Final Answer:
The linear time algorithm to find the minimum spanning tree (MST) of a connected, undirected, weighted graph is Prim's algorithm.
Step-by-step explanation:
Prim's algorithm is a greedy algorithm used to find the minimum spanning tree of a connected, undirected, weighted graph. It starts with an arbitrary node and incrementally adds the shortest edge that connects a vertex in the MST to a vertex outside the MST. This process continues until all vertices are included in the tree, forming the minimum spanning tree.
The algorithm maintains a priority queue to efficiently select the minimum-weight edge at each step. Its time complexity is O(E + V log V), making it a linear time algorithm for sparse graphs, where E is the number of edges and V is the number of vertices. Prim's algorithm is straightforward to implement and guarantees the construction of a minimum spanning tree, making it a practical choice for various applications, including network design and clustering.
Understanding the efficiency and simplicity of Prim's algorithm is crucial for solving optimization problems involving connectivity in graphs. Its linear time complexity contributes to its widespread use in real-world scenarios where finding an efficient minimum spanning tree is essential.