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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]
For other graphs, the expected weight of the random minimum spanning tree can be calculated as an integral involving the Tutte polynomial of the graph. [ 2 ] In contrast to uniformly random spanning trees of complete graphs, for which the typical diameter is proportional to the square root of the number of vertices, random minimum spanning ...
Alternatively, if the edges of the graph have positive weights, the minimum weight cycle basis may be constructed in polynomial time. In planar graphs, the set of bounded cycles of an embedding of the graph forms a cycle basis. The minimum weight cycle basis of a planar graph corresponds to the Gomory–Hu tree of the dual graph.
A cut is minimum if the size or weight of the cut is not larger than the size of any other cut. The illustration on the right shows a minimum cut: the size of this cut is 2, and there is no cut of size 1 because the graph is bridgeless.
These algorithms find the minimum spanning forest in a possibly disconnected graph; in contrast, the most basic form of Prim's algorithm only finds minimum spanning trees in connected graphs. However, running Prim's algorithm separately for each connected component of the graph, it can also be used to find the minimum spanning forest. [9]
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]
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). It is the directed analog of the minimum spanning tree problem.
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.