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(tau*(m, n)) where tau* 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 tau* are tau*(m, n) = Omega(m) and tau*(m, n) = O(m . alpha(m, n)), where alpha is a certain natural inverse of Ackermann's function. Even under the assumption that tau* 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).