The Maximum-Level Vertex in an Arrangement of Lines

Let L be a set of n lines in the plane, not necessarily general position. We present an efficient algorithm for finding all vertices arrangement $$\mathcal{A}(L)$$ maximum level, where level vertex v is number that pass strictly below v. The problem, posed Exercise 8.13 de Berg et al. (Computational Geometry. Algorithms and Applications. Springer, Berlin (2008)), appears to much harder than it seems at first sight, as this might on upper envelope lines. assume are distinct, distinguish between two cases, depending whether or contains bounded edge. In former case, we show above any $$v_0$$ only $$O(\log n)$$ . latter establish similar property holds after remove some incident single envelope. algorithms run, both optimal $$O(n\log time. then consider case distinct. This setup more challenging, computes maximum-level time $$O(n^{4/3}\log ^{3}\!n)$$ Finally, related combinatorial question degenerate arrangements, many may intersect point, but distinct: bound complexity weighted k-level such arrangement, weight through vertex. $$O(n^{4/3})$$ , which matches corresponding non-degenerate use analysis one our algorithms.