## Approximating Geometrical Graphs Via Spanners and Banyans (1998)

Citations: | 61 - 0 self |

### BibTeX

@MISC{Rao98approximatinggeometrical,

author = {Satish B. Rao and Warren D. Smith},

title = {Approximating Geometrical Graphs Via Spanners and Banyans},

year = {1998}

}

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### Abstract

The main result of this paper is an improvement of Arora's method to find (1+ ffl) approximations for geometric NP-hard problems including the Euclidean Traveling Salesman Problem and the Euclidean Steiner Minimum Tree problems. For fixed dimension d and ffl, our algorithms run in O(N log N) time. An interesting byproduct of our work is the definition and construction of banyans, a generalization of graph spanners. A (1 + ffl)-banyan for a set of points A is a set of points A 0 and line segments S with endpoints in A [ A 0 such that a 1 + ffl optimal Steiner Minimum Tree for any subset of A is contained in S. We give a construction for banyans such that the total length of the line segments in S is within a constant factor of the length of the minimum spanning tree of A, and jA 0 j = O(jAj), when ffl and d are fixed. In this abbreviated paper, we only provide proofs of these results in two dimensions. The full paper on WDS's web page (http://www.neci.nj.nec.com/homepages/wds, c...