Cutting disjoint disks by straight lines
Discrete & Computational Geometry
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
The lifting model for reconfiguration
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Reconfigurations in graphs and grids
LATIN'06 Proceedings of the 7th Latin American conference on Theoretical Informatics
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Given a pair of start and target configurations, each consisting of n pairwise disjoint disks in the plane, what is the minimum number of moves that suffice for transforming the start configuration into the target configuration? In one move a disk slides in the plane without intersecting any other disk, so that its center moves along an arbitrary (open) continuous curve. We discuss efficient algorithms for this task and estimate their number of moves under different assumptions on disk radii and disk placements. For example, with n congruent disks, $\frac{3n}{2}+O(\sqrt{n {\rm log}n})$ moves always suffice for transforming the start configuration into the target configuration; on the other hand, $(1+\frac{1}{15}){\it n} - O(\sqrt{n})$ moves are sometimes necessary.