2 research outputs found
Improved Approximation Algorithms for Earth-Mover Distance in Data Streams
For two multisets and of points in , such that , the earth-mover distance (EMD) between and is the minimum cost
of a perfect bipartite matching with edges between points in and , i.e.,
, where
ranges over all one-to-one mappings. The sketching complexity of approximating
earth-mover distance in the two-dimensional grid is mentioned as one of the
open problems in the literature. We give two algorithms for computing EMD
between two multi-sets when the number of distinct points in one set is a small
value . Our first algorithm gives a
-approximation using space and works
only in the insertion-only model. The second algorithm gives a
-approximation using
-space in the turnstile model
Nearly-optimal bounds for sparse recovery in generic norms, with applications to -median sketching
We initiate the study of trade-offs between sparsity and the number of
measurements in sparse recovery schemes for generic norms. Specifically, for a
norm , sparsity parameter , approximation factor , and
probability of failure , we ask: what is the minimal value of so that
there is a distribution over matrices with the property that
for any , given , we can recover a -sparse approximation to in
the given norm with probability at least ? We give a partial answer to
this problem, by showing that for norms that admit efficient linear sketches,
the optimal number of measurements is closely related to the doubling
dimension of the metric induced by the norm on the set of all
-sparse vectors. By applying our result to specific norms, we cast known
measurement bounds in our general framework (for the norms, ) as well as provide new, measurement-efficient schemes (for the
Earth-Mover Distance norm). The latter result directly implies more succinct
linear sketches for the well-studied planar -median clustering problem.
Finally, our lower bound for the doubling dimension of the EMD norm enables us
to address the open question of [Frahling-Sohler, STOC'05] about the space
complexity of clustering problems in the dynamic streaming model.Comment: 29 page