An Optimal Minimum Spanning Tree Algorithm
We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decision-tree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning forest of a graph with n vertices and m edges that runs in time O(T(m,n)) where T is the minimum number of edge-weight comparisons needed to determine the solution. The algorithm is quite simple and can be implemented on a pointer machine.
Although our time bound is optimal, the exact function describing it is not known at present. The current best bounds known for T are T(m,n)=Ω(m) and T(m,n)=O(itm·α(m,n)), where α is a certain natural inverse of Ackermann’s function.
Even under the assumption that T is super-linear, we show that if the input graph is selected from G n,m, our algorithm runs in linear time w.h.p., regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for G n,m similar to the edge-exposure martingale for G n,p.
KeywordsGraph algorithms minimum spanning tree optimal complexity
Unable to display preview. Download preview PDF.
- Bor26.O. Borüvka. O jistém problému minimaálním. Moravské Přírodovědecké Společnosti 3, pp. 37–58, 1926. (In Czech).Google Scholar
- Chaz97.B. Chazelle. A faster deterministic algorithm for minimum spanning trees. In FOCS’ 97, pp. 22–31, 1997.Google Scholar
- Chaz98.B. Chazelle. Car-pooling as a data structuring device: The soft heap. In ESA’ 98(Venice), pp. 35–42, LNCS 1461, Springer, 1998.Google Scholar
- Chaz99.B. Chazelle. A minimum spanning tree algorithm with inverse-Ackermann type complexity. NECI Technical Report 99-099, 1999.Google Scholar
- CHL99.K. W. Chong, Y. Han and T. W. Lam. On the parallel time complexity of undirected connectivity and minimum spanning trees. In Proc. SODA, pp. 225–234, 1999.Google Scholar
- FW90.M. Fredman, D. E. Willard. Trans-dichotomous algorithms for minimum spanning trees and shortest paths. In Proc. FOCS’ 90, pp. 719–725, 1990.Google Scholar
- Jar30.V. Jarník. O jistém problému minimaálním. Moravské Přírodovědecké Společnosti 6, pp. 57–63, 1930. (In Czech).Google Scholar
- PR99.S. Pettie, V. Ramachandran. A randomized time-work optimal parallel algorithm for finding a minimum spanning forest Proc. RANDOM’ 99, LNCS 1671, Springer, pp. 233–244, 1999.Google Scholar
- PR99b.S. Pettie, V. Ramachandran. An optimal minimum spanning tree algorithm. Tech Report TR99-17, Univ. of Texas at Austin, 1999.Google Scholar
- Pet99.S. Pettie. Finding minimum spanning trees in O(mα(m, n)) time. Tech Report TR99-23, Univ. of Texas at Austin, 1999.Google Scholar
- Prim57.R. C. Prim. Shortest connection networks and some generalizations. Bell System Technical Journal, 36:1389–1401, 1957.Google Scholar