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In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, [1] allows for multiple instances for each of its elements.The number of instances given for each element is called the multiplicity of that element in the multiset.
Python has built-in set and frozenset types since 2.4, and since Python 3.0 and 2.7, supports non-empty set literals using a curly-bracket syntax, e.g.: {x, y, z}; empty sets must be created using set(), because Python uses {} to represent the empty dictionary.
The most naïve algorithm would be to cycle through all subsets of n numbers and, for every one of them, check if the subset sums to the right number. The running time is of order (), since there are subsets and, to check each subset, we need to sum at most n elements.
Common to all versions are a set of n items, with each item having an associated profit p j and weight w j. The binary decision variable x j is used to select the item. The objective is to pick some of the items, with maximal total profit, while obeying that the maximum total weight of the chosen items must not exceed W .
The set of subsets of a given set (its power set) ordered by inclusion (see Fig. 1). Similarly, the set of sequences ordered by subsequence, and the set of strings ordered by substring. The set of natural numbers equipped with the relation of divisibility. (see Fig. 3 and Fig. 6) The vertex set of a directed acyclic graph ordered by reachability.
In the set cover optimization problem, the input is a pair (,), and the task is to find a set cover that uses the fewest sets. The decision version of set covering is NP-complete . It is one of Karp's 21 NP-complete problems shown to be NP-complete in 1972.
Another variation is the random k-labelsets (RAKEL) algorithm, which uses multiple LP classifiers, each trained on a random subset of the actual labels; label prediction is then carried out by a voting scheme. [7] A set of multi-label classifiers can be used in a similar way to create a multi-label ensemble classifier.
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 .