{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:38:08Z","timestamp":1787337488694,"version":"build-2736575974"},"reference-count":31,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[1998,1]]},"abstract":"<jats:p>\n                    We give simple randomized incremental algorithms for computing the Amk-level in an arrangement of n lines in the plane or in an arrangement of n planes in $\\Reals^3$. The expected running time of our algorithms is $O(nk+n\\alpha(n)\\log n)$ for the planarcase and O(nk\n                    <jats:sup>2<\/jats:sup>\n                    + n log\n                    <jats:sup>3<\/jats:sup>\n                    n) for the three-dimensional case. Both bounds are optimal unless k is very small. The algorithm generalizes to computing the Amk-level in an arrangement of discs or x-monotone Jordan curves in the plane. Our approach can also compute the k-level; this yields a randomized algorithm for computing the order-k Voronoi diagram of n points in the plane in expected time O(k(n-k)log n + n log\n                    <jats:sup>3<\/jats:sup>\n                    n).\n                  <\/jats:p>","DOI":"10.1137\/s0097539795281840","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"654-667","source":"Crossref","is-referenced-by-count":75,"title":["Constructing Levels in Arrangements and Higher Order Voronoi Diagrams"],"prefix":"10.1137","volume":"27","author":[{"given":"Pankaj K.","family":"Agarwal","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark","family":"de Berg","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jir\u00ed","family":"Matousek","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Otfried","family":"Schwarzkopf","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,28]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"P. K. Agarwal, B. Aronov, and M. Sharir,\n                      On levels in arrangements of lines, segments, planes, and triangles\n                      , Proc. 13th Annual Sympos. Comput. Geom., 1997, pp. 30\u201338.","DOI":"10.1145\/262839.262856"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548300000225"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/S009753979426616X"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195992000214"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(89)90032-0"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/BF01228508"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1109\/TC.1987.5009474"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/0222077"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1007\/BF02122778"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/BF02189314"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/BF02187879"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/0217052"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF02187740"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/0216005"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF02570705"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/PL00009354"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/BF02574381"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61568-9"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"P. Erdo\u02dds, L. Lov\u00e1sz, A. Simmons, and E. Straus,\n                      Dissection graphs of planar point sets\n                      , A Survey of Combinatorial Theory, (J. N. Srivastava, ed.), North\u2010Holland, Amsterdam, Netherlands, 1973, pp. 139\u2013154.","DOI":"10.1016\/B978-0-7204-2262-7.50018-1"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195996000186"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/0215019"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1007\/BF02187744"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/BF02187876"},{"key":"R24","first-page":"107","volume":"14","author":"Lov\u00e1sz L.","year":"1971","journal-title":"Ann. Univ. Sci. Budapest, Sect. Math."},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1007\/BF02574697"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1145\/102782.102785"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1007\/BF02574692"},{"key":"R28","unstructured":"K. Mulmuley,\n                      Computational Geometry: An Introduction Through Randomized Algorithms\n                      , Prentice Hall, New York, 1993."},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1007\/BF02187829"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1007\/BF02574706"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1016\/0097-3165(92)90028-S"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0097539795281840","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:13:33Z","timestamp":1787336013000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0097539795281840"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,1]]},"references-count":31,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1998,1]]}},"alternative-id":["10.1137\/S0097539795281840"],"URL":"https:\/\/doi.org\/10.1137\/s0097539795281840","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,1]]}}}