Improved Output-Sensitive Snap Rounding

Abstract This paper presents new algorithms for snap rounding an arrangement ${\mathcal{A}}$ of line segments in the plane. Snap rounding defines a set of hot pixels, which are unit squares centered on the integer grid points closest to the vertices of ${\mathcal{A}}$ . Snap rounding simplifies ${\mathcal{A}}$ by replacing every input segment by a piecewise linear curve connecting the centers of the hot pixels the segment intersects. Let ℋ be the set of all hot pixels, and for each h∈ℋ let is(h) be the number of segments with an intersection or endpoint inside h. If ${\mathcal{A}}$ contains n input segments, the running time of the first new algorithm is O(∑h∈ℋis(h)log n). This improves previous input- and output-sensitive algorithms by a factor of Θ(n) in the worst case. The second algorithm has an even better running time of O(∑h∈ℋed(h)log n); here ed(h) is the description complexity of the crossing pattern in h, which may be substantially less than is(h) and is never greater.