Disjoint Sets

Dynamic Connectivity

Connected Components

• Record sets that something belongs to
• Connected component = maximal set of mutually connected items

Quick Union

• Arrange in hierarchical tree structure
• Map items (array indices) to parents (parent indices)

Weighted Quick Union

• Modify quick-union to avoid tall trees
  • Track tree size (number of elements), also works similarly for height
  • Always link root of smaller tree to larger tree
• Max depth of any item is $\log{N}$
  • Depth of element x only increases when tree T1 that contains x linked below other tree T2
    • Occurs only when weight(T2) >= weight(T1)
    • Size of resulting tree is at least doubled
    • Depth of element x incremented only when weight(T2) >= weight(T1) & resulting tree at least doubled in size
  • Tree containing x can double in size at most $\log_{2}{N}$ times, when starting from just 1 element
    • $1 \cdot 2^x = N$
    • $x = \log_{2}{N}$
  • Max depth starts at 0 for only 1 element (root) → increments at most $\log_{2}{N}$ times, $\therefore$ max depth = $\log_{2}{N}$

Performance Summary

Path Compression

• When doing isConnected(15, 10), tie all nodes seen to the root
  • Additional cost insignificant (same order of growth)
• Kinda memoization/dynamic programming?
• Path compression results in union/connected operations very close to amortized constant time
• $M$ operations on $N$ nodes = $O(N + M \lg^{*}{N})$
• $\log^{*}{n} = \text{iterated/super logarithm}$
  • Inverse of super exponentiation
• Tighter bound: $O(N + M \alpha(N))$, where $\alpha$ is the inverse Ackermann function