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The Karmarkar–Karp (KK) bin packing algorithms are several related approximation algorithm for the bin packing problem. [1] The bin packing problem is a problem of packing items of different sizes into bins of identical capacity, such that the total number of bins is as small as possible. Finding the optimal solution is computationally hard.
Random column packing is the practice of packing a distillation column with randomly fitting filtration material in order to optimize surface area over which reactants can interact while minimizing the complexity of construction of such columns. Random column packing is an alternative to structured column packing.
Square packing in a circle is a related problem of packing n unit squares into a circle with radius as small as possible. For this problem, good solutions are known for n up to 35. Here are the minimum known solutions for up to n =12: [ 11 ] (Only the cases n=1 and n=2 are known to be optimal)
It is normally injected intramuscularly, though can also be injected intra-articularily (directly into the joint) in horses or subcutaneously in off-label uses. [2] Giving PSGAG intra-articularily requires it to be given aseptically, [8] and is sometimes supplemented by the antibiotic amikacin to prevent infection. [1] [3] [7]
An important special case of bin-packing is that which the item sizes form a divisible sequence (also called factored). A special case of divisible item sizes occurs in memory allocation in computer systems, where the item sizes are all powers of 2. In this case, FFD always finds the optimal packing. [6]: Thm.2
For example, in the case =, it is known that the optimal packing is not a tetrahedral packing like the classical packing of cannon balls, but is likely some kind of octahedral shape. [ 1 ] The sudden transition in optimal packing shape is jokingly known by some mathematicians as the sausage catastrophe (Wills, 1985). [ 4 ]
[1] [2] [3] A conjecture of Paul Erdős and Norman Oler states that, if n is a triangular number, then the optimal packings of n − 1 and of n circles have the same side length: that is, according to the conjecture, an optimal packing for n − 1 circles can be found by removing any single circle from the optimal hexagonal packing of n circles ...
Sphere packing in a sphere is a three-dimensional packing problem with the objective of packing a given number of equal spheres inside a unit sphere. It is the three-dimensional equivalent of the circle packing in a circle problem in two dimensions.