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Verification and Sensitivity Analysis Of Minimum Spanning Trees In Linear Time
 SIAM J. Comput
, 1992
"... . Koml'os has devised a way to use a linear number of binary comparisons to test whether a given spanning tree of a graph with edge costs is a minimum spanning tree. The total computational work required by his method is much larger than linear, however. We describe a lineartime algorithm for verif ..."
Abstract

Cited by 54 (2 self)
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. Koml'os has devised a way to use a linear number of binary comparisons to test whether a given spanning tree of a graph with edge costs is a minimum spanning tree. The total computational work required by his method is much larger than linear, however. We describe a lineartime algorithm for verifying a minimum spanning tree. Our algorithm combines the result of Koml'os with a preprocessing and table lookup method for small subproblems and with a previously known almostlineartime algorithm. Additionally, we present an optimal deterministic algorithm and a lineartime randomized algorithm for sensitivity analysis of minimum spanning trees. 1. Introduction. Suppose we wish to solve some problem for which we know in advance the size of the input data, using an algorithm from some welldefined class of algorithms. For example, consider sorting n numbers, when n is fixed in advance, using a binary comparison tree. Given a sufficient amount of preprocessing time and storage space, we ca...
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.
Lower Bounds for Row Minima Searching (Extended Abstract)
, 1996
"... This paper shows that finding the row minima (maxima) in an n\Thetan totally monotone matrix in the worst case requires any algorithm to make 3n \Gamma 5 comparisons or 4n \Gamma 5 matrix accesses. Where the, so called, SMAWK algorithm of Aggarwal et al . finds the row minima in no more than 5n ..."
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This paper shows that finding the row minima (maxima) in an n\Thetan totally monotone matrix in the worst case requires any algorithm to make 3n \Gamma 5 comparisons or 4n \Gamma 5 matrix accesses. Where the, so called, SMAWK algorithm of Aggarwal et al . finds the row minima in no more than 5n \Gamma 2 lg n \Gamma 6 comparisons.