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A Minimum Spanning Tree (MST) is a subset of the edges of a connected, weighted, undirected graph that connects all the vertices together without forming any cycles and with the minimum possible total edge weight. The main purpose of an MST is to ensure that every vertex is reachable from any other vertex while minimizing the cost associated with the edges. In practical applications, MSTs are used in designing efficient network layouts such as computer networks, telecommunication networks, road networks, and electrical grids, where minimizing the total wiring or distance is important. There are several algorithms to find an MST, among which Kruskal’s algorithm and Prim’s algorithm are most widely used. Kruskal’s algorithm focuses on adding edges in increasing order of their weights while avoiding cycles, whereas Prim’s algorithm grows the MST from a starting vertex by repeatedly adding the smallest edge connecting the tree to a vertex outside it. Kruskal’s algorithm is a greedy approach to find the MST. The algorithm works as follows: first, all the edges of the graph are sorted in ascending order based on their weights. Then, edges are picked one by one from the sorted list and added to the MST if they do not form a cycle with the edges already included. To detect cycles efficiently, a Disjoint Set Union (DSU) or Union-Find data ____ _______ _______ ____ ___ ___ ______ _________.
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