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Note on the structure of Kruskal’s algorithm. (English) Zbl 1219.05181
Let $$G=(V,E)$$ be a connected edge-weighted graph and let $$(V,F)$$ be its minimal spanning tree constructed by Kruskal’s algorithm [J.B. Kruskal jun., “On the shortest spanning subtree of a graph and the traveling salesman problem,” Proc. Am. Math. Soc. 7, 48–50 (1956; Zbl 0070.18404)]. We capture the evolution of the spanning forest from $$(V,\emptyset)$$ to $$(V,F)$$ by a rooted binary tree $$R$$ with leaves in $$V$$ and internal nodes in $$F$$. Let $$h(G)$$ denote the height of $$R$$. In case of Prim’s algorithm we would have $$h(G_n) = n-1$$ for every connected graph $$G_n$$ on $$n$$ vertices. In case of Kruskal’s algorithm there is a constant $$c>0$$ such that the probability of $$h(G_n) \geq cn$$ tends to $$1$$ for $$n\to\infty$$, and therefore the expected value of $$h(G_n)$$ is in $$\Theta(n)$$, for three choices of random edge-weights:
(1)
$$G_n$$ is a complete graph on $$n$$ independently uniformly distributed random points in $$[0,1]^d$$ and the edges are weighted by the Euclidean distance,
(2)
$$G_n$$ is a complete graph on $$n$$ vertices and the edge-weights are independently uniformly distributed in $$[0,1]$$,
(3)
$$G_n$$ is the Cartesian product of $$d$$ paths $$P_k$$, $$n=k^d$$, and the edge-weights are independently uniformly distributed in $$[0,1]$$.
##### MSC:
 05C85 Graph algorithms (graph-theoretic aspects) 05C80 Random graphs (graph-theoretic aspects) 68R10 Graph theory (including graph drawing) in computer science
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