Computational geometry: an introduction
Computational geometry: an introduction
Sorting Jordan sequences in linear time using level-linked search trees
Information and Control
The ultimate planar convex hull algorithm
SIAM Journal on Computing
On-line construction of the convex hull of a simple polyline
Information Processing Letters
Jordan sorting via convex hulls of certain non-simple polygons
SCG '87 Proceedings of the third annual symposium on Computational geometry
Triangulating a simple polygon in linear time
Discrete & Computational Geometry
Journal of Algorithms
An optimal algorithm for finding segments intersections
Proceedings of the eleventh annual symposium on Computational geometry
Handbook of discrete and computational geometry
Handbook of discrete and computational geometry
Adaptive algorithms for planar convex hull problems
FAW'10 Proceedings of the 4th international conference on Frontiers in algorithmics
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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, ..., pn) 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, 驴, of simple subchains. We give results that show adaptive complexity in terms of k or 驴: when k or 驴 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(驴+2)) time, and prove a matching lower bound of 驴(n log(驴 + 2)) in the algebraic decision tree model. We also prove a lower bound of 驴(n log(k/n)) for k n in the algebraic decision tree model; since 驴 驴 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 驴 min{驴k, log n} + k). This yields O(n 驴 min{驴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.