Improved Output-Sensitive Snap Rounding

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

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