in that subset. We study two well studied problems:
k-means clustering and estimating the weight of the minimum spanning tree. In
contrast to prior algorithms for the classic model, our algorithms have time,
space and sample complexity that is polynomial in the dimension and
polylogarithmic in the number of points.
  Finally, we comment on the applicability of the model and compare with
existing ones like streaming, parallel and distributed computational models.
