Before revision: We revisit the randomized incremental construction of the Trapezoidal Search DAG (TSD) for a set of{\cal   n non-crossing segments, e.g. edges from planar subdivisions.
After revision: The Randomized Incremental Construction (RIC) of search DAGs for point location in planar subdivisions, nearest-neighbor search in 2D points, and extreme point search in 3D convex hulls, are well known to take{\cal n non-crossing segments, e.g. edges from planar subdivisions.