Union-find

Chapters

1. [00:00] Einführung
2. [06:27] Wie funktioniert Union-Find nochmal?
3. [07:03] Und wie war das noch mit der asymptotischen Komplexität?

General information

Representation invariant:

1. There is a nonnegative natural number [math]\displaystyle{ N\in\mathbb{N} }[/math], which is constant throughout the lifetime of the object.
2. At any moment, there is a (dynamically changing) partition of [math]\displaystyle{ S:=\{1,\ldots,N\} }[/math] into non-empty disjoint subsets. Each element of [math]\displaystyle{ S }[/math] belongs to exactly one of these partition sets.
3. Each of these subsets [math]\displaystyle{ S'\subseteq S }[/math] is represented by an arbitrary particular member of [math]\displaystyle{ S' }[/math], which does not change as long as [math]\displaystyle{ S' }[/math] is one of the partition sets (in other words, as long as [math]\displaystyle{ S' }[/math] is not involved in any call of method unite).

Constructor: Receives [math]\displaystyle{ N }[/math] as its input and initializes the union-find object such that each [math]\displaystyle{ i\in\{1,\ldots,N\} }[/math] is a singleton [math]\displaystyle{ \{i\} }[/math], which is represented by its unique member [math]\displaystyle{ i }[/math].

Find

Input: An element [math]\displaystyle{ i\in\{1,\ldots,N\} }[/math].

Output: The element [math]\displaystyle{ j\in\{1,\ldots,N\} }[/math] that represents the subset to which [math]\displaystyle{ i }[/math] currently belongs ([math]\displaystyle{ i=j }[/math] is possible).

Precondition: none.

Postcondition: none.

Unite

Input: Two elements, [math]\displaystyle{ i,j\in\{1,\ldots,N\} }[/math].

Output: none.

Precondition: The elements [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] do not belong to the same subset. In other words, it is find[math]\displaystyle{ (i)\neq }[/math] find[math]\displaystyle{ (j) }[/math].

Postcondition: The subsets to which [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] belong are united (the representative of the united set is not specified).

Known implementations

1. Union-find with lists