A fast algorithm for Steiner trees

@article{Kou2004AFA,
  title={A fast algorithm for Steiner trees},
  author={Lawrence T. Kou and George Markowsky and Leonard Berman},
  journal={Acta Informatica},
  year={2004},
  volume={15},
  pages={141-145}
}
SummaryGiven an undirected distance graph G=(V, E, d) and a set S, where V is the set of vertices in G, E is the set of edges in G, d is a distance function which maps E into the set of nonnegative numbers and S⊑V is a subset of the vertices of V, the Steiner tree problem is to find a tree of G that spans S with minimal total distance on its edges. In this paper, we analyze a heuristic algorithm for the Steiner tree problem. The heuristic algorithm has a worst case time complexity of O(¦S¦¦V¦2… 

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References

SHOWING 1-10 OF 10 REFERENCES

Finding Minimum Spanning Trees

This paper studies methods for finding minimum spanning trees in graphs and results include relationships with other problems which might lead general lower bound for the complexity of the minimum spanning tree problem.

A note on two problems in connexion with graphs

  • E. Dijkstra
  • Mathematics, Computer Science
    Numerische Mathematik
  • 1959
A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.

On Steiner Minimal Trees with Rectilinear Distance

We consider Steiner minimal trees in the plane with rectilinear distance. The rectilinear distance $d(p_1 ,p_2 )$ between two points $p_1 $, $p_2 $ is $| {x_1 - x_2 } | + | {y_1 - y_2 } |$, where the

Some NP-complete geometric problems

We show that the STEINER TREE problem and TRAVELING SALESMAN problem for points in the plane are NP-complete when distances are measured either by the rectilinear (Manhattan) metric or by a natural

Steiner Minimal Trees

A Steiner minimal tree for given points $A_1 , \cdots ,A_n $ in the plane is a tree which interconnects these points using lines of shortest possible total length. In order to achieve minimum lengt...

Algorithm 97: Shortest path

The procedure was originally programmed in FORTRAN for the Control Data 160 desk-size computer and was limited to te t ra t ion because subroutine recursiveness in CONTROL Data 160 FORTRan has been held down to four levels in the interests of economy.

All shortest distances in a graph. An improvement to Dantzig's inductive algorithm

Reducibility Among Combinatorial Problems

  • R. Karp
  • Computer Science
    50 Years of Integer Programming
  • 1972
Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability.

Reducibility among combinatorial problems" in complexity of computer computations

In his 1972 paper, Reducibility Among Combinatorial Problems, Richard Karp used Stephen Cooks 1971 theorem that the boolean satisfiability problem is.