Excess in arrangements of segments (1996) [2 citations — 2 self]
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Abstract:
Let S be a set of n line segments in the plane. The excess of S is the number of repetitions of segments of S along the boundary of the same face of A(S), summed over all segments and faces. We show that the excess of S is at most O(n log log n), improving a previous O(n log n) bound given in [1]. In this note we study the notion, introduced in [1], of the excess of an arrangement A(S) of a set S of n line segments in the plane in general position. Intuitively, the excess counts the number of repetitions of segments along the boundary of the same face of A(S), summed over all faces. It is formally defined as follows. A side of a segment e is any one of the two halfplanes bounded by the line containing e. A 1-border is a pair (e; R), where e is a segment and R is a side of e. A 1-border (e; R) bounds a face f of A(S) if some portion e 0 of e appears as an edge of f, so that the intersection of R with a sufficiently small neighborhood of e 0 is contained in f. Let e be a segment in S. If some face f in A(S) has a 1-border of the form (e 0
Citations
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| 27 | Castles in the air revisited – Aronov, Sharir - 1994 |
