We consider the following problem. Given a polygon P, possibly with holes, and having n vertices, compute a pair of equal radius disks that do not intersect each other, are contained in P, and whose radius is maximized. Our main result is a simple randomized algorithm whose expected running time, on any input, is O(n log n). This is optimal in the algebraic decision tree model of computation.

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Journal of Discrete Algorithms
School of Computer Science

Bose, P, Morin, P, & Vigneron, A. (Antoine). (2004). Packing two disks into a polygonal environment. Journal of Discrete Algorithms, 2(3), 373–380. doi:10.1016/j.jda.2003.12.003