Comments on "Filling algorithms and analyses for layout density control"

Theorem 2 of the title paper (A.B. Kahng et al., ibid. vol. 18, pp.445-462, 1999) presents layout density bounds for any fixed r-dissection w-by-w window given the layout density of at least L and at most U for all w-by-w windows whose bottom left corners are at points (i(w/r), j(w/r)), i, j=0, 1, ..., r((n/w)-1) on an n-by-n layout plane. However, the bounds are not tight for certain combinations of U (L) and r. Here, the authors present an approach to obtaining the tight lower and upper bounds for all possible combinations of U (L) and r.