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The fair polygon partitioning problem [20] is to partition a (convex) polygon into (convex) pieces with an equal perimeter and equal area (this is a special case of fair cake-cutting). Any convex polygon can be easily cut into any number n of convex pieces with an area of exactly 1/n. However, ensuring that the pieces have both equal area and ...
In the mathematical discipline of numerical linear algebra, a matrix splitting is an expression which represents a given matrix as a sum or difference of matrices. Many iterative methods (for example, for systems of differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices.
This number is given by the 5th Catalan number. It is trivial to triangulate any convex polygon in linear time into a fan triangulation, by adding diagonals from one vertex to all other non-nearest neighbor vertices. The total number of ways to triangulate a convex n-gon by non-intersecting diagonals is the (n−2)nd Catalan number, which equals
In a bin packing problem, people are given: A container, usually a two- or three-dimensional convex region, possibly of infinite size. Multiple containers may be given depending on the problem. A set of objects, some or all of which must be packed into one or more containers. The set may contain different objects with their sizes specified, or ...
For a given set of points S = {p 1, p 2, ..., p n}, the farthest-point Voronoi diagram divides the plane into cells in which the same point of P is the farthest point. A point of P has a cell in the farthest-point Voronoi diagram if and only if it is a vertex of the convex hull of P .
If the array abstraction does not support true negative indices (as the arrays of Ada and Pascal do), then negative indices for the bounds of the slice for a given dimension are sometimes used to specify an offset from the end of the array in that dimension. In 1-based schemes, -1 generally indicates the second-to-last item, while in a 0-based ...
Then, this initial bounding box is partitioned into a grid of smaller cubes, and grid points near the boundary of the convex hull of the input are used as a coreset, a small set of points whose optimum bounding box approximates the optimum bounding box of the original input. Finally, O'Rourke's algorithm is applied to find the exact optimum ...
There is only one way (up to rotation and reflection) to divide a square into two similar rectangles. However, there are three distinct ways of partitioning a square into three similar rectangles: [1] [2] The trivial solution given by three congruent rectangles with aspect ratio 3:1.