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Almost tight upper bounds for vertical decompositions in four dimensions

Published: 01 September 2004 Publication History

Abstract

We show that the complexity of the vertical decomposition of an arrangement of n fixed-degree algebraic surfaces or surface patches in four dimensions is O(n4+ε), for any ε > 0. This improves the best previously known upper bound for this problem by a near-linear factor, and settles a major problem in the theory of arrangements of surfaces, open since 1989. The new bound can be extended to higher dimensions, yielding the bound O(n2d−4+ε), for any ε > 0, on the complexity of vertical decompositions in dimensions d ≥ 4. We also describe the immediate algorithmic applications of these results, which include improved algorithms for point location, range searching, ray shooting, robot motion planning, and some geometric optimization problems.

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cover image Journal of the ACM
Journal of the ACM  Volume 51, Issue 5
September 2004
151 pages
ISSN:0004-5411
EISSN:1557-735X
DOI:10.1145/1017460
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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 September 2004
Published in JACM Volume 51, Issue 5

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Author Tags

  1. Arrangements
  2. decompositions
  3. point location
  4. range searching
  5. ray shooting
  6. robot motion planning

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