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A planar graph and its minimum spanning tree. Each edge is labeled with its weight, which here is roughly proportional to its length. A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. [1]
In graph theory, Edmonds' algorithm or Chu–Liu/Edmonds' algorithm is an algorithm for finding a spanning arborescence of minimum weight (sometimes called an optimum branching). [1] It is the directed analog of the minimum spanning tree problem.
Initialize a tree with a single vertex, chosen arbitrarily from the graph. Grow the tree by one edge: Of the edges that connect the tree to vertices not yet in the tree, find the minimum-weight edge, and transfer it to the tree. Repeat step 2 (until all vertices are in the tree).
The quality of the tree is measured in the same way as in a graph, using the Euclidean distance between pairs of points as the weight for each edge. Thus, for instance, a Euclidean minimum spanning tree is the same as a graph minimum spanning tree in a complete graph with Euclidean edge weights.
Kruskal's algorithm [1] finds a minimum spanning forest of an undirected edge-weighted graph.If the graph is connected, it finds a minimum spanning tree.It is a greedy algorithm that in each step adds to the forest the lowest-weight edge that will not form a cycle. [2]
Example of a MST: The minimum spanning tree of a planar graph.Each edge is labeled with its weight, which here is roughly proportional to its length. The distributed minimum spanning tree (MST) problem involves the construction of a minimum spanning tree by a distributed algorithm, in a network where nodes communicate by message passing.
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In an unweighted graph, this is the spanning tree of minimum Wiener index. [1] Hu (1974) writes that the problem of constructing these trees was proposed by Francesco Maffioli. [2] It is NP-hard to construct it, even for unweighted graphs. [3] [4] However, it has a polynomial-time approximation scheme. The approximation works by choosing a ...