{"title":"Polygon triangulation in O(n log log n) time with simple data-structures","authors":"D. Kirkpatrick, M. Klawe, R. Tarjan","doi":"10.1145/98524.98533","DOIUrl":null,"url":null,"abstract":"We give a new <italic>&Ogr;</italic>(<italic>n</italic> log log <italic>n</italic>)-time deterministic linear-time algorithm for triangulating simple <italic>n</italic>-vertex polygons, which avoids the use of complicated data-structures. In addition, for polygons whose vertices have integer coordinates of polynomially bounded size, the algorithm can be modified to run in <italic>&Ogr;</italic>(<italic>n</italic> log<supscrpt>*</supscrpt> <italic>n</italic>) time. The major new techniques employed are the efficient location of horizontal visibility edges which partition the interior of the polygon into regions of approximately equal size, and a linear-time algorithm for obtaining the horizontal visibility partition of a subchain of a polygonal chain, from the horizontal visibility partition of the entire chain. This latter technique has other interesting applications, including a linear-time algorithm to convert a Steiner triangulation of a polygon into a true triangulation.\nThis research was partially supported by DIMACS and the following grants: NSERC 583584, NSERC 580485, NSF-STC88-09648, ONR-N00014-87-0467.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"6 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"49","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SCG '90","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/98524.98533","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 49
Abstract
We give a new &Ogr;(n log log n)-time deterministic linear-time algorithm for triangulating simple n-vertex polygons, which avoids the use of complicated data-structures. In addition, for polygons whose vertices have integer coordinates of polynomially bounded size, the algorithm can be modified to run in &Ogr;(n log*n) time. The major new techniques employed are the efficient location of horizontal visibility edges which partition the interior of the polygon into regions of approximately equal size, and a linear-time algorithm for obtaining the horizontal visibility partition of a subchain of a polygonal chain, from the horizontal visibility partition of the entire chain. This latter technique has other interesting applications, including a linear-time algorithm to convert a Steiner triangulation of a polygon into a true triangulation.
This research was partially supported by DIMACS and the following grants: NSERC 583584, NSERC 580485, NSF-STC88-09648, ONR-N00014-87-0467.
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