Maximizing the Area of Overlap of Two Unions of Disks Under Rigid Motion

Let A and B be two sets of n resp. m disjoint unit disks in the plane, with m ≥ n. We consider the problem of finding a translation or rigid motion of A that maximizes the total area of overlap with B. The function describing the area of overlap is quite complex, even for combinatorially equivalent translations and, hence, we turn our attention to approximation algorithms. We give deterministic (1 − )-approximation algorithms for translations and for rigid motions, which run in O((nm/ 2 ) log(m/ )) and O((n2 m2 / 3 ) log m)) time, respectively. For rigid motions, we can also compute a (1 − )-approximation in O((m2 n4/3 ∆1/3 / 3 ) log n log m) time, where ∆ is the diameter of set A. Under the condition that the maximum area of overlap is at least a constant fraction of the area of A, we give a probabilistic (1 − )-approximation algorithm for rigid motions that runs in O((m2 / 4 ) log2 (m/ ) log m) time and succeeds with high probability. Our results generalize to the case ...

• Type: Conference
• Year: 2004
• Where: SWAT
• Authors: Mark de Berg, Sergio Cabello, Panos Giannopoulos, Christian Knauer, René van Oostrum, Remco C. Veltkamp
• Keywords: Algorithms, Disjoint Unit Disks, Log M, Rigid Motions, SWAT 2004