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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]
A minimum spanning tree of a connected weighted graph is a connected subgraph, without cycles, for which the sum of the weights of all the edges in the subgraph is minimal. For a disconnected graph, a minimum spanning forest is composed of a minimum spanning tree for each connected component.
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 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.
A demo for Union-Find when using Kruskal's algorithm to find minimum spanning tree. Disjoint-set data structures model the partitioning of a set, for example to keep track of the connected components of an undirected graph. This model can then be used to determine whether two vertices belong to the same component, or whether adding an edge ...
Agarwal et al. developed a data structure that maintains the MST for a graph belonging to a minor closed family.It uses the idea of a "swap", calculating the amount by which the weight of the MST would increase if some edge in the tree e was replaced by an edge f outside the tree such that the circle induced by f in the tree contains e.
The exact solution of the minimum-diameter spanning tree problem, in the Euclidean plane, can be sped up from () to / + (), at the expense of using complicated range search data structures. The same method extends to higher dimensions, with smaller reductions in the exponent compared to the cubic algorithm.
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.