The minimum-area spanning tree problem

Authors:
Paz Carmi;Matthew J. Katz;Joseph S. B. Mitchell
Affiliations:
Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel;Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel;Department of Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY
Venue:
Computational Geometry: Theory and Applications
Year:
2006

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Abstract

Motivated by optimization problems in sensor coverage, we formulate and study the Minimum-Area Spanning Tree (MAST) problem: Given a set P of n points in the plane, find a spanning tree of P of minimum "area", where the area of a spanning tree T is the area of the union of the n - 1 disks whose diameters are the edges in T. We prove that the Euclidean minimum spanning tree of P is a constant-factor approximation for MAST. We then apply this result to obtain constant-factor approximations for the Minimum-Area Range Assignment (MARA) problem, for the Minimum-Area Connected Disk Graph (MACDG) problem, and for the Minimum-Area Tour (MAT) problem. The first problem is a variant of the power assignment problem in radio networks, the second problem is a related natural problem, and the third problem is a variant of the traveling salesman problem.