THREE PROBLEMS ABOUT DYNAMIC CONVEX HULLS

TIMOTHY M. CHAN
2012 International journal of computational geometry and applications  
We present three results related to dynamic convex hulls: • A fully dynamic data structure for maintaining a set of n points in the plane so that we can find the edges of the convex hull intersecting a query line, with expected query and amortized update time O(log 1+ε n) for an arbitrarily small constant ε > 0. This improves the previous bound of O(log 3/2 n). • A fully dynamic data structure for maintaining a set of n points in the plane to support halfplane range reporting queries in O(log n
more » ... + k) time with O(polylog n) expected amortized update time. A similar result holds for 3-dimensional orthogonal range reporting. For 3-dimensional halfspace range reporting, the query time increases to O(log 2 n/ log log n + k). • A semi-online dynamic data structure for maintaining a set of n line segments in the plane, so that we can decide whether a query line segment lies completely above the lower envelope, with query time O(log n) and amortized update time O(n ε ). As a corollary, we can solve the following problem in O(n 1+ε ) time: given a triangulated terrain in 3-d of size n, identify all faces that are partially visible from a fixed viewpoint.
doi:10.1142/s0218195912600096 fatcat:wy6uszf7vjerffnijepg7jszga