Geometric Spanner of Objects under L1 Distance

Authors:
Yongding Zhu;Jinhui Xu;Yang Yang;Naoki Katoh;Shin-Ichi Tanigawa
Affiliations:
Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, USA NY 14260;Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, USA NY 14260;Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, USA NY 14260;Department of Architecture and Architectural Systems, Kyoto University, Japan;Department of Architecture and Architectural Systems, Kyoto University, Japan
Venue:
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
Year:
2008

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Abstract

Geometric spanner is a fundamental structure in computational geometry and plays an important role in many geometric networks design applications. In this paper, we consider the following generalized geometric spanner problem under L1distance: Given a set of disjoint objects S, find a spanning network Gwith minimum size so that for any pair of points in different objects of S, there exists a path in Gwith length no more than ttimes their L1distance, where tis the stretch factor. Specifically, we focus on three types of objects: rectilinear segments, axis aligned rectangles, and rectilinear monotone polygons. By combining ideas of t-weekly dominating set, walls, aligned pairsand interval cover, we develop a 4-approximation algorithm (measured by the number of Steiner points) for each type of objects. Our algorithms run in near quadratic time, and can be easily implemented for practical applications.