6.838, Computational Geometry; Fall 2001

Point Location

Sergi Elizalde & David Pritchard

An Incremental Randomized Algorithm

How can we construct this structure, given a set of non-crossing line segments?

Algorithm TrapezoidalMap(S)

1. Determine a bounding box R that contains all of the segments of S
2. Initialize T, the trapezoidal map structure and D, the search structure:
   1. T is just the single trapezoid corresponding to the whole box R
   2. D has only one node, corresponding to that trapezoid
3. Compute a random permutation s₁, s₂, ..., s_n of the segments of S
4. For each segment s_i (in this order),
   1. Find the set of trapezoids in T properly intersected by s_i
   2. Remove them from T and replace them by the new trapezoids that appear because of the insertion of s_i
   3. Remove the leaves of D for the old trapezoids and create leaves for the new ones. Link them to the existing inner nodes by adding some new inner nodes as will be explained in a moment.

Key features of the algorithm:

Incremental: Every time we insert a segment, we have a trapezoidal map and a data structure for the segments inserted before.

It uses randomness, so there are no bad inputs.