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As a result, disjoint-set forests are both asymptotically optimal and practically efficient. Disjoint-set data structures play a key role in Kruskal's algorithm for finding the minimum spanning tree of a graph. The importance of minimum spanning trees means that disjoint-set data structures support a wide variety of algorithms.
The pseudocode below determines the lowest common ancestor of each pair in P, given the root r of a tree in which the children of node n are in the set n.children. For this offline algorithm, the set P must be specified in advance. It uses the MakeSet, Find, and Union functions of a disjoint-set data structure.
The following code is implemented with a disjoint-set data structure. It represents the forest F as a set of undirected edges, and uses the disjoint-set data structure to efficiently determine whether two vertices are part of the same tree.
Many programming techniques and data structures, including rope, lazy evaluation, class hierarchy (see below), arbitrary-precision arithmetic, CDR coding, the indirection bit, and other kinds of tagged pointers, are usually implemented using some sort of tagged union. A tagged union can be seen as the simplest kind of self-describing data format.
An efficient implementation using a disjoint-set data structure can perform each union and find operation on two sets in nearly constant amortized time (specifically, (()) time; () < for any plausible value of ), so the running time of this algorithm is essentially proportional to the number of walls available to the maze.
In that sense it is dual to the union-find data structure, which also maintains a partition into disjoint sets but in which the operations merge pairs of sets. In some applications of partition refinement, such as lexicographic breadth-first search, the data structure maintains as well an ordering on the sets in the partition.
Union-find essentially stores labels which correspond to the same blob in a disjoint-set data structure, making it easy to remember the equivalence of two labels by the use of an interface method E.g.: findSet(l). findSet(l) returns the minimum label value that is equivalent to the function argument 'l'.
Disjoint-set data structures [9] and partition refinement [10] are two techniques in computer science for efficiently maintaining partitions of a set subject to, respectively, union operations that merge two sets or refinement operations that split one set into two. A disjoint union may mean one of two things.