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The variant where variables are required to be 0 or 1, called zero-one linear programming, and several other variants are also NP-complete [2] [3]: MP1 Some problems related to Job-shop scheduling; Knapsack problem, quadratic knapsack problem, and several variants [2] [3]: MP9 Some problems related to Multiprocessor scheduling
In the ray tracing problem for a 3-dimensional system of reflective or refractive objects, determining if a ray beginning at a given position and direction eventually reaches a certain point. [ 16 ] Determining if a particle path of an ideal fluid on a three dimensional domain eventually reaches a certain region in space.
The problem of finding the smallest ball such that k disjoint open unit balls may be packed inside it has a simple and complete answer in n-dimensional Euclidean space if +, and in an infinite-dimensional Hilbert space with no restrictions. It is worth describing in detail here, to give a flavor of the general problem.
HackerRank's programming challenges can be solved in a variety of programming languages (including Java, C++, PHP, Python, SQL, and JavaScript) and span multiple computer science domains. [ 2 ] HackerRank categorizes most of their programming challenges into a number of core computer science domains, [ 3 ] including database management ...
There is an exponential increase in volume associated with adding extra dimensions to a mathematical space.For example, 10 2 = 100 evenly spaced sample points suffice to sample a unit interval (try to visualize a "1-dimensional" cube) with no more than 10 −2 = 0.01 distance between points; an equivalent sampling of a 10-dimensional unit hypercube with a lattice that has a spacing of 10 −2 ...
Fair cake-cutting is a kind of fair division problem. The problem involves a heterogeneous resource, such as a cake with different toppings, that is assumed to be divisible – it is possible to cut arbitrarily small pieces of it without destroying their value. The resource has to be divided among several partners who have different preferences ...
A simple example of an NP-hard problem is the subset sum problem. Informally, if H is NP-hard, then it is at least as difficult to solve as the problems in NP . However, the opposite direction is not true: some problems are undecidable , and therefore even more difficult to solve than all problems in NP, but they are probably not NP-hard ...
The Z-ordering can be used to efficiently build a quadtree (2D) or octree (3D) for a set of points. [4] [5] The basic idea is to sort the input set according to Z-order.Once sorted, the points can either be stored in a binary search tree and used directly, which is called a linear quadtree, [6] or they can be used to build a pointer based quadtree.