Three problems about dynamic convex hulls

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(log1+ε n) for an arbitrarily small constant ε > 0. This improves the previous bound of O(log3/2 n). • A fully dynamic data structure for maintaining a set of n points in the plane to support halfplane range reporting queries in O(logn+k) time with O(polylogn) 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(log2 n/ loglogn+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(logn) and amortized update time O(nε ). As a corollary, we can ...

• Type: Journal
• Year: 2011
• Where: COMPGEOM
• Authors: Timothy M. Chan
• Keywords: COMPGEOM 2011, Discrete Geometry, Dynamic Data Structure, Line Segments, Query Line