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Partitions of a 4-element set ordered by refinement. A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarser than α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentation of ρ. In that case, it is written that ...
The 3-partition problem - a different and harder problem, in which the number of subsets is not considered a fixed parameter, but is determined by the input (the number of sets is the number of integers divided by 3).
In computational complexity theory, the set splitting problem is the following decision problem: given a family F of subsets of a finite set S, decide whether there exists a partition of S into two subsets S 1, S 2 such that all elements of F are split by this partition, i.e., none of the elements of F is completely in S 1 or S 2.
Balanced number partitioning is a variant of multiway number partitioning in which there are constraints on the number of items allocated to each set. The input to the problem is a set of n items of different sizes, and two integers m, k. The output is a partition of the items into m subsets, such that the number of items in each subset is at ...
Partitions of sets can be arranged in a partial order, showing that each partition of a set of size n "uses" one of the partitions of a set of size n − 1. The 52 partitions of a set with 5 elements. In general, is the number of partitions of a set of size . A partition of a set is defined as a family of nonempty, pairwise disjoint subsets of ...
Assign each pair in turn to different sets: the largest in the pair to the set with the smallest sum, and the smallest in the pair to the set with the largest sum. For two-way partitioning, when inputs are uniformly-distributed random variables, the expected difference between largest and smallest sum is O ( log n / n 2 ) {\displaystyle O ...
A partition of a set S is a set of non-empty, pairwise disjoint subsets of S, called "parts" or "blocks", whose union is all of S.Consider a finite set that is linearly ordered, or (equivalently, for purposes of this definition) arranged in a cyclic order like the vertices of a regular n-gon.
Partition refinement also forms a key step in lexicographic breadth-first search, a graph search algorithm with applications in the recognition of chordal graphs and several other important classes of graphs. Again, the disjoint set elements are vertices and the set X represents sets of neighbors, so the algorithm takes linear time. [8] [9]