Three Problems About Dynamic Convex Hulls

Abstract

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 + 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.

On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures

“…As mentioned in Section 2.2; in the special case where the points are uniformly distributed within a square, Golin gave a randomized algorithm that uses 1n + O n 6 7 log 5 n expected number of point comparisons [32]. Clarkson later improved the running time to 1n + O n 5 8 expected number of point comparisons [18].…”

Computing the Fréchet Distance with a Retractable Leash

“…Since the minimum of the upper envelope occurs when the slopes change from negative to nonnegative, it dualizes to the line segment on the convex hull that intersects the y-axis. The fastest known data structure for this problem is due to Chan [12]: for h lines, it has an O(log 1+τ h) query and amortized update time, for any τ > 0.…”

Processing a large number of continuous preference top- k queries

“…• Connection to halfspace range queries : We draw the connection between halfspace range queries [1,6,13,16] and reverse top-k queries. This connection allows us to leverage results in computational geometry on halfspace range searching to devise an index for reverse top-k queries, which, in addition to being of independent interest, serves as a critical component of our solution to the scalable continuous top-k query processing problem.…”

“…The known lower bounds [12] suggest that these bounds are close to optimal. I/Oefficient indexing schemes for halfspace range queries were given in [2]; dynamic schemes were presented in [6,13]; see also [15].…”