Minimum spanning trees connect all vertices with minimal weight from "summary" of Data Structures and Algorithms in Python by Michael T. Goodrich,Roberto Tamassia,Michael H. Goldwasser
A minimum spanning tree for a connected, undirected graph is a spanning tree that connects all vertices of the graph with the smallest possible sum of edge weights. In other words, it is a tree that includes all the vertices of the graph and has the minimum total weight among all possible trees that can be formed from the graph. For example, in a city map where vertices represent locations and edges represent roads between locations, a minimum spanning tree would be the set of roads that connect all locations with the smallest total distance. To find a minimum spanning tree for a given graph, we can use algorithms such as Prim's algorithm or Kruskal's algorithm. Prim's algorithm starts with an arbitrary vertex and grows the minimum spanning tree one edge at a time by adding the shortest edge that connects a vertex in the tree to a vertex outside the tree. Kruskal's algorithm, on the other hand, starts with all vertices being separate trees and merges them into a single tree by adding the shortest edge that connects two separate trees. Both algorithms guarantee that the resulting tree is a minimum spanning tree, as they add edges in a way that maintains the minimal total weight property. While Prim's algorithm operates by growing the tree from a single vertex, Kruskal's algorithm operates by merging smaller trees into larger ones. Despite their different approaches, both algorithms achieve the same goal of finding the minimum spanning tree for a given graph.- Such as network design, circuit design, and clustering algorithms. By connecting all vertices with minimal weight, minimum spanning trees provide an efficient way to establish connections between points while minimizing the overall cost. In summary, minimum spanning trees are a fundamental concept in graph theory that enables us to find the most efficient way to connect all vertices in a graph with minimal total weight.
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