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Spanning trees and shortest paths in Monge graphs. (English) Zbl 1088.90539
A Monge matrix of order n is an n×n matrix (c ij ) n×n such that c ij +c kl c il +c kj for all 1i<kn, 1j<ln, ij, kl, il, kj. A Monge graph is a complete, undirected weighted graph whose distance matrix is a Monge matrix. In this paper, the authors investigate the following three problems on Monge graphs: (1) the minimum spanning tree problem, (2) the problem of computing all-pairs shortest paths, and (3) the problem of determining a minimum weighted 1-to-all shortest path tree. For all three problems best possible algorithms (in terms of complexity) are presented; the complexity of each of them is linear or the square of n, the number of vertices of the Monge graphs.
MSC:
90C35Programming involving graphs or networks
90C27Combinatorial optimization
05C05Trees
05C12Distance in graphs
05C38Paths; cycles
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