Stochastic Modeling by Nicolas Lanchier

Author:Nicolas Lanchier
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

In the case of four cities, of the six possible roads, exactly three must be constructed to connect the cities. In addition, looking at the subgraphs with three edges, there are exactly four triangles for a total of 6 choose 3 possible subgraphs, so the fraction of such subgraphs that are triangles is

As previously, let X i be the cost of the ith cheapest road. Because each of the six possible roads is equally likely to be the one with cost X i , the probability that the three cheapest roads do not form a triangle is 4/5, in which case these three roads connect all four cities. In the case where the three cheapest roads form a triangle, the minimum spanning tree is obtained by choosing the two cheapest roads and the fourth cheapest one. In particular, the expected minimal cost is

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