5 research outputs found
A Fast Minimum Degree Algorithm and Matching Lower Bound
The minimum degree algorithm is one of the most widely-used heuristics for
reducing the cost of solving large sparse systems of linear equations. It has
been studied for nearly half a century and has a rich history of bridging
techniques from data structures, graph algorithms, and scientific computing. In
this paper, we present a simple but novel combinatorial algorithm for computing
an exact minimum degree elimination ordering in time, which improves on
the best known time complexity of and offers practical improvements
for sparse systems with small values of . Our approach leverages a careful
amortized analysis, which also allows us to derive output-sensitive bounds for
the running time of , where is
the number of unique fill edges and original edges that the algorithm
encounters and is the maximum degree of the input graph.
Furthermore, we show there cannot exist an exact minimum degree algorithm
that runs in time, for any , assuming
the strong exponential time hypothesis. This fine-grained reduction goes
through the orthogonal vectors problem and uses a new low-degree graph
construction called -fillers, which act as pathological inputs and cause any
minimum degree algorithm to exhibit nearly worst-case performance. With these
two results, we nearly characterize the time complexity of computing an exact
minimum degree ordering.Comment: 17 page
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