Almost tight bounds for eliminating depth cycles in three dimensions
Boris S. Aronov, Micha Sharir · 2016
Given n non-vertical lines in 3-space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into O(n3/2polylog n) pieces, such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. As a consequence, we deduce that the number of pairwise non-overlapping cycles, namely, cycles whose xy-projections do not overlap, is O(n3/2polylog n); this bound too is almost tight in the worst case.