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In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation.
In geometry, space partitioning is the process of dividing an entire space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set). In other words, space partitioning divides a space into non-overlapping regions. Any point in the space can then be identified to lie in exactly one of the regions.
For example, 4 can be partitioned in five distinct ways: 4 3 + 1 2 + 2 2 + 1 + 1 1 + 1 + 1 + 1. The only partition of zero is the empty sum, having no parts. The order-dependent composition 1 + 3 is the same partition as 3 + 1, and the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition as 2 + 1 + 1.
Generally, a partition is a division of a whole into non-overlapping parts. Among the kinds of partitions considered in mathematics are partition of a set or an ordered partition of a set,
A polygon partition problem is a problem of finding a partition which is minimal in some sense, for example a partition with a smallest number of units or with units of smallest total side-length. Polygon partitioning is an important class of problems in computational geometry. There are many different polygon partition problems, depending on ...
If there is a remainder in solving a partition problem, the parts will end up with unequal sizes. For example, if 52 cards are dealt out to 5 players, then 3 of the players will receive 10 cards each, and 2 of the players will receive 11 cards each, since = +.
Equal-cardinality partition is a variant in which both parts should have an equal number of items, in addition to having an equal sum. This variant is NP-hard too. [5]: SP12 Proof. Given a standard Partition instance with some n numbers, construct an Equal-Cardinality-Partition instance by adding n zeros. Clearly, the new instance has an equal ...
In geometry, a dissection problem is the problem of partitioning a geometric figure (such as a polytope or ball) into smaller pieces that may be rearranged into a new figure of equal content. In this context, the partitioning is called simply a dissection (of one polytope into another). It is usually required that the dissection use only a ...