  Given a persistence diagram with $n$ points, we give an algorithm that
produces a sequence of $n$ persistence diagrams converging in bottleneck
distance to the input diagram, the $i$th of which has $i$ distinct (weighted)
points and is a $2$-approximation to the closest persistence diagram with that
many distinct points. For each approximation, we precompute the optimal
matching between the $i$th and the $(i+1)$st. Perhaps surprisingly, the entire
sequence of diagrams as well as the sequence of matchings can be represented in
$O(n)$ space. The main approach is to use a variation of the greedy permutation
of the persistence diagram to give good Hausdorff approximations and assign
weights to these subsets. We give a new algorithm to efficiently compute this
permutation, despite the high implicit dimension of points in a persistence
diagram due to the effect of the diagonal. The sketches are also structured to
permit fast (linear time) approximations to the Hausdorff distance between
diagrams -- a lower bound on the bottleneck distance. For approximating the
bottleneck distance, sketches can also be used to compute a linear-size
neighborhood graph directly, obviating the need for geometric data structures
used in state-of-the-art methods for bottleneck computation.
