187 Prim's algorithm

187 Prim's algorithm

Prim's algorithm is a typical method for finding minimum global trees in graph theory, especially applied to weighted undirected graphs. A minimum global tree is a structure that connects all vertices while having no closed paths and minimizing the sum of the weights of all edges. The distinctive feature of the Prim method is that it is based on the greedy method, which accumulates the best alternatives at a given point in time, one by one.

In this algorithm, the process begins with any one vertex in the graph. Among the edges directly connected to that vertex, the one with the lowest weight is chosen, and the vertex at the connected end is added as a new part of the "tree. After that, it always adds the vertex that is already in the tree and the vertex that is not yet in the tree by selecting the edge with the lowest weight among the edges connecting them. This process is repeated until all vertices are included, resulting in a minimum global tree.

The prim method is used in a variety of situations where the entire area needs to be connected while minimizing costs, such as laying electrical wires, designing communication networks, and routing roads and railroads. A similar method is the Kruskal method, but it is different in nature because the prim method expands the tree around the vertices, whereas the Kruskal method selects the edges in order of their weight. The prim method tends to perform particularly well on dense graphs with many edges.