An almost-linear time algorithm for uniform random spanning tree generation

Schild, Aaron

doi:10.1145/3188745.3188852

Cited by 49 publications

(33 citation statements)

References 26 publications

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“…Random spanning trees. Algorithms for sampling random spanning trees have a long history, but only recently have they explicitly used matrix concentration [DKP + 17, DPPR17,Sch18]. The matrix concentration arguments in these papers, however, deal mostly with how modifying a graph results in changes to the distribution of random spanning trees in the graph.…”

Section: Establishes a Related Results For K-homogeneousmentioning

confidence: 99%

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A Matrix Chernoff Bound for Strongly Rayleigh Distributions and Spectral Sparsifiers from a few Random Spanning Trees

Kyng

Song

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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Strongly Rayleigh distributions are a class of negatively dependent distributions of binaryvalued random variables [Borcea, Brändén, Liggett JAMS 09]. Recently, these distributions have played a crucial role in the analysis of algorithms for fundamental graph problems, e.g. Traveling Salesman Problem [Gharan, Saberi, Singh FOCS 11]. We prove a new matrix Chernoff bound for Strongly Rayleigh distributions.As an immediate application, we show that adding together the Laplacians of ǫ −2 log 2 n random spanning trees gives an (1 ± ǫ) spectral sparsifiers of graph Laplacians with high probability. Thus, we positively answer an open question posed in [Baston, Spielman, Srivastava, Teng JACM 13]. Our number of spanning trees for spectral sparsifier matches the number of spanning trees required to obtain a cut sparsifier in [Fung, Hariharan, Harvey, Panigraphi STOC 11]. The previous best result was by naively applying a classical matrix Chernoff bound which requires ǫ −2 n log n spanning trees. For the tree averaging procedure to agree with the original graph Laplacian in expectation, each edge of the tree should be reweighted by the inverse of the edge leverage score in the original graph. We also show that when using this reweighting of the edges, the Laplacian of single random tree is bounded above in the PSD order by the original graph Laplacian times a factor log n with high probability, i.e. L T O(log n)L G .We show a lower bound that almost matches our last result, namely that in some graphs, with high probability, the random spanning tree is not bounded above in the spectral order by log n log log n times the original graph Laplacian. We also show a lower bound that in ǫ −2 log n spanning trees are necessary to get a (1 ± ǫ) spectral sparsifier.

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Section: Establishes a Related Results For K-homogeneousmentioning

confidence: 99%

“…Algorithms for sampling of random spanning trees have been studied extensively, [Gue83, Bro89, Ald90, Kul90, Wil96, CMN96, KM09, MST15, HX16, DKP + 17, DPPR17,Sch18], and a random spanning tree can now be sampled in almost linear time [Sch18].…”

Section: Introductionmentioning

confidence: 99%

A Matrix Chernoff Bound for Strongly Rayleigh Distributions and Spectral Sparsifiers from a few Random Spanning Trees

Kyng

Song

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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“…For efficiency purposes, SubspaceSparsifier(G, S, ǫ) receives martingale-difference-minimizing edges from a steady oracle O with the additional guarantee that differences remain small after many edge updates. This oracle is similar to the stable oracles given in Section 10 of [Sch17]. and outputs a set Z ⊆ E(I).…”

Section: Existence Of Sparsifiersmentioning

confidence: 99%

Spectral Subspace Sparsification

Schild

2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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We introduce a new approach to spectral sparsification that approximates the quadratic form of the pseudoinverse of a graph Laplacian restricted to a subspace. We show that sparsifiers with a near-linear number of edges in the dimension of the subspace exist. Our setting generalizes that of Schur complement sparsifiers. Our approach produces sparsifiers by sampling a uniformly random spanning tree of the input graph and using that tree to guide an edge elimination procedure that contracts, deletes, and reweights edges. In the context of Schur complement sparsifiers, our approach has two benefits over prior work. First, it produces a sparsifier in almost-linear time with no runtime dependence on the desired error. We directly exploit this to compute approximate effective resistances for a small set of vertex pairs in faster time than prior work (Durfee-Kyng-Peebles-Rao-Sachdeva '17). Secondly, it yields sparsifiers that are reweighted minors of the input graph. As a result, we give a near-optimal answer to a variant of the Steiner point removal problem.A key ingredient of our algorithm is a subroutine of independent interest: a nearlinear time algorithm that, given a chosen set of vertices, builds a data structure from which we can query a multiplicative approximation to the decrease in the effective resistance between two vertices after identifying all vertices in the chosen set to a single vertex with inverse polynomial additional additive error in near-constant time.

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“…The seminal work of Spielman and Teng [ST04] gave the first nearly-linear time algorithm for solving the weighted ℓ 2 version to highaccuracy (a (1+ε)-approximate solution in time O(m·log 1 ε ) 1 ). The work of Spielman-Teng and the several followup-works have led to the fastest algorithms for maximum matching [Mad13], shortest paths with negative weights [Coh+17b], graph partitioning [OSV12], sampling random spanning trees [KM09; MST15;Sch18], matrix scaling [Coh+17a; All+17], and resulted in dramatic progress on the problem of computing maximum flows.…”

Section: Introductionmentioning

confidence: 99%

Faster p-norm minimizing flows, via smoothed q-norm problems

Adil¹,

Sachdeva²

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We present faster high-accuracy algorithms for computing ℓ p -norm minimizing flows. On a graph with m edges, our algorithm can compute a (1 + 1/poly(m))-approximate unweighted ℓ p -norm minimizing flow with pm 1+ 1 p−1 +o(1) operations, for any p ≥ 2, giving the best bound for all p 5.24. Combined with the algorithm from the work of Adil et al. (SODA '19), we can now compute such flows for any 2 ≤ p ≤ m o(1) in time at most O(m 1.24 ). In comparison, the previous best running time was Ω(m 1.33 ) for large constant p. For p ∼ δ −1 log m, our algorithm computes a (1+δ)-approximate maximum flow on undirected graphs using m 1+o(1) δ −1 operations, matching the current best bound, albeit only for unit-capacity graphs.We also give an algorithm for solving general ℓ p -norm regression problems for large p. Our algorithm makes pm 1 3 +o(1) log 2 (1/ε) calls to a linear solver. This gives the first high-accuracy algorithm for computing weighted ℓ p -norm minimizing flows that runs in time o(m 1.5 ) for some p = m Ω(1) .Our key technical contribution is to show that smoothed ℓ p -norm problems introduced by Adil et al., are interreducible for different values of p. No such reduction is known for standard ℓ p -norm problems. an iterative refinement scheme for ℓ p -norm regression, giving a running time of O(p O(p) ·m p−2 3p−2 +1 ) ≤ O(p O(p) ·m 4 /3 ). Building on the work of Adil et al., Kyng et al.. [Kyn+19] designed an algorithmApproximating Max-Flow. For p ≥ log m, ℓ p norms approximate ℓ ∞ , and hence the above algorithm returns an approximate maximum-flow. For p = Θ log m δ , this gives a m 1+o(1) δ −1operations algorithm for computing a (1 + δ)-approximation to maximum-flow problem on unitcapacity graphs.Corollary 1.2. Given an (undirected) graph G with m edges with unit capacities, a demand vector d , and δ > 0, we can compute a flow f that satisfies the demands, i.e., B ⊤ f = d such thatThis gives another approach for approximating maximum flow with a δ −1 dependence on the approximation achieved in the recent works of Sherman [She17] and Sidford-Tian [ST18], albeit only for unit-capacity graphs, and with a m o(1) factor instead of poly(log m). To compute maxflow essentially exactly on unit-capacity graphs, one needs to compute p-norm minimizing flows for p = m.

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An almost-linear time algorithm for uniform random spanning tree generation

Cited by 49 publications

References 26 publications

A Matrix Chernoff Bound for Strongly Rayleigh Distributions and Spectral Sparsifiers from a few Random Spanning Trees

A Matrix Chernoff Bound for Strongly Rayleigh Distributions and Spectral Sparsifiers from a few Random Spanning Trees

Spectral Subspace Sparsification

Faster p-norm minimizing flows, via smoothed q-norm problems

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