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  2. Partition of a set - Wikipedia

    en.wikipedia.org/wiki/Partition_of_a_set

    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 ...

  3. Multiway number partitioning - Wikipedia

    en.wikipedia.org/wiki/Multiway_number_partitioning

    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).

  4. Set splitting problem - Wikipedia

    en.wikipedia.org/wiki/Set_splitting_problem

    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.

  5. Balanced number partitioning - Wikipedia

    en.wikipedia.org/wiki/Balanced_number_partitioning

    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 ...

  6. Bell number - Wikipedia

    en.wikipedia.org/wiki/Bell_number

    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 ...

  7. Largest differencing method - Wikipedia

    en.wikipedia.org/wiki/Largest_differencing_method

    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 ...

  8. Noncrossing partition - Wikipedia

    en.wikipedia.org/wiki/Noncrossing_partition

    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.

  9. Partition refinement - Wikipedia

    en.wikipedia.org/wiki/Partition_refinement

    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]