Efficient Snap Rounding with Integer Arithmetic

In this paper we present a slightly modified definition of snap rounding, and provide two efficient algorithms that perform this rounding. The first algorithm takes n line segments as input and generates the set of snapped segments in O(|I| + Σc is(c) log n + |I∗ m|), where |I| is the complexity of the unrounded arrangement I, is(c) is the number of segments that have an intersection or endpoint in pixel column c, and I∗ m is the multiset of snapped segment fragments. The second algorithm generates the rounded arrangement of segments in O(|I| + Σc is(c) log n + |I∗| log n), where |I∗| is the complexity of the rounded arrangement I∗. Both use simple integer arithmetic to compute the rounded arrangement by sweeping a strip of unit width through the arrangement, are robust, and are practical to implement. They improve upon existing algorithms, since existing running times either include a logarithmic factor in |I|, (i.e., |I| log n), or depend upon the number of segments interacting within a particular hot pixel (is(h) and ed(h) [7], or |h| [3]), whereas ours are linear in |I| and depend upon the number of segments interacting in an entire hot column (is(c)), which is a much coarser partition of the plane.