Analysis

Union by Rank and Path Compression:

O(m * $\alpha$(m,n)) worst case running time

$\alpha$(m,n) is inverse of Ackermann's function A(i,j)

$\alpha$(m,n) = min{i $\geq$ 1 $\mid$ A(i, ) > lg n}

Ackermann's Function A(i,j)

• A(1, j) = 2^j $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$for j $\geq$ 1
• A(i, 1) = A(i-1, 2) $\;\;\;\;\;$ $\;\;\;\;\;$for i $\geq$ 2
  
• A(i, j) = A(i-1, A(i, j-1)) $\;\;\;\;\;$for i, j $\geq$ 2

Note: A(i,j) is strictly increasing and since m $\geq$ n.

Therefore A(4, ) $\geq$ A(4,1) = A(3,2)

A(3,2) = 2 raised to the power 2 16 times

10⁸⁰ = the number of atoms in the observable universe

= 4 for practical uses since lg n is typically less than 10⁸⁰

Thus, T(m) = O(m).

O(1) amortized cost per operation