Abstract
We study some fundamental computational geometry problems with the goal to exploit structure in input data that is given as a sequence C = (p 1,p 2,...,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(nlog(Χ + 2)) time, and prove a matching lower bound of ω(nlog(Χ + 2)) in the algebraic decision tree model. We also prove a lower bound of ω(nlog(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, logn} + 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.
Partially supported by HRL Labs (DARPA subcontract), NASA Ames Research, NSF (CCR-0098172), U.S.-Israel Binational Science Foundation.
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Levcopoulos, C., Lingas, A., Mitchell, J.S.B. (2002). Adaptive Algorithms for Constructing Convex Hulls and Triangulations of Polygonal Chains. In: Penttonen, M., Schmidt, E.M. (eds) Algorithm Theory — SWAT 2002. SWAT 2002. Lecture Notes in Computer Science, vol 2368. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45471-3_9
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DOI: https://doi.org/10.1007/3-540-45471-3_9
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