{"month":"01","page":"341 - 363","date_updated":"2022-02-01T11:03:07Z","volume":15,"date_created":"2018-12-11T12:06:58Z","_id":"4105","title":"Constructing arrangements of lines and hyperplanes with applications","author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"full_name":"O'Rourke, Joseph","last_name":"O'Rourke","first_name":"Joseph"},{"full_name":"Seidel, Raimund","last_name":"Seidel","first_name":"Raimund"}],"type":"journal_article","year":"1986","publication_status":"published","oa_version":"None","article_type":"original","acknowledgement":"We thank Emmerich Welzl for discussions on Theorem 2.7. We also thank Friedrich Huber for implementing the \r\nconstruction of arrangements in arbitrary dimensions, and Gerd Stoeckl for implementing the algorithms presented in §§\r\n4.1 and 4.3. The third author wishes to thank Jack Edmonds for the many enlightening discussions.\r\n","intvolume":"        15","extern":"1","publication":"SIAM Journal on Computing","quality_controlled":"1","date_published":"1986-01-01T00:00:00Z","publication_identifier":{"issn":["0097-5397"],"eissn":["1095-7111"]},"day":"01","status":"public","citation":{"ista":"Edelsbrunner H, O’Rourke J, Seidel R. 1986. Constructing arrangements of lines and hyperplanes with applications. SIAM Journal on Computing. 15(2), 341–363.","mla":"Edelsbrunner, Herbert, et al. “Constructing Arrangements of Lines and Hyperplanes with Applications.” <i>SIAM Journal on Computing</i>, vol. 15, no. 2, SIAM, 1986, pp. 341–63, doi:<a href=\"https://doi.org/10.1137/0215024\">10.1137/0215024</a>.","short":"H. Edelsbrunner, J. O’Rourke, R. Seidel, SIAM Journal on Computing 15 (1986) 341–363.","apa":"Edelsbrunner, H., O’Rourke, J., &#38; Seidel, R. (1986). Constructing arrangements of lines and hyperplanes with applications. <i>SIAM Journal on Computing</i>. SIAM. <a href=\"https://doi.org/10.1137/0215024\">https://doi.org/10.1137/0215024</a>","chicago":"Edelsbrunner, Herbert, Joseph O’Rourke, and Raimund Seidel. “Constructing Arrangements of Lines and Hyperplanes with Applications.” <i>SIAM Journal on Computing</i>. SIAM, 1986. <a href=\"https://doi.org/10.1137/0215024\">https://doi.org/10.1137/0215024</a>.","ama":"Edelsbrunner H, O’Rourke J, Seidel R. Constructing arrangements of lines and hyperplanes with applications. <i>SIAM Journal on Computing</i>. 1986;15(2):341-363. doi:<a href=\"https://doi.org/10.1137/0215024\">10.1137/0215024</a>","ieee":"H. Edelsbrunner, J. O’Rourke, and R. Seidel, “Constructing arrangements of lines and hyperplanes with applications,” <i>SIAM Journal on Computing</i>, vol. 15, no. 2. SIAM, pp. 341–363, 1986."},"doi":"10.1137/0215024","scopus_import":"1","user_id":"ea97e931-d5af-11eb-85d4-e6957dddbf17","article_processing_charge":"No","language":[{"iso":"eng"}],"issue":"2","publisher":"SIAM","publist_id":"2017","abstract":[{"text":"A finite set of lines partitions the Euclidean plane into a cell complex. Similarly, a finite set of $(d - 1)$-dimensional hyperplanes partitions $d$-dimensional Euclidean space. An algorithm is presented that constructs a representation for the cell complex defined by $n$ hyperplanes in optimal $O(n^d )$ time in $d$ dimensions. It relies on a combinatorial result that is of interest in its own right. The algorithm is shown to lead to new methods for computing $\\lambda $-matrices, constructing all higher-order Voronoi diagrams, halfspatial range estimation, degeneracy testing, and finding minimum measure simplices. In all five applications, the new algorithms are asymptotically faster than previous results, and in several cases are the only known methods that generalize to arbitrary dimensions. The algorithm also implies an upper bound of $2^{cn^d } $, $c$ a positive constant, for the number of combinatorially distinct arrangements of $n$ hyperplanes in $E^d $.\r\n© 1986 Society for Industrial and Applied Mathematics","lang":"eng"}]}