Minimum Spanning Tree: A Classic Application of Greedy Algorithm, Introduction to Prim's Algorithm

This paper introduces spanning trees, Minimum Spanning Trees (MST), and the Prim algorithm. A spanning tree is an acyclic subgraph of a connected undirected graph that includes all vertices; an MST is the spanning tree with the minimum sum of edge weights, which is suitable for the greedy algorithm (selecting locally optimal choices at each step to achieve a globally optimal solution). The core steps of the Prim algorithm are: selecting a starting vertex, repeatedly choosing the edge with the smallest weight from the edges between the selected and unselected vertices, adding the corresponding vertex to the selected set, until all vertices are included in the set. The key is to use an adjacency matrix or adjacency list to record the graph structure. In the algorithm's pseudocode, the `key` array records the minimum edge weight, and the `parent` array records the parent node. The time complexity is O(n²) using an adjacency matrix, and can be optimized to O(m log n). The Prim algorithm is based on the greedy choice, and the cut property and cycle property ensure that the total weight is minimized. It is applied in scenarios requiring the minimum-cost connection of all nodes, such as network wiring and circuit design. In summary, MST is a classic application of the greedy algorithm, and Prim efficiently constructs the optimal spanning tree by incrementally expanding and selecting the smallest edge.