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Three problems about dynamic convex hulls

Published: 13 June 2011 Publication History

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(log1+εn) for an arbitrarily small constant ε>0. This improves the previous bound of O(log3/2n). 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.

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  • (2012)Processing a large number of continuous preference top-k queriesProceedings of the 2012 ACM SIGMOD International Conference on Management of Data10.1145/2213836.2213882(397-408)Online publication date: 20-May-2012

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cover image ACM Conferences
SoCG '11: Proceedings of the twenty-seventh annual symposium on Computational geometry
June 2011
532 pages
ISBN:9781450306829
DOI:10.1145/1998196
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 13 June 2011

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Author Tags

  1. convex hulls
  2. dynamic data structures
  3. halfspace range searching
  4. lower envelopes
  5. orthogonal range searching

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SoCG '11
SoCG '11: Symposium on Computational Geometry
June 13 - 15, 2011
Paris, France

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Cited By

View all
  • (2012)Processing a large number of continuous preference top-k queriesProceedings of the 2012 ACM SIGMOD International Conference on Management of Data10.1145/2213836.2213882(397-408)Online publication date: 20-May-2012

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