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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.
When used to implement a set of stacks, the structure is called a spaghetti stack, cactus stack or saguaro stack (after the saguaro, a kind of cactus). [1] Parent pointer trees are also used as disjoint-set data structures. The structure can be regarded as a set of singly linked lists that share part of their structure, in particular, their ...
Once sorted, it is possible to loop through the edges in sorted order in constant time per edge. Next, use a disjoint-set data structure, with a set of vertices for each component, to keep track of which vertices are in which components. Creating this structure, with a separate set for each vertex, takes V operations and O(V) time. The final ...
This graph becomes disconnected when the right-most node in the gray area on the left is removed This graph becomes disconnected when the dashed edge is removed.. In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more ...
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 the online version of the problem, vertices and edges are added (but not removed) dynamically, and a data structure must maintain the biconnected components. Jeffery Westbrook and Robert Tarjan (1992) [ 4 ] developed an efficient data structure for this problem based on disjoint-set data structures .
In the disjoint-set data structure, the structure represents a collection of disjoint sets; there is an update operation, called Union, which unites two sets, and a query operation, called Find, which identifies the set to which a given element belongs.