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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]
The shortest-path tree from this point to all vertices in the graph is a minimum-diameter spanning tree of the graph. [2] The absolute 1-center problem was introduced long before the first study of the minimum-diameter spanning tree problem, [ 2 ] [ 3 ] and in a graph with n {\displaystyle n} vertices and m {\displaystyle m} edges it can be ...
However since T is a minimum spanning tree then T − f + e has the same weight as T, otherwise we get a contradiction and T would not be a minimum spanning tree. So T − f + e is a minimum spanning tree containing F + e and again P holds. Therefore, by the principle of induction, P holds when F has become a spanning tree, which is only ...
Repeat the steps above and we will eventually obtain a minimum spanning tree of graph P that is identical to tree Y. This shows Y is a minimum spanning tree. The minimum spanning tree allows for the first subset of the sub-region to be expanded into a larger subset X, which we assume to be the minimum.
The set of these minimum spanning trees is called a minimum spanning forest, which contains every vertex in the graph. This algorithm is a greedy algorithm, choosing the best choice given any situation. It is the reverse of Kruskal's algorithm, which is another greedy algorithm to find a minimum spanning tree. Kruskal’s algorithm starts with ...
It returns a spanning arborescence rooted at of minimum weight, where the weight of an arborescence is defined to be the sum of its edge weights, () = (). The algorithm has a recursive description. Let f ( D , r , w ) {\displaystyle f(D,r,w)} denote the function which returns a spanning arborescence rooted at r {\displaystyle r} of minimum weight.
In computer science, the minimum routing cost spanning tree of a weighted graph is a spanning tree minimizing the sum of pairwise distances between vertices in the tree. It is also called the optimum distance spanning tree, shortest total path length spanning tree, minimum total distance spanning tree, or minimum average distance spanning tree.
Example of rectilinear minimum spanning tree from random points. In graph theory, the rectilinear minimum spanning tree (RMST) of a set of n points in the plane (or more generally, in ) is a minimum spanning tree of that set, where the weight of the edge between each pair of points is the rectilinear distance between those two points.