Minimum Spanning Tree Algorithms for Graphs

Minimum Spanning Tree Algorithms for Graphs

Description
The Minimum Spanning Tree (MST) is a classic problem in graph theory. It refers to finding a tree (an acyclic connected subgraph) that includes all vertices in a weighted, undirected, connected graph, such that the sum of the weights of its edges is minimized. This problem has important applications in communication network design, circuit routing, and other fields. The main algorithms include Prim's algorithm and Kruskal's algorithm, both based on greedy strategies but with different implementation approaches. This section will focus on explaining the principle and implementation of Prim's algorithm.

Core Concepts

  1. Spanning Tree: A tree that contains all vertices of the graph, with the number of edges being one less than the number of vertices (|V| - 1).
  2. Minimum Spanning Tree: The spanning tree with the smallest total edge weight among all spanning trees (may not be unique, but the weight sum is unique).
  3. Greedy Strategy: At each step, select the edge with the smallest current weight, ensuring no cycle is formed.

Detailed Explanation of Prim's Algorithm
Algorithm Idea: Start from any vertex and gradually expand the spanning tree. Each time, select the edge with the minimum weight connecting the selected vertex set and the unselected vertex set, adding the new vertex to the selected set until all vertices are covered.

Step-by-Step Breakdown (operating on vertices):

  1. Initialization:

    • Choose any starting vertex (e.g., vertex 0) and add it to the selected set visited.
    • Maintain an array key[] to record the minimum edge weight from each vertex to the selected set (initially, the starting vertex key=0, others key=∞).
    • Maintain an array parent[] to record the connection relationships of edges in the spanning tree.

  2. Iterative Expansion:

    • Repeat the following steps until all vertices are added to visited:
      a. From the unselected vertices, select the vertex u with the smallest key value (i.e., the vertex closest to the selected set).
      b. Add u to visited.
      c. Traverse all adjacent vertices v of u:
      • If v is unvisited and the edge weight w(u, v) is less than key[v], then update key[v] = w(u, v) and set parent[v] = u.

  3. Output Result:

    • The final parent[] array defines the edges of the spanning tree, and the total weight is the sum of all key[] values.

Example Demonstration
Take the following graph as an example (number of vertices=5, edge weights as shown):

       2
    A —— B
    |3   |4
    C —— D
    |1   |5
    E ——

Step-by-Step Execution (starting from A):

  1. Initial: visited = {A}, key = [A:0, B:∞, C:∞, D:∞, E:∞].
  2. Update A's adjacent points: B's key is updated to 2, C's key is updated to 3.
  3. Select vertex B with the smallest key: add to visited, update D's key (adjacent to B) to 4.
  4. Select vertex C with the smallest key: add to visited, update E's key to 1 (via edge C-E).
  5. Select vertex E with the smallest key: add to visited, no new adjacent points need updating.
  6. Select vertex D with the smallest key: add to visited, done.
    Spanning tree edges: A-B, A-C, C-E, B-D (total weight = 2+3+1+4=10).

Complexity Analysis

  • Time Complexity:
    • Naive implementation: Each time traverse all vertices to find the minimum key, complexity O(V²), suitable for dense graphs.
    • Binary heap optimization: Use a priority queue (min-heap) to store keys, complexity O((V+E)logV), suitable for sparse graphs.
  • Space Complexity: O(V) for storing key, parent, and other arrays.

Key Points Summary

  1. Prim's algorithm always maintains a connected subtree and expands it step by step.
  2. It must ensure that each selected edge is the minimum edge connecting the selected set and the unselected set (to avoid forming cycles).
  3. Difference from Kruskal's algorithm: Kruskal selects edges globally by sorting edge weights and uses a union-find data structure to detect cycles; Prim expands by vertices and does not require explicit cycle detection.

Through the above steps, the Minimum Spanning Tree problem for undirected connected graphs can be solved efficiently. In actual coding, choose the appropriate implementation based on graph density.