1. 233-234233-234 Sedgewick & Wayne (2004); Chazelle (2005) Sedgewick & Wayne (2004); Chazelle (2005)

10. How to check if adding an edge creates a cycle? Label i all nodes of i-th tree 1 1 1 2 2 2 2

11. How to check if adding an edge creates a cycle? Label i all nodes of i-th tree 1 1 1 2 2 2 2

12. How to check if adding an edge creates a cycle? Label i all nodes of i-th tree 1 1 1 2 2 2 2 Relabel smaller tree

13. How to check if adding an edge creates a cycle? Label i all nodes of i-th tree 2 2 2 2 2 2 2 Relabel smaller tree

14. How to check if adding an edge creates a cycle? Label i all nodes of i-th tree 2 2 2 2 2 2 2 Each node relabeled at most log V times

16. 3 7 9 8 edge of the MST correctness A nodes B nodes

17. 3 A B

18. 3 8 A A B A B A

19. How much time ? 1. Sorting: 2. Cycle detection:

20. How much time ? 1. Sorting: 2. Cycle detection: Time proportional to

21. Advanced MST Algorithms Yao/Cheriton-Tarjan (1976) Fredman-Tarjan (1984) Gabow-Galil-Spencer- Tarjan (1986) Chazelle (2000)

22. number of particles in the universe If E = 2 Ackermann’s function

23. Two times x Two to the x-th Tower of x Tower of height equal to tower of x