TY - GEN

T1 - Three problems about dynamic convex hulls

AU - Chan, Timothy M.

PY - 2011

Y1 - 2011

N2 - 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(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(log2 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(nε). As a corollary, we can solve the following problem in O(n1+ε) time: given a triangulated terrain in 3-d of size n, identify all faces that are partially visible from a fixed viewpoint.

AB - 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(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(log2 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(nε). As a corollary, we can solve the following problem in O(n1+ε) time: given a triangulated terrain in 3-d of size n, identify all faces that are partially visible from a fixed viewpoint.

KW - Convex hull

KW - Dynamic data structures

KW - Halfspace range searching

KW - Lower envelopes

KW - Orthogonal range searching

UR - http://www.scopus.com/inward/record.url?scp=79960201117&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960201117&partnerID=8YFLogxK

U2 - 10.1145/1998196.1998201

DO - 10.1145/1998196.1998201

M3 - Conference contribution

AN - SCOPUS:79960201117

SN - 9781450306829

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 27

EP - 36

BT - Proceedings of the 27th Annual Symposium on Computational Geometry, SCG'11

T2 - 27th Annual ACM Symposium on Computational Geometry, SCG'11

Y2 - 13 June 2011 through 15 June 2011

ER -