# Fast Generation of Random Spanning Trees and the Effective Resistance Metric

@inproceedings{Madry2015FastGO,
title={Fast Generation of Random Spanning Trees and the Effective Resistance Metric},
author={Aleksander Madry and Damian Straszak and Jakub Tarnawski},
booktitle={ACM-SIAM Symposium on Discrete Algorithms},
year={2015}
}
• Published in
ACM-SIAM Symposium on…
1 January 2015
• Computer Science
We present a new algorithm for generating a uniformly random spanning tree in an undirected graph. Our algorithm samples such a tree in expected $\tilde{O}(m^{4/3})$ time. This improves over the best previously known bound of $\min(\tilde{O}(m\sqrt{n}),O(n^{\omega}))$ -- that follows from the work of Kelner and M\k{a}dry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] -- whenever the input graph is sufficiently sparse. At a high level, our result stems from carefully exploiting the…
59 Citations

## Figures from this paper

• Computer Science, Mathematics
STOC
• 2017
An algorithm is presented that, with high probability, generates a random spanning tree from an edge-weighted undirected graph in (n5/3 m1/3) time, 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).
• Computer Science, Mathematics
ArXiv
• 2018
A data-structure is given that maintains $(1+\epsilon)-approximations to all-pair effective resistances of a fully-dynamic unweighted, undirected multi-graph$G$with expected amortized update and query time, against an oblivious adversary. An m1+o(1)βo( 1)-time algorithm for generating uniformly random spanning trees in weighted graphs with max-to-min weight ratio β is given and it is shown that most random walk steps occur far away from an unvisited vertex. • Computer Science, Mathematics ESA • 2018 This work gives a fully dynamic algorithm that maintains$(1+\varepsilon)-approximations of the all-pairs effective resistances of an $n$-vertex graph undergoing edge insertions and deletions with worst-case update time and query time, and shows that this algorithm is close to optimal.
• Computer Science, Mathematics
2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
• 2016
This paper provides faster algorithms for computing various fundamental quantities associated with random walks on a directed graph, including the stationary distribution, personalized PageRank vectors, hitting times, and escape probabilities, and shows how to compute each quantity in time Õ(m3/4n + mn2/3), where the Ó notation suppresses polylog factors in n.
• Mathematics, Computer Science
ITCS
• 2022
It is proved that no algorithm with subconstant error given probe access to an input d-regular graph can have runtime better than Ω( √ n/ log(n) per query in expectation when the input graph is drawn from G(n, d), obtaining a nearly matching lower bound.
• Mathematics, Computer Science
2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)
• 2022
This work achieves the optimal limit on domain sparsification for strongly Rayleigh distributions and improves the state of the art for obtaining a single sample from a determinantal point process from the prior runtime of ${\widetilde{O}(k)$.
The Kelner-M ↪ adry algorithm (FOCS 2009) is presented which builds on these connections to sample a random spanning tree from the uniform distribution in Õ(m) time.
• Computer Science, Mathematics
ArXiv
• 2019
This work uses queueing networks to present a new approach to solving Laplacian systems that can be used to adapt the approach by Kelner and M\k{a}dry (2009) to give the first distributed algorithm to compute approximate random spanning trees efficiently.
• Computer Science
Electron. Colloquium Comput. Complex.
• 2020
An inherent tradeoff is demonstrated between the amount of (bounded) independence used in the edge sampling algorithm, denoted by $k$ above, and the resulting sparsity that can be achieved.

## References

SHOWING 1-10 OF 49 REFERENCES

• Computer Science, Mathematics
2009 50th Annual IEEE Symposium on Foundations of Computer Science
• 2009
A new approach to the problem that integrates discrete random walk-based techniques with continuous linear algebraic methods is introduced, and the use of electrical networks and sparse linear system solvers in conjunction with random walks and combinatorial partitioning techniques is used.
• Computer Science, Mathematics
SODA
• 2009
It is proved that for any bounded-degree n-vertex graph, the union of two random spanning trees approximates the expansion of every cut of the graph to within a factor of O(log n).
• A. Broder
• Computer Science, Mathematics
30th Annual Symposium on Foundations of Computer Science
• 1989
It is shown that the Markov chain on the space of all spanning trees of a given graph where the basic step is an edge swap is rapidly mixing.
This paper gives a new algorithm for generating random spanning trees of an undirected graph that is easy to code up, has small running time constants, and has a nice proof that it generates trees with the right probabilities.
• Computer Science, Mathematics
2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
• 2011
This work gives a (3/2-\eps_0)-approximation algorithm that finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or T-join) of that tree.
Dedicated to the marvelous random walk of Paul Erd} os through universities, c ontinents, and mathematics Various aspects of the theory of random walks on graphs are surveyed. In particular,
• Mathematics
STOC '89
• 1989
Known relations between random walks and electrical networks are extended by showing that resistance in this network is intimately connected with the lengths of random walks on the graph, and bounds on cover time obtained are better than those obtained from previous techniques such as the eigenvalues of the adjacency matrix.
• Computer Science
Proceedings 41st Annual Symposium on Foundations of Computer Science
• 2000
The upper and lower bounds and an approximation algorithm for the cover time of the random walk on a graph are proved and it is proved that M/2<C= O(M(lnlnn)/sup 2/).
• Computer Science
20th Annual Symposium on Foundations of Computer Science (sfcs 1979)
• 1979
Results are derived suggesting that the undirected reachability problem is structurally different from, and easier than, the directed version of NSPACE(logn), an affirmative answer to a question of S. Cook.