{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T13:25:54Z","timestamp":1648905954045},"reference-count":27,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Comput. Geom. Appl."],"published-print":{"date-parts":[[2012,2]]},"abstract":"<jats:p> The stretch factor and maximum detour of a graph G embedded in a metric space measure how well G approximates the minimum complete graph containing G and the metric space, respectively. In this paper we show that computing the stretch factor of a rectilinear path in L<jats:sub>1<\/jats:sub> plane has a lower bound of \u03a9(n log n) in the algebraic computation tree model and describe a worst-case O(\u03c3n log <jats:sup>2<\/jats:sup> n) time algorithm for computing the stretch factor or maximum detour of a path embedded in the plane with a weighted fixed orientation metric defined by \u03c3 \u2265 2 vectors and a worst-case O(n log <jats:sup>d<\/jats:sup> n) time algorithm to d \u2265 3 dimensions in L<jats:sub>1<\/jats:sub>-metric. We generalize the algorithms to compute the stretch factor or maximum detour of trees and cycles in O(\u03c3n log <jats:sup>d+1<\/jats:sup> n) time. We also obtain an optimal O(n) time algorithm for computing the maximum detour of a monotone rectilinear path in L<jats:sub>1<\/jats:sub> plane. <\/jats:p>","DOI":"10.1142\/s0218195912600035","type":"journal-article","created":{"date-parts":[[2012,9,12]],"date-time":"2012-09-12T05:32:56Z","timestamp":1347427976000},"page":"45-60","source":"Crossref","is-referenced-by-count":0,"title":["COMPUTING THE STRETCH FACTOR AND MAXIMUM DETOUR OF PATHS, TREES, AND CYCLES IN THE NORMED SPACE"],"prefix":"10.1142","volume":"22","author":[{"given":"CHRISTIAN","family":"WULFF-NILSEN","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Copenhagen, Denmark"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ANSGAR","family":"GR\u00dcNE","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Informatik I, Universit\u00e4t Bonn, Bonn, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ROLF","family":"KLEIN","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Informatik I, Universit\u00e4t Bonn, Bonn, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ELMAR","family":"LANGETEPE","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Informatik I, Universit\u00e4t Bonn, Bonn, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"D. T.","family":"LEE","sequence":"additional","affiliation":[{"name":"Institute of Information Science, Academia Sinica, Nankang, Taipei, Taiwan"},{"name":"Department of Computer Science and Engineering, National Chung-Hsing University, Taichung, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"TIEN-CHING","family":"LIN","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, National Chung-Hsing University, Taichung, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"SHEUNG-HUNG","family":"POON","sequence":"additional","affiliation":[{"name":"Department of Computer Science, National Tsing Hua University, Hsinchu, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"TENG-KAI","family":"YU","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2012,9,12]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-007-9019-9"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(00)00233-X"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1016\/S0925-7721(03)00046-4"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-0255-1_1"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1002\/net.20082"},{"key":"rf9","volume-title":"Introduction to Algorithms","author":"Cormen T. H.","year":"2001"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04245-8"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01386390"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1016\/B978-044482537-7\/50010-3"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1137\/0216064"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1145\/28869.28874"},{"key":"rf16","volume-title":"Handbook of Approximation Algorithm and Metaheuristics","author":"Gudmundsson J.","year":"2007"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1137\/0114025"},{"key":"rf18","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.1996.0054"},{"key":"rf19","doi-asserted-by":"publisher","DOI":"10.1137\/0130013"},{"key":"rf20","volume-title":"Annals of Discrete Mamathematics 53","author":"Hwang F. K.","year":"1992"},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004198003016"},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1142\/S021819590700232X"},{"key":"rf24","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539799361671"},{"key":"rf25","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511546884"},{"key":"rf27","doi-asserted-by":"publisher","DOI":"10.1145\/359131.359132"},{"key":"rf28","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1098-6"},{"key":"rf29","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100071875"},{"key":"rf30","doi-asserted-by":"publisher","DOI":"10.1109\/43.159995"},{"key":"rf31","doi-asserted-by":"publisher","DOI":"10.1016\/B978-044482537-7\/50021-8"},{"key":"rf32","doi-asserted-by":"publisher","DOI":"10.1137\/0216049"},{"key":"rf34","doi-asserted-by":"publisher","DOI":"10.1137\/0220041"}],"container-title":["International Journal of Computational Geometry &amp; Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218195912600035","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T15:00:39Z","timestamp":1565103639000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218195912600035"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,2]]},"references-count":27,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2012,9,12]]},"published-print":{"date-parts":[[2012,2]]}},"alternative-id":["10.1142\/S0218195912600035"],"URL":"https:\/\/doi.org\/10.1142\/s0218195912600035","relation":{},"ISSN":["0218-1959","1793-6357"],"issn-type":[{"value":"0218-1959","type":"print"},{"value":"1793-6357","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,2]]}}}