Kruskal's Algorithm
Kruskal's Algorithm is a greedy algorithm used to find the Minimum Spanning Tree (MST) of a graph. A spanning tree is a subset of the edges that connects all vertices in a graph without any cycles. The minimum spanning tree is the spanning tree with the least total edge weight.
Key Concepts:
Undirected, Weighted Graph: Kruskal's algorithm works on undirected, weighted graphs, where each edge has a weight (or cost).
Minimum Spanning Tree: A tree that spans all vertices with the minimum possible total edge weight and no cycles.
Greedy Approach: Kruskal's algorithm repeatedly adds the smallest edge that does not form a cycle, ensuring a globally optimal solution.
Union-Find (Disjoint Set): A data structure used to efficiently manage and merge different sets of vertices, helping detect cycles as edges are added.
Key Points:
Greedy Strategy: Kruskal's algorithm builds the MST by selecting the smallest edge at each step, ensuring that the total weight of the MST is minimized.
Cycle-Free: Using the Union-Find data structure ensures that no cycles are formed, so the resulting spanning tree is valid.
Multiple Components: Kruskal’s algorithm naturally handles disconnected graphs by producing a minimum spanning forest (MST for each connected component).
Real-World Applications:
Network Design: Kruskal's algorithm can be used to design network infrastructures (e.g., telecommunications, computer networks) where we need to connect a set of nodes (e.g., cities or computers) with the minimum total wiring or connection cost.
Example: Laying out fiber-optic cables between cities to minimize the total cost.
Transportation Networks: Used in planning road systems, railways, or airline routes where the goal is to connect multiple locations with the least total distance or cost.
Electrical Grid Design: Kruskal’s algorithm can help minimize the total wiring needed to connect different components of an electrical grid.