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The required output is a partition of S into k subsets, such that the sums in the subsets are as nearly equal as possible. Greedy algorithms process the numbers sequentially, and insert the next number into a bin in which the sum of numbers is currently smallest.
The input to the problem is a multiset S of numbers (usually integers), whose sum is k*T. The associated decision problem is to decide whether S can be partitioned into k subsets such that the sum of each subset is exactly T. There is also an optimization problem: find a partition of S into k subsets, such that the k sums are "as near as ...
The variant in which all inputs are positive, and the target sum is exactly half the sum of all inputs, i.e., = (+ +). This special case of SSP is known as the partition problem . SSP can also be regarded as an optimization problem : find a subset whose sum is at most T , and subject to that, as close as possible to T .
In the subset sum problem, the goal is to find a subset of S whose sum is a certain target number T given as input (the partition problem is the special case in which T is half the sum of S). In multiway number partitioning , there is an integer parameter k , and the goal is to decide whether S can be partitioned into k subsets of equal sum ...
The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences.
For minimizing the largest sum, they present an EPTAS for constant k, and FPTAS for constant m. For maximizing the smallest sum, they present a 1/(k − 1) approximation algorithm for the general case, and an EPTAS for constant k. They also study a more general objective: minimizing the lp-norm of the vector of sums. They prove that the layered ...
The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances; see partition function (mathematics) for
A Riemann sum of over [,] with partition is defined as = = (), where = and [,]. [1] One might produce different Riemann sums depending on which x i ∗ {\displaystyle x_{i}^{*}} 's are chosen. In the end this will not matter, if the function is Riemann integrable , when the difference or width of the summands Δ x i {\displaystyle \Delta x_{i ...