9 research outputs found
Graph Sketching Against Adaptive Adversaries Applied to the Minimum Degree Algorithm
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 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.Comment: 58 pages, 3 figures. This is a substantially revised version of
arXiv:1711.08446 with an emphasis on the underlying theoretical problem
Covering and packing for pairs
When a v-set can be equipped with a set of k-subsets so that every 2-subset of the v-set appears in exactly (or at most, or at least) one of the chosen k-subsets, the result is a balanced incomplete block design (or packing, or covering, respectively). For each k, balanced incomplete block designs are known to exist for all sufficiently large values of v that meet certain divisibility conditions. When these conditions are not met, one can ask for the packing with the most blocks and/or the covering with the fewest blocks. Elementary necessary conditions furnish an upper bound on the number of blocks in a packing and a lower bound on the number of blocks in a covering. In this paper it is shown that for all sufficiently large values of v, a packing and a covering on v elements exist whose numbers of blocks differ from the basic bounds by no more than an additive constant depending only on k