# 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} }

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

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## References

SHOWING 1-10 OF 49 REFERENCES

### Faster Generation of Random Spanning Trees

- Computer Science, Mathematics2009 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.

### Expanders via random spanning trees

- Computer Science, MathematicsSODA
- 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).

### Generating random spanning trees

- Computer Science, Mathematics30th 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.

### Generating random spanning trees more quickly than the cover time

- Computer ScienceSTOC '96
- 1996

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.

### A Randomized Rounding Approach to the Traveling Salesman Problem

- Computer Science, Mathematics2011 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.

### Random Walks on Graphs: a Survey

- Mathematics

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,…

### The electrical resistance of a graph captures its commute and cover times

- MathematicsSTOC '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.

### The cover time, the blanket time, and the Matthews bound

- Computer ScienceProceedings 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/).

### Random walks, universal traversal sequences, and the complexity of maze problems

- Computer Science20th 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.