Day 22: Minimum Spanning Trees
Minimum Spanning Trees #
Welcome to Day 22 of our 60 Days of Coding Algorithm Challenge! Today, we’ll explore Minimum Spanning Trees (MST) and two fundamental algorithms for finding them: Kruskal’s Algorithm and Prim’s Algorithm. These algorithms are crucial for solving problems involving network design, clustering, and other optimization tasks.
Introduction to Minimum Spanning Trees #
A Minimum Spanning Tree of an undirected, connected, weighted graph is a tree that spans all vertices of the graph with the minimum total edge weight. In other words, it’s a subset of the edges that forms a tree including every vertex, where the total weight of all the edges in the tree is minimized.
Key properties of a Minimum Spanning Tree:
- It includes all vertices of the graph.
- It forms a tree (no cycles).
- Among all possible spanning trees, it has the minimum total edge weight.
Kruskal’s Algorithm #
Kruskal’s algorithm builds the minimum spanning tree by adding edges one by one, always choosing the edge with the lowest weight that doesn’t create a cycle.
Algorithm: #
- Sort all edges in non-decreasing order of their weight. …
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