Which algorithm guarantees that a minimum spanning tree is found?

Prim's Algorithm is a well-established method for finding the minimum spanning tree (MST) of a connected, undirected graph with weighted edges. The algorithm starts with a single vertex and grows the spanning tree by continuously adding the smallest edge that connects a vertex in the tree to a vertex outside the tree. This greedy approach ensures that at each step, the edge added to the spanning tree is the least weight among all possible options, which ultimately leads to a minimum overall weight for the spanning tree.

The reason Prim's Algorithm guarantees the discovery of a minimum spanning tree lies in its construction process. By always selecting the minimum edge that connects the tree to the remaining vertices, the algorithm maintains the minimally weighted structure. This characteristic differentiates it from other algorithms that do not focus specifically on minimum spanning trees.

In contrast, Dijkstra's Algorithm is designed to find the shortest path from a single source to other vertices in a graph, rather than constructing a minimum spanning tree. Quick Sort and Binary Search serve different purposes entirely, with Quick Sort being a sorting algorithm and Binary Search being a search algorithm applicable to sorted data, neither of which is aimed at finding spanning trees in graphs.