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In computer science, a disjoint-set data structure, also called a union–find data structure or merge–find set, is a data structure that stores a collection of disjoint (non-overlapping) sets. Equivalently, it stores a partition of a set into disjoint subsets .
Set covering is equivalent to the hitting set problem. That is seen by observing that an instance of set covering can be viewed as an arbitrary bipartite graph, with the universe represented by vertices on the left, the sets represented by vertices on the right, and edges representing the membership of elements to sets. The task is then to find ...
Formally, the nearest-neighbor (NN) search problem is defined as follows: given a set S of points in a space M and a query point q ∈ M, find the closest point in S to q. Donald Knuth in vol. 3 of The Art of Computer Programming (1973) called it the post-office problem, referring to an application of assigning to a residence the nearest post ...
The complexity of the best known algorithms is exponential in the smaller of the two parameters n and L. The problem is NP-hard even when all input integers are positive (and the target-sum T is a part of the input). This can be proved by a direct reduction from 3SAT. [2] It can also be proved by reduction from 3-dimensional matching (3DM): [3]
Set packing is a classical NP-complete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP-complete problems. Suppose one has a finite set S and a list of subsets of S. Then, the set packing problem asks if some k subsets in the list are pairwise disjoint (in other words, no two of them share an element).
The complexity of the dynamic closest pair algorithm cited above is exponential in the dimension , and therefore such an algorithm becomes less suitable for high-dimensional problems. An algorithm for the dynamic closest-pair problem in d {\displaystyle d} dimensional space was developed by Sergey Bespamyatnikh in 1998. [ 10 ]
The maximal independent set problem was originally thought to be non-trivial to parallelize due to the fact that the lexicographical maximal independent set proved to be P-Complete; however, it has been shown that a deterministic parallel solution could be given by an reduction from either the maximum set packing or the maximal matching problem ...
Set subtraction complexity: To manage the many identities involving set subtraction, this section is divided based on where the set subtraction operation and parentheses are located on the left hand side of the identity.