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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 456739, 14 pages
Research Article

Incremental Construction of Generalized Voronoi Diagrams on Pointerless Quadtrees

College of Information System and Management, National University of Defense Technology, Changsha, Hunan 410073, China

Received 21 October 2013; Revised 5 January 2014; Accepted 6 January 2014; Published 26 February 2014

Academic Editor: Piermarco Cannarsa

Copyright © 2014 Quanjun Yin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In robotics, Generalized Voronoi Diagrams (GVDs) are widely used by mobile robots to represent the spatial topologies of their surrounding area. In this paper we consider the problem of constructing GVDs on discrete environments. Several algorithms that solve this problem exist in the literature, notably the Brushfire algorithm and its improved versions which possess local repair mechanism. However, when the area to be processed is very large or is of high resolution, the size of the metric matrices used by these algorithms to compute GVDs can be prohibitive. To address this issue, we propose an improvement on the current algorithms, using pointerless quadtrees in place of metric matrices to compute and maintain GVDs. Beyond the construction and reconstruction of a GVD, our algorithm further provides a method to approximate roadmaps in multiple granularities from the quadtree based GVD. Simulation tests in representative scenarios demonstrate that, compared with the current algorithms, our algorithm generally makes an order of magnitude improvement regarding memory cost when the area is larger than . We also demonstrate the usefulness of the approximated roadmaps for coarse-to-fine pathfinding tasks.