Fast Generation of Random Spanning Trees and the Effective Resistance Metric

Mądry, Aleksander; Straszak, Damian; Tarnawski, Jakub

doi:10.1137/1.9781611973730.134

Cited by 48 publications

(45 citation statements)

References 42 publications

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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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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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“…The effective resistance of an edge is a fundamental quantity. It and its extensions have a variety of connections in the analysis of networks [SM07,Sar10], combinatorics [Lov93,DFGX18] and the design of better graph algorithms [CKM + 11, MST15,Sch17].…”

Section: Estimating Effective-resistancesmentioning

confidence: 99%

Graph Sparsification, Spectral Sketches, and Faster Resistance Computation, via Short Cycle Decompositions

Chu

Gao

Peng

et al. 2018

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

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We develop a framework for graph sparsification and sketching, based on a new tool, short cycle decomposition -a decomposition of an unweighted graph into an edge-disjoint collection of short cycles, plus a small number of extra edges. A simple observation gives that every graph G on n vertices with m edges can be decomposed in O(mn) time into cycles of length at most 2 log n, and at most 2n extra edges. We give an m 1+o(1) time algorithm for constructing a short cycle decomposition, with cycles of length n o(1) , and n 1+o(1) extra edges. Both the existential and algorithmic variants of this decomposition enable us to make progress on several open problems in randomized graph algorithms.1. We present an algorithm that runs in time m 1+o(1) ε −1.5 and returns (1±ε)-approximations to effective resistances of all edges, improving over the previous best of O (min{mε −2 , n 2 ε −1 }). This routine in turn gives an algorithm to approximate the determinant of a graph Laplacian up to a factor of (1 ± ε) in m 1+o(1) + n 15 /8+o(1) ε − 7 /4 time.2. We show existence and efficient algorithms for constructing graphical spectral sketchesa distribution over sparse graphs H such that for a fixed vector x , we havewhere L is the graph Laplacian and L + is its pseudoinverse. This implies the existence of resistance-sparsifiers with about nε −1 edges that preserve the effective resistances between every pair of vertices up to (1 ± ε). By combining short cycle decompositions with known tools in graph sparsification, weshow the existence of nearly-linear sized degree-preserving spectral sparsifiers, as well as significantly sparser approximations of directed graphs. The latter is critical to recent breakthroughs on faster algorithms for solving linear systems in directed Laplacians.The running time and output qualities of our spectral sketch and degree-preserving (directed) sparsification algorithms are limited by the efficiency of our routines for constructing short cycle decompositions. Improved algorithms for short cycle decompositions will lead to improvements for each of these algorithms.2. What if we only want to preserve the effective resistance 2 between all pairs of vertices? Dinitz, Krauthgamer, and Wagner [DKW15] define such a graph H as a resistance sparsifier of G, and show their existence for regular expanders with degree Ω(n). They conjecture that every graph admits an ε-resistance sparsifier with O (nε −1 ) edges.3. An ε-spectral sparsifier preserves weighted vertex degrees up to (1±ε). Do there exist spectral sparsifiers that exactly preserve weighted degrees? Dinitz et al. [DKW15] also explicitly pose a related question -does every dense regular expander contain a sparse regular expander? 4. What about sparsification for directed graphs? The above sparsification notions, and algorithms are difficult to generalize to directed graphs. Cohen et al. [CKP + 16] developed a notion of sparsification for Eulerian directed graphs (directed graphs with all vertices having in-degree equal to out-degree), and gave the first a...

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“…This result was improved to O(V 2.373 ) by Colbourn et al [22,23], where the exponent corresponds to the fastest algorithm to compute matrix multiplication. Improvements on the random walk approach were obtained by Kelner and Mądry [24], and Mądry [25], culminating in anÕ(E o(1)+4/3 ) time algorithm by Madry et al [26], which relies on insight provided by the effective resistance metric. Interestingly, the initial work by Broader [16] contains a reference to the edge-swapping chain we presented in this paper (Section 5, named the swap chain).…”

Section: Related Workmentioning

confidence: 99%

Linking and Cutting Spanning Trees

2018

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Abstract:We consider the problem of uniformly generating a spanning tree for an undirected connected graph. This process is useful for computing statistics, namely for phylogenetic trees. We describe a Markov chain for producing these trees. For cycle graphs, we prove that this approach significantly outperforms existing algorithms. For general graphs, experimental results show that the chain converges quickly. This yields an efficient algorithm due to the use of proper fast data structures. To obtain the mixing time of the chain we describe a coupling, which we analyze for cycle graphs and simulate for other graphs.

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Fast Generation of Random Spanning Trees and the Effective Resistance Metric

Cited by 48 publications

References 42 publications

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

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

Graph Sparsification, Spectral Sketches, and Faster Resistance Computation, via Short Cycle Decompositions

Linking and Cutting Spanning Trees

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