Union-Find

1. Union-Find • Application in Kruskal’s Algorithm • Optimizing Union and Find Methods

2. Minimum Spanning Trees • Tree that connects all vertices of a graph • Sum of the edge weights is a minimum

3. Kruskal’s Algorithm • Sort edges in order of weights • Start adding edges to sub-graph: • Start from lowest weight • Skip edge if it makes the sub-graph cyclic

4. Kruskal’s Algorithm 1 6 3 4 4 5 5 2 8 6

5. Union-Find & Kruskal’s Algorithm • Vertices grouped in sets • Can only add edges linking vertices not in same set

6. Non-Optimal Solution • Array of labels • Change labels for a union • O (n) for each union • O (n^2)

7. Union-Find Methods • makeSet ( x ) • union ( x , y ) • find ( x )

8. Optimizing Union(x,y) • Sets of vertices stored in trees • Root of tree is label of set • union(x,y) by joining two trees • Root of smaller tree points to root of larger tree

9. x Union(x,y) Illustration y

10. Path Compression • Nodes from ‘x’ to root have same label • Change these parent-pointers to the root

11. a g c b e d f Path Compression Illustration a c b e d f g

12. Time Efficiency • Sorting is O(e log e) • Find maximum is O(log n) • Path compression makes future finds O(1) • Calling find many times gives O(1) average • Union is 2 finds and a pointer change: O(1) • Kruskal becomes O(e log e)