Abstract—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 polylogarithmic 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 (approx imate) minimum fill degree. This leads to a nearlylinear time algorithm for generating approximate greedy minimum degree orderings. Despite extensive studies ofmore »
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 O(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 widelyused heuristic for producing elimination orderings for sparse matrices by repeatedly eliminating the vertex with (approximate) minimum fill degree. This leads to a nearlylinear 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 nearlylinear 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 more »
 Award ID(s):
 1637566
 Publication Date:
 NSFPAR ID:
 10113891
 Journal Name:
 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)
 Page Range or eLocationID:
 101 to 112
 Sponsoring Org:
 National Science Foundation
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