{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,27]],"date-time":"2023-10-27T01:40:14Z","timestamp":1698370814377},"reference-count":21,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":7619,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1985,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Based on the probabilistic approach introduced by Karp [14], we present two partitioning algorithms for the approximate solution of large instances of the rectilinear Steiner problem in the plane. The algorithms subdivide a set of given points into small groups, construct a minimum rectilinear Steiner tree for each small group, and then patch the subtrees together to form a near\u2010optimum rectilinear Steiner tree for the given points. Suppose there are <jats:italic>n<\/jats:italic> given points, uniformly distributed over the unit square [0,1] \u00d7 [0,1] and spanned by a minimum rectilinear Steiner tree T<jats:sub><jats:italic>o<\/jats:italic><\/jats:sub>. Then for any given integer <jats:italic>t<\/jats:italic> &gt; 0, the first algorithm runs in O(<jats:italic>f<\/jats:italic>(<jats:italic>t<\/jats:italic>)<jats:italic>n<\/jats:italic> + <jats:italic>n<\/jats:italic> log <jats:italic>n<\/jats:italic>) time and produces a rectilinear Steiner tree <jats:italic>T<\/jats:italic><jats:sub>1<\/jats:sub> such that <jats:styled-content>\\documentclass{article}\\pagestyle{empty}\\begin{document}$ \\left({|T_1 |\/|T_0 |} \\right) \\le 1 + O\\left({\\sqrt {1\/t}} \\right) $\\end{document}<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-ueqn-1.gif\" xlink:title=\"equation image\" \/><\/jats:styled-content> with probability approaching 1 as <jats:italic>n<\/jats:italic> \u2192 \u221e, while the second algorithm has an expected running time of <jats:italic>O<\/jats:italic>(<jats:italic>g<\/jats:italic>(<jats:italic>t<\/jats:italic>)<jats:italic>n<\/jats:italic>) and it also produces a rectilinear Steiner tree T<jats:sub>2<\/jats:sub> such that <jats:styled-content>\\documentclass{article}\\pagestyle{empty}\\begin{document}$ \\left({|T_2 |\/|T_0 |} \\right) \\le 1 + O\\left({\\sqrt {1\/t}} \\right) $\\end{document}<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-ueqn-2.gif\" xlink:title=\"equation image\" \/><\/jats:styled-content> with probability approaching 1 as <jats:italic>n<\/jats:italic> \u2192 \u221e.<\/jats:p>","DOI":"10.1002\/net.3230150403","type":"journal-article","created":{"date-parts":[[2007,5,11]],"date-time":"2007-05-11T20:18:12Z","timestamp":1178914692000},"page":"413-423","source":"Crossref","is-referenced-by-count":16,"title":["Probabilistic partitioning algorithms for the rectilinear steiner problem"],"prefix":"10.1002","volume":"15","author":[{"given":"J\u00e1nos","family":"Koml\u00f3s","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M. T.","family":"Shing","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230070104"},{"key":"e_1_2_1_3_2","volume-title":"The Design and Analysis of Computer Algorithms","author":"Aho A.","year":"1974"},{"key":"e_1_2_1_4_2","volume-title":"Graph Theory: An Algorithmic Approach","author":"Christofides N.","year":"1975"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230010302"},{"key":"e_1_2_1_6_2","unstructured":"L. R.Foulds andV. J.Rayward\u2010Smith Steiner problem in graphs: Algorithms and applications. Tech. Report CSA2 School of Computing Studies and Accountancy University of East Anglia Norwich NR4 7TJ England 1983."},{"key":"e_1_2_1_7_2","first-page":"19","volume-title":"Large Scale Networks: Theory and Design","author":"Frank H.","year":"1976"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1137\/0132071"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.1137\/0114025"},{"key":"e_1_2_1_10_2","volume-title":"Design Automation of Digital Systems","author":"Hanan M.","year":"1972"},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.1975.5.1.85"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1137\/0130013"},{"issue":"4","key":"e_1_2_1_13_2","first-page":"303","article-title":"The rectilinear Steiner problem","volume":"2","author":"Hwang F. K.","year":"1978","journal-title":"Design Automation and Fault\u2010Tolerant Comput"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1109\/TCS.1979.1084551"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2.3.209"},{"key":"e_1_2_1_16_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1956-0078686-7"},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1109\/TCS.1976.1084243"},{"key":"e_1_2_1_18_2","doi-asserted-by":"publisher","DOI":"10.1080\/03052157908902401"},{"key":"e_1_2_1_19_2","doi-asserted-by":"publisher","DOI":"10.1080\/03052158008902421"},{"key":"e_1_2_1_20_2","first-page":"21","article-title":"Heuristic algorithms for rectilinear Steiner trees","volume":"7","author":"Servit M.","year":"1981","journal-title":"Digital Proc."},{"key":"e_1_2_1_21_2","series-title":"Proc. IEEE Int. Symp. Circuit Theory","first-page":"154","author":"Yang Y. Y.","year":"1972"},{"key":"e_1_2_1_22_2","unstructured":"A. C.Yao Lecture notes on average\u2010case analysis of algorithms. Private communication Computer Science Dept. Stanford University CA. 1980."}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230150403","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230150403","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,20]],"date-time":"2023-10-20T20:06:33Z","timestamp":1697832393000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230150403"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,12]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1985,12]]}},"alternative-id":["10.1002\/net.3230150403"],"URL":"https:\/\/doi.org\/10.1002\/net.3230150403","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,12]]}}}