The document discusses different techniques for improving the efficiency of union-find algorithms, including union-by-size, path compression, and union-by-height. Union-by-size works by making the root of the smaller tree the child of the larger tree during a union operation, keeping tree heights small. Path compression further optimizes find operations by updating the parent pointers along the search path. Together these optimizations allow union-find algorithms to run in almost linear time for practical purposes.

1. Class No.30 Data Structures http://ecomputernotes.com

5. Union by Size Eight elements, initially in different sets. 1 2 3 4 5 6 7 8 -1 -1 -1 -1 -1 -1 -1 -1 1 2 3 4 5 6 7 8 http://ecomputernotes.com

6. Union by Size Union(4,6) 1 2 3 4 5 6 7 8 -1 -1 -1 -2 -1 4 -1 -1 1 2 3 4 5 6 7 8 http://ecomputernotes.com

7. Union by Size Union(2,3) 1 2 3 4 5 6 7 8 -1 -2 2 -2 -1 4 -1 -1 1 2 3 4 5 6 7 8 http://ecomputernotes.com

8. Union by Size Union(1,4) 1 2 3 4 5 6 7 8 4 -2 2 -3 -1 4 -1 -1 1 2 3 4 5 6 7 8 http://ecomputernotes.com

9. Union by Size Union(2,4) 1 2 3 4 5 6 7 8 4 4 2 -5 -1 4 -1 -1 1 2 3 4 5 6 7 8 http://ecomputernotes.com

10. Union by Size Union(5,4) 1 2 3 4 5 6 7 8 4 4 2 -6 4 4 -1 -1 1 2 3 4 5 6 7 8 http://ecomputernotes.com