应用于最小度算法的抵抗自适应对手的图草图
Graph Sketching Against Adaptive Adversaries Applied to the Minimum Degree Algorithm
AI总结：
受矩阵消元顺序研究启发，本文利用图草图和局部采样构建数据结构，提出了生成近似贪心最小度消元顺序的近线性时间算法，并通过额外采样解决了自适应对手模型中的随机依赖问题。
摘要：
受组合科学计算中矩阵消元顺序研究的启发，我们利用图草图和局部采样构建了一种数据结构，该结构能在每次消元和查询 $O(\ ext{polylog}(n))$ 的时间内，提供对正在进行消元的矩阵的近似填充度访问。随后，我们研究了在最小度算法中使用该数据结构的问题，最小度算法是一种广泛使用的启发式方法，通过反复消除具有（近似）最小填充度的顶点来为稀疏矩阵生成消元顺序。这导出了一种生成近似贪心最小度顺序的近线性时间算法。尽管组合科学计算中的消元顺序算法已被广泛研究，但我们的结果是在该背景下首次严格引入随机化工具，也是首个具有可证明近似保证的产生消元顺序的近线性时间算法。虽然我们的草图数据结构在遗忘对手模型中能直接工作，但通过反复查询和贪心更新自身，它进入了自适应对手模型，在该模型中，底层草图由于与其内部随机性的依赖问题而容易失效。我们展示了如何使用额外的采样程序来规避此问题并创建独立的访问序列。我们对这种随机化数据结构中交错查询和更新的去相关技术可能具有独立的研究价值。
英文摘要：
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 $O(\text{polylog}(n))$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
