inGRASS：基于低电阻直径分解的增量图谱稀疏化
inGRASS: Incremental Graph Spectral Sparsification via Low-Resistance-Diameter Decomposition
AI总结：
本文提出 inGRASS 增量图谱稀疏化算法，利用低电阻直径分解与低维电阻嵌入实现近线性初始化和 O(log N) 更新，实验中最高加速逾 200 倍且质量相当。
摘要：
本工作提出 inGRASS，这是一种为大型无向图的增量谱稀疏化设计的新型算法。所提出的 inGRASS 算法具有高度可扩展性和并行友好性，其设置阶段具有近线性时间复杂度，并且对于包含 N 个节点的原始图，每次增量变化都能在 O(log N) 时间内更新谱稀疏化器。inGRASS 设置阶段的一个关键组成部分是一个多层电阻嵌入框架，用于高效识别谱关键边并有效检测冗余边；这是通过利用低电阻直径分解（LRD）方案，将初始稀疏化器分解为许多具有有界有效电阻直径的节点簇来实现的。inGRASS 的更新阶段利用低维节点嵌入向量，高效估计每条新加入边的重要性和唯一性。大量实验表明，在对来自电路仿真、有限元分析和社交网络等各种数据集的图进行增量谱稀疏化时，inGRASS 实现了最高超过 200 倍的加速，同时保持了相当的求解质量。
英文摘要：
This work presents inGRASS, a novel algorithm designed for incremental spectral sparsification of large undirected graphs. The proposed inGRASS algorithm is highly scalable and parallel-friendly, having a nearly-linear time complexity for the setup phase and the ability to update the spectral sparsifier in $O(\log N)$ time for each incremental change made to the original graph with $N$ nodes. A key component in the setup phase of inGRASS is a multilevel resistance embedding framework introduced for efficiently identifying spectrally-critical edges and effectively detecting redundant ones, which is achieved by decomposing the initial sparsifier into many node clusters with bounded effective-resistance diameters leveraging a low-resistance-diameter decomposition (LRD) scheme. The update phase of inGRASS exploits low-dimensional node embedding vectors for efficiently estimating the importance and uniqueness of each newly added edge. As demonstrated through extensive experiments, inGRASS achieves up to over $200 \times$ speedups while retaining comparable solution quality in incremental spectral sparsification of graphs obtained from various datasets, such as circuit simulations, finite element analysis, and social networks.
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