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 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 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.
Joint work with Matthew Fahrbach, Gary L. Miller, Saurabh Sawlani, Junxing Wang, and Shen Chen Xu. Manuscript available at:
2018-07-30 14:00 ~ 15:00
Richard Peng, Georgia Institute of Technology
Room 602, School of Information Management & Engineering, Shanghai University of Finance & Economics