Adaptive algorithms for constructing convex hulls and triangulations of polygonal chains

We study some fundamental computational geometry problems with the goal to exploit structure in input data that is given as a sequence C = (p₁, p₂,…, p_n) of points that are “almost sorted” in the sense that the polygonal chain they define has a possibly small number, k, of self-intersections, or the chain can be partitioned into a small number, χ, of simple subchains. We give results that show adaptive complexity in terms of k or χ: when k or χ is small compared to n, we achieve time bounds that approach the linear-time (O(n)) bounds known for the corresponding problems on simple polygonal chains. In particular, we show that the convex hull of C can be computed in O(n log(χ+2)) time, and prove a matching lower bound of Ω(n log(χ + 2)) in the algebraic decision tree model. We also prove a lower bound of Ω(n log(k/n)) for k > n in the algebraic decision tree model; since χ ≤ k, the upper bound of O(n log(k + 2)) follows. We also show that a polygonal chain with k proper intersections can be transformed into a polygonal chain without proper intersections by adding at most 2k new vertices in time O(n · min{√ k, log n} + k). This yields O(n · min{√k, log n} + k)-time algorithms for triangulation, in particular the constrained Delaunay triangulation of a polygonal chain where the proper intersection points are also regarded as vertices.