Approximating Minimum Manhattan Networks in Higher Dimensions
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by
Aparna Das, Emden R. Gansner, Michael Kaufmann, Stephen Kobourov,
Joachim Spoerhase, Alexander Wolff
2012
Abstract
We study the minimum Manhattan network problem, which is defined as follows.
Given a set of points called terminals in ^d, find a minimum-length
network such that each pair of terminals is connected by a set of axis-parallel
line segments whose total length is equal to the pair's Manhattan (that is,
L_1-) distance. The problem is NP-hard in 2D and there is no PTAS for 3D
(unless P= NP). Approximation algorithms are known for 2D, but
not for 3D.
We present, for any fixed dimension d and any >0, an
O(n^)-approximation algorithm. For 3D, we also give a
4(k-1)-approximation algorithm for the case that the terminals are contained
in the union of k > 2 parallel planes.
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