Sedgewick & Wayne (2004); Chazelle (2005) Sedgewick & Wayne (2004); Chazelle (2005)

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

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

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

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

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

edge of the MST correctness A nodes B nodes

3 A B

3 8 A A B A B A

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

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

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

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

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