We initiate the study of fast dynamic algorithms for graph sparsification problems and obtain fully dynamic algorithms, allowing both edge insertions and edge deletions, that take polylogarithmic time after each update in the graph. Our three main results are as follows. First, we give a fully dynamic algorithm for maintaining a (1 ± )-spectral sparsifier with amortized update time poly(log n, −1 ). Second, we give a fully dynamic algorithm for maintaining a (1 ± )-cut sparsifier with worst-case update time poly(log n, −1 ). Both sparsifiers have size n · poly(log n, −1 ). Third, we apply our dynamic sparsifier algorithm to obtain a fully dynamic algorithm for maintaining a (1 + )-approximation to the value of the maximum flow in an unweighted, undirected, bipartite graph with amortized update time poly(log n, −1 ).

We show variants of spectral sparsification routines can preserve the total spanning tree counts of graphs, which by Kirchhoff's matrix-tree theorem, is equivalent to determinant of a graph Laplacian minor, or equivalently, of any SDDM matrix. Our analyses utilizes this combinatorial connection to bridge between statistical leverage scores / effective resistances and the analysis of random graphs by [Janson, Combinatorics, Probability and Computing '94]. This leads to a routine that in quadratic time, sparsifies a graph down to about n 1.5 edges in ways that preserve both the determinant and the distribution of spanning trees (provided the sparsified graph is viewed as a random object). Extending this algorithm to work with Schur complements and approximate Choleksy factorizations leads to algorithms for counting and sampling spanning trees which are nearly optimal for dense graphs.We give an algorithm that computes a (1 ± δ) approximation to the determinant of 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 computing determinants of arbitrary matrices. We also give an algorithm that generates in about n 2 δ −2 time a spanning tree of a weighted undirected graph from a distribution with total variation distance of δ from the w -uniform distribution .

We study dynamic algorithms for maintaining spectral vertex sparsifiers of graphs with respect to a set of terminals T of our choice. Such objects preserve pairwise resistances, solutions to systems of linear equations, and energy of electrical flows between the terminals in T . We give a data structure that supports insertions and deletions of edges, and terminal additions, all in sublinear time. We then show the applicability of our result to the following problems.(1) A data structure for dynamically maintaining solutions to Laplacian systems Lx = b, where L is the graph Laplacian matrix and b is a demand vector. For a bounded degree, unweighted graph, we support modifications to both L and b while providing access to ǫapproximations to the energy of routing an electrical flow with demand b, as well as query access to entries of a vectorx such that x − L † b L ≤ ǫ L † b L inÕ(n 11/12 ǫ −5 ) expected amortized update and query time.(2) A data structure for maintaining fully dynamic All-Pairs Effective Resistance. For an intermixed sequence of edge insertions, deletions, and resistance queries, our data structure returns (1 ± ǫ)-approximation to all the resistance queries against an oblivious adversary with high probability. Its expected amortized update and query times areÕ(min(m 3/4 , n 5/6 ǫ −2 )ǫ −4 ) on an unweighted graph, andÕ(n 5/6 ǫ −6 ) on weighted graphs.The key ingredients in these results are (1) the intepretation of Schur complement as a sum of random walks, and (2) a suitable choice of terminals based on the behavior of these random walks to make sure that the majority of walks are local, even when the graph itself is highly connected and (3) maintenance of these local walks and numerical solutions using data structures.These results together represent the first data structures for maintaining key primitives from the Laplacian paradigm for graph algorithms in sublinear time without assumptions on the underlying graph topologies. The importance of routines such as effective resistance, electrical flows, and Laplacian solvers in the static setting make us optimistic that some of our components can provide new building blocks for dynamic graph algorithms.

We present an algorithm that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in O(n 4/3 m 1/2 + n 2 ) time 1 . The tree is sampled from a distribution where the probability of each tree is proportional to the product of its edge weights. This improves upon the previous best algorithm due to Colbourn et al. that runs in matrix multiplication time, O(n ω ). For the special case of unweighted graphs, this improves upon the best previously known running time ofÕ(min{n ω , m √ n, m 4/3 }) for m n 5/3 (Colbourn et al. '96, Kelner-Madry '09, Madry et al. '15).The effective resistance metric is essential to our algorithm, as in the work of Madry et al., but we eschew determinant-based and random walk-based techniques used by previous algorithms. Instead, our algorithm is based on Gaussian elimination, and the fact that effective resistance is preserved in the graph resulting from eliminating a subset of vertices (called a Schur complement). As part of our algorithm, we show how to compute -approximate effective resistances for a set S of vertex pairs via approximate Schur complements in O(m + (n + |S|) −2 ) time, without using the Johnson-Lindenstrauss lemma which requires O(min{(m + |S|) −2 , m + n −4 + |S| −2 }) time. We combine this approximation procedure with an error correction procedure for handing edges where our estimate isn't sufficiently accurate.