Dynamic Matrix Inverse: Improved Algorithms and Matching Conditional Lower Bounds
Dynamic Matrix Inverse: Improved Algorithms and Matching Conditional Lower Bounds
复制标题
动态矩阵逆：改进的算法和匹配条件下界
DOI：
10.1109/focs.2019.00036
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Thatchaphol Saranurak
中科院分区：
文献类型：
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作者：
Jan van den Brand;Danupon Nanongkai;Thatchaphol Saranurak
动态矩阵逆问题是维持正在进行元素和列更新的矩阵的倒数。矩阵的决定因素并保持响应，距离，最大匹配大小和k-paths/cycles in agraph中。在本文中，我们（i）改进了动态矩阵逆的算法及其扩展到某些增量/look-aphap-apap-apap-apher的变体，以及（ii）在线矩阵 - 矢量猜想的变体[Henzinger〜Et〜15]。这意味着这些算法很紧，我们的算法会自动导致合理问题的动态算法，其中一些算法在我们的猜想下也很紧。大小（实际上，Abboud和V Williams [FOCS'14]询问了这两个问题的差距）。 07某些转换矩阵以前只能保持组合（即没有快速矩阵乘法）。但是，动态矩阵逆和一些相关的问题。进一步下降。
The dynamic matrix inverse problem is to maintain the inverse of a matrix undergoing element and column updates. It is the main subroutine behind the best algorithms for many dynamic problems whose complexity is not yet well-understood, such as maintaining the largest eigenvalue, rank and determinant of a matrix and maintaining reachability, distances, maximum matching size, and k-paths/cycles in a graph. Understanding the complexity of dynamic matrix inverse is a key to understand these problems. In this paper, we present (i) improved algorithms for dynamic matrix inverse and their extensions to some incremental/look-ahead variants, and (ii) variants of the Online Matrix-Vector conjecture [Henzinger~et~al. STOC'15] that, if true, imply that these algorithms are tight. Our algorithms automatically lead to faster dynamic algorithms for the aforementioned problems, some of which are also tight under our conjectures, e.g. reachability and maximum matching size (closing the gaps for these two problems was in fact asked by Abboud and V. Williams [FOCS'14]). Prior best bounds for most of these problems date back to more than a decade ago [Sankowski FOCS'04, COCOON'05, SODA'07; Kavitha FSTTCS'08; Mucha and Sankowski Algorithmica'10; Bosek et~al. FOCS'14]. Our improvements stem mostly from the ability to use fast matrix multiplication “one more time'', to maintain a certain transformation matrix which could be maintained only combinatorially previously (i.e. without fast matrix multiplication). Oddly, unlike other dynamic problems where this approach, once successful, could be repeated several times (“bootstrapping''), our conjectures imply that this is not the case for dynamic matrix inverse and some related problems. However, when a small additional “look-ahead'' information is provided we can perform such repetition to drive the bounds down further.
DOI：
10.1109/focs.2015.71
发表时间：
2015-04
期刊：
2015 IEEE 56th Annual Symposium on Foundations of Computer Science
影响因子：
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作者：
R. Clifford;A. Jørgensen;Kasper Green Larsen
通讯作者：
R. Clifford;A. Jørgensen;Kasper Green Larsen