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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 ...
A minimum single-trunk Steiner tree (MSTST) may be found in O(n log n) time. However simply finding all its edges requires linear time . The idea is that STSTs for a given point set essentially have only one "degree of freedom", which is the position of the horizontal trunk.
Minimum degree spanning tree; Minimum k-cut; Minimum k-spanning tree; Minor testing (checking whether an input graph contains an input graph as a minor); the same holds with topological minors; Steiner tree, or Minimum spanning tree for a subset of the vertices of a graph. [2] (The minimum spanning tree for an entire graph is solvable in ...
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]
Therefore, the k-minimum spanning tree must be formed by combining the optimal Steiner tree with enough of the zero-weight edges of the added trees to make the total tree size large enough. [ 2 ] Even for a graph whose edge weights belong to the set {1, 2, 3 }, testing whether the optimal solution value is less than a given threshold is NP ...
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.
The main motivation for studying the Hanan grid stems from the fact that it is known to contain a minimum length rectilinear Steiner tree for S. [1] It is named after Maurice Hanan, who was first [2] to investigate the rectilinear Steiner minimum tree and introduced this graph. [3]
For a unit square, the perimeter is 4, the perimeter minus the longest edge is 3, and the length of the minimum Steiner tree is +. However, a shorter, disconnected opaque forest is known, with length +. It consists of the minimum Steiner tree of three of the square's vertices, together with a line segment connecting the fourth vertex to the center.