Abstract
We consider the problem of sampling a uniformly random spanning tree of a graph. This is a classic algorithmic problem for which several exact and approximate algorithms are known. Random spanning trees have several connections to Laplacian matrices; this leads to algorithms based on fast matrix multiplication. The best algorithm for dense graphs can produce a uniformly random spanning tree of an n-vertex graph in time \(O(n^{2.38})\). This algorithm is intricate and requires explicitly computing the LU-decomposition of the Laplacian.
We present a new algorithm that also runs in time \(O(n^{2.38})\) but has several conceptual advantages. First, whereas previous algorithms need to introduce directed graphs, our algorithm works only with undirected graphs. Second, our algorithm uses fast matrix inversion as a black-box, thereby avoiding the intricate details of the LU-decomposition.
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Notes
- 1.
\(P := I - \mathbf {1}\mathbf {1}^T/n\)
References
Aldous, D.: The random walk construction of uniform spanning trees and uniform labelled trees. SIAM J. Discrete Math. 3, 450–465 (1990)
Asadpour, A., Goemans, M., Madry, A., Gharan, S.O., Saberi, A.: An \({O}\)(log \(n\)/log log \(n\))-approximation algorithm for the asymmetric traveling salesman problem. In: Proceedings of SODA (2010)
Broder, A.: Generating random spanning trees. In: Proceedings of FOCS, pp. 442–447 (1989)
Bunch, J.R., Hopcroft, J.E.: Triangular factorization and inversion by fast matrix multiplication. Math. Comput. 28, 231–236 (1974)
Campbell, S.L., Meyer, C.D.: Generalized Inverses of Linear Transformations. SIAM (1973)
Chekuri, C., Vondrak, J., Zenklusen, R.: Dependent randomized rounding via exchange properties of combinatorial structures. In: Proceedings of FOCS (2010)
Colbourn, C.J., Debroni, B.M., Myrvold, W.J.: Estimating the coefficients of the reliability polynomial. Congr. Numer. 62, 217–223 (1988)
Colbourn, C.J., Day, R.P.J., Nel, L.D.: Unranking and ranking spanning trees of a graph. J. Algor. 10, 271–286 (1989)
Colbourn, C.J., Myrvold, W.J., Neufeld, E.: Two algorithms for unranking arborescences. J. Algor. 20, 268–281 (1996)
Gharan, S.O., Saberi, A., Singh, M.: A randomized rounding approach to the traveling salesman problem. In: Proceedings of FOCS (2011)
Goyal, N., Rademacher, L., Vempala, S.: Expanders via random spanning trees. In: Proceedings of SODA (2009)
Guénoche, A.: Random spanning tree. J. Algor. 4, 214–220 (1983)
Harvey, N.J.A.: Algebraic algorithms for matching and matroid problems. SIAM J. Comput. 39, 679–702 (2009)
Harvey, N.J.A., Olver, N.: Pipage rounding, pessimistic estimators and matrix concentration. In: Proceedings of SODA (2014)
Kelner, J.A., Madry, A.: Faster generation of random spanning trees. In: Proceedings of FOCS (2009)
Kirchhoff, G.: Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird. Ann. Phys. und Chem. 72, 497–508 (1847)
Koutis, I., Miller, G.L., Peng, R.: A fast solver for a class of linear systems. Commun. ACM 55(10), 99–107 (2012)
Kulkarni, V.G.: Generating random combinatorial objects. J. Algor. 11(2), 185–207 (1990)
Lyons, R., Peres, Y.: Probability on Trees and Networks. Cambridge University Press (in preparation). Current version available at http://pages.iu.edu/~rdlyons/
Madry, A., Straszak, D., Tarnawski, J.: Fast generation of random spanning trees and the effective resistance metric. In: Proceedings of SODA (2015)
Wilson, D.B.: Generating random spanning trees more quickly than the cover time. In: Proceedings of STOC (1996)
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Harvey, N.J.A., Xu, K. (2016). Generating Random Spanning Trees via Fast Matrix Multiplication. In: Kranakis, E., Navarro, G., Chávez, E. (eds) LATIN 2016: Theoretical Informatics. LATIN 2016. Lecture Notes in Computer Science(), vol 9644. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-49529-2_39
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DOI: https://doi.org/10.1007/978-3-662-49529-2_39
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