Space-efficient algorithms for computing the convex hull of a simple polygonal line in linear time
We present space-efficient algorithms for computing the convex hull of a simple polygonal line in-place, in linear time. It turns out that the problem is as hard as in-place stable partition, i.e., if there were a truly simple solution then in-place stable partition would also have a truly simple so...
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| Veröffentlicht in: | Computational geometry : theory and applications Jg. 34; H. 2; S. 75 - 82 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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Elsevier B.V
01.05.2006
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| ISSN: | 0925-7721 |
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| Abstract | We present space-efficient algorithms for computing the convex hull of a simple polygonal line in-place, in linear time. It turns out that the problem is as hard as in-place stable partition, i.e., if there were a truly simple solution then in-place stable partition would also have a truly simple solution, and vice versa. Nevertheless, we present a simple self-contained solution that uses
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space, and indicate how to improve it to
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space with the same techniques used for stable partition. If the points inside the convex hull can be discarded, then there is a truly simple solution that uses a single call to stable partition, and even that call can be spared if only extreme points are desired (and not their order). If the polygonal line is closed, the problem admits a very simple solution which does not call for stable partitioning at all. |
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| AbstractList | We present space-efficient algorithms for computing the convex hull of a simple polygonal line in-place, in linear time. It turns out that the problem is as hard as in-place stable partition, i.e., if there were a truly simple solution then in-place stable partition would also have a truly simple solution, and vice versa. Nevertheless, we present a simple self-contained solution that uses
O
(
log
n
)
space, and indicate how to improve it to
O
(
1
)
space with the same techniques used for stable partition. If the points inside the convex hull can be discarded, then there is a truly simple solution that uses a single call to stable partition, and even that call can be spared if only extreme points are desired (and not their order). If the polygonal line is closed, the problem admits a very simple solution which does not call for stable partitioning at all. |
| Author | Brönnimann, Hervé Chan, Timothy M. |
| Author_xml | – sequence: 1 givenname: Hervé surname: Brönnimann fullname: Brönnimann, Hervé email: hbr@poly.edu organization: Computer and Information Science Department, Polytechnic University, Brooklyn, NY 11201, USA – sequence: 2 givenname: Timothy M. surname: Chan fullname: Chan, Timothy M. email: tmchan@uwaterloo.ca organization: School of Computer Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada |
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| Cites_doi | 10.1016/j.tcs.2003.05.004 10.1016/S0304-3975(98)00162-5 10.1007/BF01994842 10.1016/0020-0190(87)90086-X 10.1007/BF00993195 10.1007/BF02017344 10.1007/BF01178508 |
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| Keywords | Computational geometry Space-efficient algorithm Convex hull |
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| References | Brönnimann, Iacono, Katajainen, Morin, Morrison, Toussaint (bib004) 2004; 321 Katajainen, Pasanen (bib009) 1994; 31 Katajainen, Pasanen, Teuhola (bib010) 1996; 3 Cormen, Leiserson, Rivest, Stein (bib005) 2001 Brönnimann, Chen, Chan (bib003) 2004 Aloupis (bib001) Lee (bib011) 1983; 12 Katajainen, Pasanen (bib008) 1982; 32 Melkman (bib012) 1987; 25 Boissonnat, Yvinec (bib002) 1998 Chen, Chan (bib006) 2003 Geffert, Katajainen, Pasanen (bib007) 2000; 237 Munro, Raman, Salowe (bib013) 1990; 30 Melkman (10.1016/j.comgeo.2005.11.005_bib012) 1987; 25 Katajainen (10.1016/j.comgeo.2005.11.005_bib009) 1994; 31 Geffert (10.1016/j.comgeo.2005.11.005_bib007) 2000; 237 Chen (10.1016/j.comgeo.2005.11.005_bib006) 2003 Aloupis (10.1016/j.comgeo.2005.11.005_bib001) Cormen (10.1016/j.comgeo.2005.11.005_bib005) 2001 Brönnimann (10.1016/j.comgeo.2005.11.005_bib004) 2004; 321 Brönnimann (10.1016/j.comgeo.2005.11.005_bib003) 2004 Katajainen (10.1016/j.comgeo.2005.11.005_bib008) 1982; 32 Katajainen (10.1016/j.comgeo.2005.11.005_bib010) 1996; 3 Boissonnat (10.1016/j.comgeo.2005.11.005_bib002) 1998 Munro (10.1016/j.comgeo.2005.11.005_bib013) 1990; 30 Lee (10.1016/j.comgeo.2005.11.005_bib011) 1983; 12 |
| References_xml | – volume: 3 start-page: 27 year: 1996 end-page: 40 ident: bib010 article-title: Practical in-place mergesort publication-title: Nordic J. Comput. – start-page: 239 year: 2004 end-page: 246 ident: bib003 article-title: Towards in-place geometric algorithms and data structures publication-title: Proc. 20th Annual ACM Symposium on Computational Geometry – volume: 32 start-page: 580 year: 1982 end-page: 585 ident: bib008 article-title: Stable minimum space partitioning in linear time publication-title: BIT – volume: 237 start-page: 159 year: 2000 end-page: 181 ident: bib007 article-title: Asymptotically efficient in-place merging publication-title: Theoret. Comput. Sci. – volume: 12 start-page: 87 year: 1983 end-page: 98 ident: bib011 article-title: On finding the convex hull of a simple polygon publication-title: Int. J. Computing Info. Sci. – ident: bib001 article-title: A history of linear-time convex hull algorithms for simple polygons – year: 2001 ident: bib005 article-title: Introduction to Algorithms – volume: 31 start-page: 410 year: 1994 end-page: 421 ident: bib009 article-title: Sorting multiset stably in minimum space publication-title: Acta Informatica – year: 1998 ident: bib002 article-title: Algorithmic Geometry – start-page: 68 year: 2003 end-page: 71 ident: bib006 article-title: A space-efficient algorithm for segment intersection publication-title: Proc. 15th Canadian Conference on Computational Geometry – volume: 25 start-page: 11 year: 1987 end-page: 12 ident: bib012 article-title: On-line construction of the convex hull of a simple polygon publication-title: Inform. Process. Lett. – volume: 30 start-page: 220 year: 1990 end-page: 234 ident: bib013 article-title: Stable in situ sorting and minimum data movement publication-title: BIT – volume: 321 start-page: 25 year: 2004 end-page: 40 ident: bib004 article-title: Space-efficient planar convex hull algorithms publication-title: Theoret. Comput. Sci. – volume: 3 start-page: 27 year: 1996 ident: 10.1016/j.comgeo.2005.11.005_bib010 article-title: Practical in-place mergesort publication-title: Nordic J. Comput. – volume: 321 start-page: 25 issue: 1 year: 2004 ident: 10.1016/j.comgeo.2005.11.005_bib004 article-title: Space-efficient planar convex hull algorithms publication-title: Theoret. Comput. Sci. doi: 10.1016/j.tcs.2003.05.004 – start-page: 239 year: 2004 ident: 10.1016/j.comgeo.2005.11.005_bib003 article-title: Towards in-place geometric algorithms and data structures – ident: 10.1016/j.comgeo.2005.11.005_bib001 – volume: 237 start-page: 159 year: 2000 ident: 10.1016/j.comgeo.2005.11.005_bib007 article-title: Asymptotically efficient in-place merging publication-title: Theoret. Comput. Sci. doi: 10.1016/S0304-3975(98)00162-5 – volume: 32 start-page: 580 year: 1982 ident: 10.1016/j.comgeo.2005.11.005_bib008 article-title: Stable minimum space partitioning in linear time publication-title: BIT doi: 10.1007/BF01994842 – start-page: 68 year: 2003 ident: 10.1016/j.comgeo.2005.11.005_bib006 article-title: A space-efficient algorithm for segment intersection – volume: 25 start-page: 11 year: 1987 ident: 10.1016/j.comgeo.2005.11.005_bib012 article-title: On-line construction of the convex hull of a simple polygon publication-title: Inform. Process. Lett. doi: 10.1016/0020-0190(87)90086-X – year: 1998 ident: 10.1016/j.comgeo.2005.11.005_bib002 – volume: 12 start-page: 87 issue: 2 year: 1983 ident: 10.1016/j.comgeo.2005.11.005_bib011 article-title: On finding the convex hull of a simple polygon publication-title: Int. J. Computing Info. Sci. doi: 10.1007/BF00993195 – year: 2001 ident: 10.1016/j.comgeo.2005.11.005_bib005 – volume: 30 start-page: 220 issue: 2 year: 1990 ident: 10.1016/j.comgeo.2005.11.005_bib013 article-title: Stable in situ sorting and minimum data movement publication-title: BIT doi: 10.1007/BF02017344 – volume: 31 start-page: 410 year: 1994 ident: 10.1016/j.comgeo.2005.11.005_bib009 article-title: Sorting multiset stably in minimum space publication-title: Acta Informatica doi: 10.1007/BF01178508 |
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