  Motivated by the study of matrix elimination orderings in combinatorial
scientific computing, we utilize graph sketching and local sampling to give a
data structure that provides access to approximate fill degrees of a matrix
undergoing elimination in $O(\text{polylog}(n))$ time per elimination and
query. We then study the problem of using this data structure in the minimum
degree algorithm, which is a widely-used heuristic for producing elimination
orderings for sparse matrices by repeatedly eliminating the vertex with
(approximate) minimum fill degree. This leads to a nearly-linear time algorithm
for generating approximate greedy minimum degree orderings. Despite extensive
studies of algorithms for elimination orderings in combinatorial scientific
computing, our result is the first rigorous incorporation of randomized tools
in this setting, as well as the first nearly-linear time algorithm for
producing elimination orderings with provable approximation guarantees.
  While our sketching data structure readily works in the oblivious adversary
model, by repeatedly querying and greedily updating itself, it enters the
adaptive adversarial model where the underlying sketches become prone to
failure due to dependency issues with their internal randomness. We show how to
use an additional sampling procedure to circumvent this problem and to create
an independent access sequence. Our technique for decorrelating the interleaved
queries and updates to this randomized data structure may be of independent
interest.
