![]() The lowest network length L for N > 5 is the circumference less one side. Minimum Steiner trees of vertices of regular polygons with N = 3 to 8 sides. Despite the pessimistic worst-case complexity, several Steiner tree problem variants, including the Steiner tree problem in graphs and the rectilinear Steiner tree problem, can be solved efficiently in practice, even for large-scale real-world problems. Most versions of the Steiner tree problem are NP-hard, but some restricted cases can be solved in polynomial time. However, practical applications usually require variations, giving rise to a multitude of Steiner tree problem variants. The Steiner tree problem in graphs has applications in circuit layout or network design. However, while both the non-negative shortest path and the minimum spanning tree problem are solvable in polynomial time, the decision variant of the Steiner tree problem in graphs is NP-complete (which implies that the optimization variant is NP-hard) in fact, the decision variant was among Karp's original 21 NP-complete problems. ![]() If, on the other hand, all vertices are terminals, the Steiner tree problem in graphs is equivalent to the minimum spanning tree. If a Steiner tree problem in graphs contains exactly two terminals, it reduces to finding the shortest path. The Steiner tree problem in graphs can be seen as a generalization of two other famous combinatorial optimization problems: the (non-negative) shortest path problem and the minimum spanning tree problem. Further well-known variants are the Euclidean Steiner tree problem and the rectilinear minimum Steiner tree problem. Given an undirected graph with non-negative edge weights and a subset of vertices, usually referred to as terminals, the Steiner tree problem in graphs requires a tree of minimum weight that contains all terminals (but may include additional vertices). One well-known variant, which is often used synonymously with the term Steiner tree problem, is the Steiner tree problem in graphs. 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 objective function. 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. Solution for four points-there are two Steiner points, S 1 and S 2
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