Journal ArticleDOI
A fast algorithm for Steiner trees
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TLDR
The heuristic algorithm has a worst case time complexity of O(¦S¦¦V¦2) on a random access computer and it guarantees to output a tree that spans S with total distance on its edges no more than 2(1−1/l) times that of the optimal tree.Abstract:
Given 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) on a random access computer and it guarantees to output a tree that spans S with total distance on its edges no more than 2(1?1/l) times that of the optimal tree, where l is the number of leaves in the optimal tree.read more
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References
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Journal ArticleDOI
Finding Minimum Spanning Trees
TL;DR: 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.
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On Steiner Minimal Trees with Rectilinear Distance
TL;DR: In this article, the authors consider Steiner minimal trees in the plane with rectilinear distance and show that the rectilINear distance between two points is at most 2.3.
Proceedings ArticleDOI
Some NP-complete geometric problems
TL;DR: It is shown 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 discretized version of the Euclidean metric.
Journal ArticleDOI
An 0(|E|loglog|V|) algorithm for finding minimum spanning trees☆
TL;DR: The algorithm is a modification of an algorithm by SoWn that works by successively enlarging, components al the MST, by shrinking each group of vertices to node, to obtain a new graph with at most odes.
Journal ArticleDOI
All shortest distances in a graph. An improvement to Dantzig's inductive algorithm
TL;DR: Dantzig's inductive algorithm is one of the best solutions to find all shortest distances in a graph and a change in the order of computations makes it possible to improve that algorithm by exploiting nonexistent arcs.