Kruskal's Algorithm

Kruskal's Algorithm
Runtime	O(E log E)
Class	Minimum Spanning Tree, Greedy Algorithm

Approach: Greedy

The approach is to try to add the smallest edges as long as they do not create a cycle. Unlike Prim's algorithm, which prevents cycles by only choosing edges that crosses a cut of nodes already in the tree and nodes that aren't, Kruskal prevents cycles using a data structure known as disjoint set (aka. union-find).

Given the MST of $V_{n-m}=v_{1},v_{2},\ldots ,v_{n-m}$, the MST of $V$ should be that of $V_{n-1}$ plus the edge that connects to $v_{n}$ that is the shortest.

$OPT(n)=OPT(n-1)+min((v_{i},v_{n})\in E)$

Implementation

Kruskal(G):
    let A be an empty graph
    for each v in V:
        makeSet(v)
    sort E in nondecreasing order by weight

    for each (u,v) in E:
        if findSet(u) != findSet(v):
            A = A U {(u,v)}
            Union(u,v)
    return A

Analysis

Sort edges + E (cycle?) + (V - 1) adding edge

Disjoint set allows O(log V) cycle checking and O(log V) edge adding.

Sorting takes E log E

For weighted disjoint set, checking cycle takes log V, and adding edge takes log V

For fast-find, where all members have the same ID, fast-set-id needs O(1) and union needs O(n)