Greedy Algorithms Questions Medium
Kruskal's algorithm is a greedy algorithm used to find a minimum spanning tree in a graph. It works by considering the edges of the graph in ascending order of their weights and adding them to the spanning tree if they do not create a cycle.
Here is a step-by-step explanation of how Kruskal's algorithm works:
1. Sort all the edges of the graph in non-decreasing order of their weights.
2. Initialize an empty set called the "spanning tree" to store the final minimum spanning tree.
3. Iterate through each edge in the sorted order:
a. If adding the current edge to the spanning tree does not create a cycle, add it to the spanning tree.
b. To check for cycles, we can use a disjoint set data structure. Initially, each vertex is in its own disjoint set. When adding an edge, check if the two vertices of the edge belong to the same disjoint set. If they do, adding the edge will create a cycle, so we skip it. Otherwise, we merge the two disjoint sets into one.
4. Repeat step 3 until all edges have been considered or the spanning tree has V-1 edges (V is the number of vertices in the graph).
5. The resulting spanning tree is the minimum spanning tree of the graph.
Kruskal's algorithm guarantees that the resulting spanning tree will have the minimum total weight among all possible spanning trees. It achieves this by greedily selecting the edges with the smallest weights that do not create cycles.