### Almost tight upper bounds for vertical decompositions in four dimensionsAlmost tight upper bounds for vertical decompositions in four dimensions

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 Author Koltun, Vladlen Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2004 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Arrangements ♦ Decompositions ♦ Point location ♦ Range searching ♦ Ray shooting ♦ Robot motion planning Abstract We show that the complexity of the vertical decomposition of an arrangement of $\textit{n}$ fixed-degree algebraic surfaces or surface patches in four dimensions is $O(n^{4+ϵ}),$ 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 $O(n^{2d™4+ϵ}),$ for any ϵ > 0, on the complexity of vertical decompositions in dimensions $\textit{d}$ ≥ 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. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2004-09-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 51 Issue Number 5 Page Count 32 Starting Page 699 Ending Page 730

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Source: ACM Digital Library