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In some cases a multiset in this counting sense may be generalized to allow negative values, as in Python. C++'s Standard Template Library implements both sorted and unsorted multisets. It provides the multiset class for the sorted multiset, as a kind of associative container, which implements this multiset using a self-balancing binary search ...
Also, a monomial is a multiset of indeterminates; for example, the monomial x 3 y 2 corresponds to the multiset {x, x, x, y, y}. A multiset corresponds to an ordinary set if the multiplicity of every element is 1. An indexed family (a i) i∈I, where i varies over some index set I, may define a multiset, sometimes written {a i}.
In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class. A collection F {\displaystyle F} of subsets of a given set S {\displaystyle S} is called a family of subsets of S {\displaystyle S} , or a family of sets over S ...
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number.
In C++, associative containers are a group of class templates in the standard library of the C++ programming language that implement ordered associative arrays. [1] Being templates , they can be used to store arbitrary elements, such as integers or custom classes.
C++'s Standard Template Library provides the multimap container for the sorted multimap using a self-balancing binary search tree, [1] and SGI's STL extension provides the hash_multimap container, which implements a multimap using a hash table. [2] As of C++11, the Standard Template Library provides the unordered_multimap for the unordered ...
Throughout this article, capital letters (such as ,,,,, and ) will denote sets.On the left hand side of an identity, typically, will be the leftmost set, will be the middle set, and
The 3-partition problem remains NP-complete even when the integers in S are bounded above by a polynomial in n.In other words, the problem remains NP-complete even when representing the numbers in the input instance in unary. i.e., 3-partition is NP-complete in the strong sense or strongly NP-complete.