disjoint-set data structure

A simple approach to creating a disjoint-set data structure is to create a linked list for each set.

Detecting cycles is possible in near-constant time with the aid of a disjoint-set data structure.

The following code is implemented with disjoint-set data structure:

Sethi also shows how to implement the level assignment stage of the algorithm efficiently by using a disjoint-set data structure.

Next, we use a disjoint-set data structure (Union&Find) to keep track of which vertices are in which components.

As in Kruskal's algorithm, tracking components of T can be done efficiently using a disjoint-set data structure.

A similar data structure appears in disjoint-set forests, a type of disjoint-set data structure.

Because it supports these two operations, a disjoint-set data structure is sometimes called a union-find data structure or merge-find set.

Already during his years at Michigan, Fischer studied disjoint-set data structures together with Bernard Galler.

This inverse appears in the time complexity of some algorithms, such as the disjoint-set data structure and Chazelle's algorithm for minimum spanning trees.

Disjoint-set data structures model the partitioning of a set, for example to keep track of the connected components of an undirected graph.

Another significant contribution was the analysis of the disjoint-set data structure; he was the first to prove the optimal runtime involving the inverse Ackermann function.

Jeffery Westbrook and Robert Tarjan (1992) developed an efficient data structure for this problem based on disjoint-set data structures.

In fact, this is asymptotically optimal: Fredman and Saks showed in 1989 that words must be accessed by any disjoint-set data structure per operation on average.

More generally, it is easy to determine computationally whether a graph is connected (for example, by using a disjoint-set data structure), or to count the number of connected components.

In computing, a disjoint-set data structure is a data structure that keeps track of a set of elements partitioned into a number of disjoint (nonoverlapping) subsets.

Even a simple disjoint-set data structure such as disjoint-set forests with union by rank can perform O(E) operations in O(E log V) time.

There are also efficient algorithms to dynamically track the connected components of a graph as vertices and edges are added, as a straightforward application of disjoint-set data structures.

Some natural examples include consistency statements from logic, several statements related to Ramsey theory such as Szemerédi's lemma and the graph minor theorem, and Tarjan's algorithm for the disjoint-set data structure.

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).

In the disjoint-set data structure, m represents the number of operations while n represents the number of elements; in the minimum spanning tree algorithm, m represents the number of edges while n represents the number of vertices.

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.