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The precise analysis of the performance of a disjoint-set forest is somewhat intricate. However, there is a much simpler analysis that proves that the amortized time for any m Find or Union operations on a disjoint-set forest containing n objects is O(m log * n), where log * denotes the iterated logarithm. [12] [13] [14] [15]
Two disjoint sets. In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set. [1] For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of two ...
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.
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 iteration through all edges performs two find operations and possibly one union operation per edge.
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
The interval scheduling problem is 1-dimensional – only the time dimension is relevant. The Maximum disjoint set problem is a generalization to 2 or more dimensions. This generalization, too, is NP-complete. Another variation is resource allocation, in which a set of intervals s are scheduled using resources k such that k is minimized. That ...
In theory, many other abstract data structures can be viewed as set structures with additional operations and/or additional axioms imposed on the standard operations. For example, an abstract heap can be viewed as a set structure with a min( S ) operation that returns the element of smallest value.
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.