Kruskal's Algorithm

External

• https://www.coursera.org/learn/algorithms-greedy/lecture/PLdBf/kruskals-mst-algorithm

Internal

• The Minimum Spanning Tree Problem
• Greedy Algorithms
• Graph Concepts
• Union-Find

Overview

Kruskal's algorithms is a greedy algorithm that computes the minimum cost spanning tree for an undirected connected graph. It has a different approach than Prim's algorithm, in that it does not grow the connected explored territory one node at a time, but it grows the trees in parallel with a lot of simultaneous little pieces, unconnected at first, which eventually merge into the minimum cost spanning tree. It does that by sorting the edges in the ascending order of their cost, and picking edges in that order, one by one, careful not to create cycles. The greediness of the algorithm comes from the fact that at each step, it chooses an edge by attempting to minimize a greedy score, which is the cost of the edge.

Non-Optimized Implementation

sort edges in the order of increasing cost
initialize T = ∅ # the tree we are growing
for edge e = (u,v) in the order of increasing cost: # at most m iteration
  if T ⋃ e has no cycles # there is no path in T from u to v
    add e to T

We could count the edges added to the spanning tree, and once we have n-1 edges, which means the spanning tree is complete, we can abort the loop.

One critical step of the algorithm is checking whether a new edge e=(u,v) introduces no cycles in T, and this can be checked in several ways. The non-optimized way is to run a BFS or a DFS starting from u, attempting to find a path from u to v. A better way would be to use a union-find.

Non-Optimized Implementation Running Time

Sorting the edges will take O(m log m) time. However, since we assume that the number of edges is at most quadratic of the number of vertices, m = O(n²), so O(log m) = O(log n²)=2O(log n)=O(log n). In consequence, the running time of the edge sorting step is O(m log n).

The main loop has at most m iterations. In each iteration, we need to ensure that the current edge e=(u,v) does not form a cycle in T, which means we need to ensure that there is no path from u to v.

The non-optimized solution checks whether there is no path from u to v using either BFS or DFS starting from u. Both methods have a running time O(n) linear in the number of n.

Correctness Proof

https://www.coursera.org/learn/algorithms-greedy/lecture/U3ukN/correctness-of-kruskals-algorithm

Optimized Implementation

https://www.coursera.org/learn/algorithms-greedy/lecture/e0TJP/implementing-kruskals-algorithm-via-union-find-i