Determinant-Preserving Sparsification of SDDM Matrices with Applications to Counting and Sampling Spanning Trees
Determinant-Preserving Sparsification of SDDM Matrices with Applications to Counting and Sampling Spanning Trees
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DOI：
10.1109/focs.2017.90
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发表时间：
2017-05
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影响因子：
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通讯作者：
D. Durfee;John Peebles;Richard Peng;Anup B. Rao
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文献类型：
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作者：
D. Durfee;John Peebles;Richard Peng;Anup B. Rao
我们显示的光谱稀疏例程的变体可以保留图形的盖树计数，该图由Kirchhoffs Matrix-Tree Theorem，其等效符号，可以决定laplacian Minor或等效于任何SDDM Matrix的图形。我们的分析利用这种组合连接到统计级别分数 /有效电阻之间的桥接和随机Graphsby的分析[Janson，Combinatorics，概率和计算94]。这导致了一个例程，在二次时期，将图形放在大约n^（1.5）边的情况下，以保持跨越树的行列式和分布的方式（前提是将稀疏的图视为随机对象）。将此算法扩展到与Schur进行补充和大约atecholesky的因素化的算法，导致计数和采样跨越树木的算法，这些算法几乎是密度图最佳的。概率约为n^2 /δ^2时间。这是图形的第一个例程，该图表优于计算任意矩阵的计算确定因子的通用例程。我们还提供了一种算法，该算法在大约n^2 /δ^2时生成一个从w-均匀分布的总变异δ的分布中的跨越未取向的图形。
We show variants of spectral sparsification routines can preserve the totalspanning tree counts of graphs, which by Kirchhoffs matrix-tree theorem, isequivalent to determinant of a graph Laplacian minor, or equivalently, of any SDDM matrix. Our analyses utilizes this combinatorial connection to bridge between statisticalleverage scores / effective resistances and the analysis of random graphsby [Janson, Combinatorics, Probability and Computing 94]. This leads to a routine that in quadratic time, sparsifies a graph down to aboutn^(1.5) edges in ways that preserve both the determinant and the distributionof spanning trees (provided the sparsified graph is viewed as a random object). Extending this algorithm to work with Schur complements and approximateCholesky factorizations leads to algorithms for counting andsampling spanning trees which are nearly optimal for dense graphs.We give an algorithm that computes a (1 +/- δ) approximation to the determinantof any SDDM matrix with constant probability in about n^2 / δ^2 time. This is the first routine for graphs that outperforms general-purpose routines for computingdeterminants of arbitrary matrices. We also give an algorithm that generates in about n^2 / δ^2 time a spanning tree ofa weighted undirected graph from a distribution with total variationdistance of δ from the w-uniform distribution.