Computing and verifying depth orders

A depth order on a set of objects is an order such that object <italic>a</italic> comes before object <italic>a′</italic> in the order when <italic>a′</italic> lies behind <italic>a′</italic>, or, in other words, when <italic>a</italic> is (partially) hidden by <italic>a′</italic> by <italic>a′</italic>. We present efficient algorithms for the computation and verification of depth orders of sets of <italic>n</italic> rods in 3–space. Our algorithms run in time <italic>O</italic>(<italic>n</italic><supscrpt>4/3+ε</supscrpt>), for any fixed ε > 0). If all rods are axis-parallel, or, more generally, have only a constant number of different orientations, then the sorting algorithm runs in <italic>O</italic>(<italic>n</italic> log<supscrpt>2</supscrpt> <italic>n</italic>) time. The algorithms can be generalized to handle triangles and other polygons instead of rods. They are based on a general framework for computing and verifying linear extensions of implicitly defined binary relations.