# 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…

## 848 Citations

### A faster approximation algorithm for the Steiner problem in graphs

- Mathematics, Computer ScienceActa Informatica
- 2004

The essence of the algorithm is to find a generalized minimum spanning tree of a graph in one coherent phase as opposed to the previous multiple steps approach.

### An Effective Construction Algorithm for the Steiner Tree Problem Based on Edge Betweenness

- Mathematics, Computer Science
- 2016

The improved KMB algorithm, which is an efficient construction method for Steiner tree problems, is enhanced by considering edge betweenness, and the results of numerical simulations indicate that the algorithm shows good performances for various types of benchmark Steiner Tree problems.

### Approximation Algorithms for Steiner Trees 7.1 the Problem and Its Complexity

- Computer Science

It is likely that any algorithm ever designed for nding Steiner minimal trees will be ineecient, which would give eecient algorithms for solving all NP-complete problems.

### Approximation Algorithms for Steiner Trees

- Computer Science

The problem, even with unit costs on the edges, is NP-hard, so an eecient algorithm for nding an optimal solution would give ineecient algorithms for solving all NP-complete problems.

### FasterDSP: A Faster Approximation Algorithm for Directed Steiner Tree Problem

- Computer Science, MathematicsJ. Inf. Sci. Eng.
- 2006

A faster approximation algorithm improving Charikar et al.'s DSP algorithm with a better time complexity, O(n'k' + n 2 k + nm), where m is the number of edges, and an amended √8k - S In k factor for the 2-level Steiner tree, where δ = √6 - 2 = 0.4494.

### On the Integrality Gap of Directed Steiner Tree Problem

- Mathematics, Computer Science
- 2014

A polynomial-time O(log |X|)-approximation for quasi-bipartite instances of Directed Steiner Tree is presented and bounds the integrality gap of the natural LP relaxation by the same quantity.

### Node-weighted Steiner tree approximation in unit disk graphs

- Computer Science, MathematicsJ. Comb. Optim.
- 2009

This paper is the first to show that even though for unit disk graphs, the node-weighted Steiner tree problem is still NP-hard and it has a polynomial time constant approximation, and presents a 2.5ρ-approximation where ρ is the best known performance ratio for polynometric time approximation of classical Steiner minimum tree problem in graphs.

### Steiner Problems with Limited Number of Branching Nodes

- Computer Science, MathematicsSIROCCO
- 2013

A polynomial algorithm when the input graph is acyclic and an other algorithm when k is fixed in an input graph of bounded treewidth are proposed, and an n e -inapproximability proof is provided, for any e < 1.

### A PTAS for Node-Weighted Steiner Tree in Unit Disk Graphs

- Mathematics, Computer ScienceCOCOA
- 2009

This paper studies the node-weighted Steiner tree problem in unit disk graphs and presents a (1+*** )-approximation algorithm for any *** > 0, when the given set of vertices is c -local.

## References

SHOWING 1-10 OF 10 REFERENCES

### Finding Minimum Spanning Trees

- Mathematics, Computer ScienceSIAM J. Comput.
- 1976

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.

### An O(|E| log log |V|) Algorithm for Finding Minimum Spanning Trees

- Computer ScienceInf. Process. Lett.
- 1975

### A note on two problems in connexion with graphs

- Mathematics, Computer ScienceNumerische 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

- Mathematics
- 1976

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

- MathematicsSTOC '76
- 1976

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

- Mathematics
- 1968

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

- ChemistryCommun. ACM
- 1962

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

- Computer ScienceDiscret. Math.
- 1973

### Reducibility Among Combinatorial Problems

- Computer Science50 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

- Computer Science
- 1972

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