Kruskal's Algorithm
2. Building the Forest
Kruskal's algorithm takes a decidedly "edge-first" approach. It's like a diligent construction worker who meticulously picks the cheapest materials first. The algorithm starts by sorting all the edges in the graph in ascending order of their weights. Then, it iterates through the sorted edges, adding each edge to the MST unless adding that edge would create a cycle. Imagine slowly building a forest of trees, connecting them one by one until a single, connected tree remains. That's essentially Kruskal's in a nutshell.
One of the key data structures that Kruskal's relies on is the "disjoint set data structure" (also known as the "union-find" data structure). This structure efficiently keeps track of which vertices belong to which connected component. When considering a new edge, Kruskal's uses the disjoint set to check if the two vertices connected by the edge are already in the same set (i.e., already connected). If they are, adding the edge would create a cycle, so it's skipped. If they aren't, the edge is added, and the two sets are merged.
The beauty of Kruskal's is its simplicity and its ability to handle disconnected graphs gracefully. It doesn't matter if the graph starts as a single, connected piece or as a bunch of isolated islands; Kruskal's will eventually connect them all into a single, minimum spanning tree, as long as a spanning tree exists, of course! The time complexity hinges largely on the sorting step (usually O(E log E), where E is the number of edges), and the efficiency of the disjoint set operations.
Think of it as building a bridge across a series of islands. You start by connecting the two closest islands with the shortest bridge. Then, you look for the next shortest bridge and so on, carefully making sure you don't build any redundant bridges that would create unnecessary loops in your network. Smart, right?
