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In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a number of settings, they all require an optimal interconnect for a given set of objects and a predefined ...
In particular, because the Steiner tree problem is NP-hard to approximate to an approximation ratio better than 96/95, [3] the same is true for the k-minimum spanning tree problem. The best approximation known for the general problem achieves an approximation ratio of 2, and is by Garg (2005) . [ 4 ]
A number of algorithms exist which start from the rectilinear minimum spanning tree (RMST; the minimum spanning tree in the plane with rectilinear distance) and try to decrease its length by introducing Steiner points. The RMST itself may be up to 1.5 times longer than MRST. [6]
Minimum Steiner trees of vertices of regular polygons with N = 3 to 8 sides. The lowest network length L for N > 5 is the circumference less one side. Squares represent Steiner points. The Steiner tree of a subset of the vertices is the minimum tree that spans the given subset. Finding the Steiner tree is NP-Complete. [16]
The Steiner tree problem again seeks a system of line segments connecting all given points, but without requiring the segments to start and end only at given points. In this problem, additional points may be added as segment endpoints, allowing the Steiner tree to be shorter than the minimum spanning tree. [12]
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According to the National Christmas Tree Association, the average cost of a natural Christmas tree last year was around $70.
Example of rectilinear minimum spanning tree from random points. In graph theory, the rectilinear minimum spanning tree (RMST) of a set of n points in the plane (or more generally, in ) is a minimum spanning tree of that set, where the weight of the edge between each pair of points is the rectilinear distance between those two points.
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