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An optimal minimum spanning tree algorithm
 J. ACM
, 2000
"... Abstract. We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decisiontree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T ∗ (m, n)) where T ∗ is ..."
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Cited by 46 (10 self)
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Abstract. We establish that the algorithmic complexity of the minimum spanning tree problem is equal to its decisiontree complexity. Specifically, we present a deterministic algorithm to find a minimum spanning tree of a graph with n vertices and m edges that runs in time O(T ∗ (m, n)) where T ∗ is the minimum number of edgeweight 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) = �(m) and T ∗ (m, n) = O(m · α(m, n)), where α is a certain natural inverse of Ackermann’s function. Even under the assumption that T ∗ is superlinear, we show that if the input graph is selected from Gn,m, our algorithm runs in linear time with high probability, regardless of n, m, or the permutation of edge weights. The analysis uses a new martingale for Gn,m similar to the edgeexposure martingale for Gn,p.
A randomized timework optimal parallel algorithm for finding a minimum spanning forest
 SIAM J. COMPUT
, 1999
"... We present a randomized algorithm to find a minimum spanning forest (MSF) in an undirected graph. With high probability, the algorithm runs in logarithmic time and linear work on an exclusive read exclusive write (EREW) PRAM. This result is optimal w.r.t. both work and parallel time, and is the fi ..."
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Cited by 12 (3 self)
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We present a randomized algorithm to find a minimum spanning forest (MSF) in an undirected graph. With high probability, the algorithm runs in logarithmic time and linear work on an exclusive read exclusive write (EREW) PRAM. This result is optimal w.r.t. both work and parallel time, and is the first provably optimal parallel algorithm for this problem under both measures. We also give a simple, general processor allocation scheme for treelike computations.
Finding Minimum Spanning Trees in O(m α(m,n)) Time
, 1999
"... We describe a deterministic minimum spanning tree algorithm running in time O(m α(m; n)), where α is a natural inverse of Ackermann's function and m and n are the number of edges and vertices, respectively. This improves upon the O(m α(m; n) log α(m; n)) bound established by Chazelle in 1997. A sim ..."
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We describe a deterministic minimum spanning tree algorithm running in time O(m α(m; n)), where α is a natural inverse of Ackermann's function and m and n are the number of edges and vertices, respectively. This improves upon the O(m α(m; n) log α(m; n)) bound established by Chazelle in 1997. A similar O(m α(m; n))time algorithm was discovered independently by Chazelle, predating the algorithm presented here by many months. This paper may still be of interest for its alternative exposition.