Concepts inMinimum-weight triangulation is NP-hard
Minimum-weight triangulation
In computational geometry and computer science, the minimum-weight triangulation problem is the problem of finding a triangulation of minimal total edge length. That is, an input polygon or the convex hull of an input point set must be subdivided into triangles that meet edge-to-edge and vertex-to-vertex, in such a way as to minimize the sum of the perimeters of the triangles. The problem is NP-hard for point set inputs, but may be approximated to any desired degree of accuracy.
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NP-hard
NP-hard, in computational complexity theory, is a class of problems that are, informally, "at least as hard as the hardest problems in NP". A problem H is NP-hard if and only if there is an NP-complete problem L that is polynomial time Turing-reducible to H (i.e. , L¿¿¿TH). In other words, L can be solved in polynomial time by an oracle machine with an oracle for H.
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Computer-assisted proof
A computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program.
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Decision problem
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem. The answer can be either 'yes' or 'no', and depends upon the values of x and y.
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Dynamic programming
In mathematics and computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller and optimal substructure (described below). When applicable, the method takes far less time than naive methods. The key idea behind dynamic programming is quite simple.
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N-skeleton
This article is not about the topological skeleton concept of computer graphics In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex refers to the subspace Xn that is the union of the simplices of X (resp. cells of X) of dimensions m ¿ n. In other words, given an inductive definition of a complex, the n-skeleton is obtained by stopping at the n-th step. These subspaces increase with n.
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Planar graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e. , it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. A planar graph already drawn in the plane without edge intersections is called a plane graph or planar embedding of the graph.
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Reduction (complexity)
In computability theory and computational complexity theory, a reduction is a transformation of one problem into another problem. Depending on the transformation used this can be used to define complexity classes on a set of problems. Intuitively, problem A is reducible to problem B if solutions to B exist and give solutions to A whenever A has solutions. Thus, solving A cannot be harder than solving B.
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