Kruskal’s Algorithm: MST for Weighted Graphs
Graph algorithms play a crucial role in the world of computer science and networking. One such algorithm, known for its simplicity and efficiency, is Kruskal’s Algorithm. It helps in finding a Minimum Spanning Tree (MST) from a given weighted undirected graph, ensuring that all vertices are connected with the smallest possible total edge weight.
Understanding Minimum Spanning Tree (MST)
Before diving into Kruskal’s Algorithm, it’s essential to understand the concept of a Minimum Spanning Tree. An MST is a subset of the edges of a connected, edge-weighted graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight.
Steps of Kruskal’s Algorithm
- Sort all edges: Start by sorting all the edges of the graph in non-decreasing order of their weight.
- Initialize MST: Initialize the Minimum Spanning Tree as an empty set.
- Edge Selection: For every sorted edge, do the following:
- Pick the smallest edge. Check if by including this edge in the MST, it forms a cycle with the MST constructed so far. If no cycle is formed, include this edge; otherwise, discard it.
- End: Repeat step 3 until there are (V-1) edges in the spanning tree, where V is the number of vertices.
Cycle Detection
A critical part of Kruskal’s Algorithm is determining whether adding an edge results in a cycle. This can be achieved using the Union-Find algorithm. The algorithm keeps track of connected components (sets of vertices) and can quickly determine if two vertices belong to the same component.
JavaScript Example
function kruskal(graph) {
graph.edges.sort((a, b) => a.weight - b.weight);
let mst = [];
let unionFind = new UnionFind(graph.vertices);
for(let edge of graph.edges) {
if(unionFind.union(edge.start, edge.end)) {
mst.push(edge);
}
}
return mst;
}
class UnionFind {
constructor(vertices) {
this.parent = {};
vertices.forEach(v => this.parent[v] = v);
}
find(vertex) {
if (this.parent[vertex] === vertex) {
return vertex;
}
return this.find(this.parent[vertex]);
}
union(start, end) {
let rootStart = this.find(start);
let rootEnd = this.find(end);
if (rootStart !== rootEnd) {
this.parent[rootStart] = rootEnd;
return true;
}
return false;
}
}
Conclusion
Kruskal’s Algorithm provides an efficient way to find the Minimum Spanning Tree of a graph, which can be pivotal in various practical scenarios from network design to resource optimization. With its simplicity and efficiency, it remains a fundamental topic in graph theory and computer science courses.