Almost Tight Bounds for Eliminating Depth Cycles in Three Dimensions

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(n(3/2) polylogn) 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(n(3/2) polylogn); this bound too is almost tight in the worst case. Previous results on this topic could only handle restricted cases of the problem (such as handling only triangular cycles, by Aronov, Koltun, and Sharir, or only cycles in grid-like patterns, by Chazelle et al.), and the bounds were considerably weaker much closer to the trivial quadratic bound. Our proof uses a recent variant of the polynomial partitioning technique, due to Guth, and some simple tools from algebraic geometry. It is much more straightforward than the previous "purely combinatorial" methods. Our approach extends to eliminating all cycles in the depth relation among segments, and among constant-degree algebraic arcs. We hope that a suitable extension of this technique could be used to handle the much more difficult case of pairwise-disjoint triangles as well. Our results almost completely settle a long-standing (35 years old) open problem in computational geometry, motivated by hidden-surface removal in computer graphics.