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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}.
Compared to other parallel-processing models, Linda is more orthogonal in treating process coordination as a separate activity from computation, and it is more general in being able to subsume various levels of concurrency—uniprocessor, multi-threaded multiprocessor, or networked—under a single model.
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.
There is an optimization version of the partition problem, which is to partition the multiset S into two subsets S 1, S 2 such that the difference between the sum of elements in S 1 and the sum of elements in S 2 is minimized. The optimization version is NP-hard, but can be solved efficiently in practice. [4]
Let C i (for i between 1 and k) be the sum of subset i in a given partition. Instead of minimizing the objective function max(C i), one can minimize the objective function max(f(C i)), where f is any fixed function. Similarly, one can minimize the objective function sum(f(C i)), or maximize min(f(C i)), or maximize sum(f(C i)).
Sometimes the items derive from a common type; even deriving from the most general type of a programming language such as object or variant. Although easily confused with implementations in programming languages, collection, as an abstract concept, refers to mathematical concepts which can be misunderstood when the focus is on an implementation.
As an example, a term rewriting system for "multiplying out" mathematical expressions could contain a rule x*(y+z) → (x*y) + (x*z). In order to prove termination, a reduction ordering (>) must be found with respect to which the term x*(y+z) is greater than the term (x*y)+(x*z). This is not trivial, since the former term contains both fewer ...