Kruskal's algorithm is a greedy algorithm used to find the minimum spanning tree (MST) of a connected weighted graph. The minimum spanning tree is a subset of the graph's edges that connects all the vertices with the minimum possible total edge weight.
Here's how Kruskal's algorithm works:
Initialize an empty set called the MST, which will eventually contain the minimum spanning tree.
Sort all the edges of the graph in non-decreasing order of their weights.
Iterate through the sorted edges and consider them in ascending order of weights.
For each edge, check if including it in the MST creates a cycle or not. This can be done by checking if the two vertices of the edge belong to the same connected component in the current MST. If they do not belong to the same component, add the edge to the MST.
Repeat step 4 until all the edges have been considered or the MST contains V-1 edges (where V is the number of vertices in the graph).
The algorithm terminates when the MST contains V-1 edges (where V is the number of vertices) or when all the edges have been considered. At the end of the algorithm, the MST will contain the subset of edges that form the minimum spanning tree.
Kruskal's algorithm can be implemented using a disjoint-set data structure to efficiently check if adding an edge creates a cycle. The disjoint-set data structure allows efficient union and find operations on disjoint sets.
Overall, Kruskal's algorithm is an efficient way to find the minimum spanning tree of a graph and has a time complexity of O(E log E), where E is the number of edges in the graph.
Certainly! Here are some additional details about Kruskal's algorithm:
Graph Representation: Kruskal's algorithm works on a connected, undirected graph with weighted edges. The graph can be represented using an adjacency matrix or an adjacency list.
Sorting the Edges: Before starting the algorithm, all the edges of the graph need to be sorted in non-decreasing order of their weights. This sorting step ensures that we consider the edges with the lowest weights first during the iteration.
Disjoint-Set Data Structure: Kruskal's algorithm relies on the disjoint-set data structure to efficiently check for cycles in the MST construction process. The disjoint-set data structure maintains a collection of disjoint sets, each represented by a representative element. The two primary operations on disjoint sets are:
Find(x): Given an element x, it returns the representative element of the set that x belongs to. Union(x, y): Given two elements x and y, it merges the sets containing x and y into a single set. The disjoint-set data structure allows us to determine if adding an edge between two vertices will create a cycle in the MST.
Cycle Detection: To check if adding an edge to the MST will create a cycle, we perform a find operation on the two vertices of the edge. If the find operation returns different representatives for the two vertices, it means they belong to different sets and adding the edge will not create a cycle. In this case, we can safely add the edge to the MST and perform a union operation on the two sets.
Building the MST: Starting with an empty MST, we iterate through the sorted edges in ascending order of their weights. For each edge, we check if it creates a cycle or not. If it does not create a cycle, we add the edge to the MST. We repeat this process until the MST contains V-1 edges (where V is the number of vertices) or all the edges have been considered.
Final MST: At the end of the algorithm, the MST will contain the subset of edges that form the minimum spanning tree. The MST is a tree that connects all the vertices of the graph with the minimum total edge weight.
Kruskal's algorithm is widely used in various applications, such as network design, clustering, and finding the shortest path in a graph with weighted edges. Its time complexity of O(E log E) makes it efficient for large graphs.