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.
Read MoreFigure: A Beginner's Guide to the Basic Concepts and Adjacency List Representation of Graphs
A graph consists of vertices (nodes) and edges (connections). Vertices are the basic units, and edges can be directed (digraph) or undirected. Weighted graphs have edges with weights (e.g., distances), while unweighted graphs only retain connection relationships. The adjacency list is an efficient representation method that solves the space waste problem of the adjacency matrix in sparse graphs (where the number of edges is much less than the square of the number of vertices). Its core is that each vertex stores a list of directly connected vertices. For an undirected graph, if vertex 0 is connected to 1, 2, and 3, its adjacency list is [1, 2, 3]. For a weighted graph, the adjacency list can store tuples of "neighbor + weight". The space complexity of an adjacency list is O(V + E) (where V is the number of vertices and E is the number of edges), making it suitable for sparse graphs. It facilitates traversal of neighboring vertices but requires traversing the adjacency list to check if an edge exists between two vertices. Mastering the adjacency list is fundamental for algorithms such as graph traversal and shortest path finding.
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