{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,19]],"date-time":"2025-06-19T04:23:22Z","timestamp":1750307002449,"version":"3.41.0"},"reference-count":23,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2012,7,1]],"date-time":"2012-07-01T00:00:00Z","timestamp":1341100800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Algorithms"],"published-print":{"date-parts":[[2012,7]]},"abstract":"<jats:p>\n            A left-to-right maximum in a sequence of\n            <jats:italic>n<\/jats:italic>\n            numbers\n            <jats:italic>s<\/jats:italic>\n            <jats:sub>1<\/jats:sub>\n            , \u2026,\n            <jats:italic>\n              s\n              <jats:sub>n<\/jats:sub>\n            <\/jats:italic>\n            is a number that is strictly larger than all preceding numbers. In this article we present a smoothed analysis of the number of left-to-right maxima in the presence of additive random noise. We show that for every sequence of\n            <jats:italic>n<\/jats:italic>\n            numbers\n            <jats:italic>\n              s\n              <jats:sub>i<\/jats:sub>\n            <\/jats:italic>\n            \u2208 [0,1] that are perturbed by uniform noise from the interval [-\u03f5,\u03f5], the expected number of left-to-right maxima is \u0398(\u221a\n            <jats:italic>n<\/jats:italic>\n            \/\u03f5 + log\n            <jats:italic>n<\/jats:italic>\n            ) for \u03f5&gt;1\/\n            <jats:italic>n<\/jats:italic>\n            . For Gaussian noise with standard deviation \u03c3 we obtain a bound of\n            <jats:italic>O<\/jats:italic>\n            ((log\n            <jats:sup>3\/2<\/jats:sup>\n            <jats:italic>n<\/jats:italic>\n            )\/\u03c3 + log\n            <jats:italic>n<\/jats:italic>\n            ).\n          <\/jats:p>\n          <jats:p>\n            We apply our results to the analysis of the smoothed height of binary search trees and the smoothed number of comparisons in the quicksort algorithm and prove bounds of \u0398(\u221a\n            <jats:italic>n<\/jats:italic>\n            \/\u03f5 + log\n            <jats:italic>n<\/jats:italic>\n            ) and \u0398(\n            <jats:italic>n<\/jats:italic>\n            \/\u03f5+1\u221a\n            <jats:italic>n<\/jats:italic>\n            \/\u03f5 +\n            <jats:italic>n<\/jats:italic>\n            log\n            <jats:italic>n<\/jats:italic>\n            ), respectively, for uniform random noise from the interval [-\u03f5,\u03f5]. Our results can also be applied to bound the smoothed number of points on a convex hull of points in the two-dimensional plane and to\n            <jats:italic>smoothed motion complexity,<\/jats:italic>\n            a concept we describe in this article. We bound how often one needs to update a data structure storing the smallest axis-aligned box enclosing a set of points moving in\n            <jats:italic>d<\/jats:italic>\n            -dimensional space.\n          <\/jats:p>","DOI":"10.1145\/2229163.2229174","type":"journal-article","created":{"date-parts":[[2012,7,26]],"date-time":"2012-07-26T14:41:09Z","timestamp":1343313669000},"page":"1-28","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":7,"title":["Smoothed analysis of left-to-right maxima with applications"],"prefix":"10.1145","volume":"8","author":[{"given":"Valentina","family":"Damerow","sequence":"first","affiliation":[{"name":"Heinz Nixdorf Institute, University of Paderborn, Germany"}]},{"given":"Bodo","family":"Manthey","sequence":"additional","affiliation":[{"name":"University of Twente, Netherlands"}]},{"given":"Friedhelm Meyer Auf Der","family":"Heide","sequence":"additional","affiliation":[{"name":"Heinz Nixdorf Institute, University of Paderborn, Germany"}]},{"given":"Harald","family":"R\u00e4cke","sequence":"additional","affiliation":[{"name":"University of Warwick, UK"}]},{"given":"Christian","family":"Scheideler","sequence":"additional","affiliation":[{"name":"University of Paderborn, Germany"}]},{"given":"Christian","family":"Sohler","sequence":"additional","affiliation":[{"name":"Technische Universit\u00e4t Dortmund, Germany"}]},{"given":"Till","family":"Tantau","sequence":"additional","affiliation":[{"name":"Universit\u00e4t zu L\u00fcbeck, Germany"}]}],"member":"320","published-online":{"date-parts":[[2012,7,24]]},"reference":[{"volume-title":"Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA)","author":"Agarwal P. K.","key":"e_1_2_1_1_1","unstructured":"Agarwal , P. K. and Har-Peled , S. 2001. Maintaining approximate extent measures of moving points . In Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA) . SIAM , 148--157. Agarwal, P. K. and Har-Peled, S. 2001. Maintaining approximate extent measures of moving points. In Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 148--157."},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-45138-9_14"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.1998.0988"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/262839.262998"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-39658-1_17"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/765568.765572"},{"key":"e_1_2_1_7_1","first-page":"117","article-title":"Profile and height of random binary search trees","volume":"3","author":"Drmota M.","year":"2004","unstructured":"Drmota , M. 2004 . Profile and height of random binary search trees . J. Iran. Statist. Soc. 3 , 2, 117 -- 138 . Drmota, M. 2004. Profile and height of random binary search trees. J. Iran. Statist. Soc. 3, 2, 117--138.","journal-title":"J. Iran. Statist. Soc."},{"key":"e_1_2_1_8_1","volume-title":"Probability: Theory and Examples","author":"Durrett R.","year":"1991","unstructured":"Durrett , R. 1991 . Probability: Theory and Examples . Duxburry Press . Durrett, R. 1991. Probability: Theory and Examples. Duxburry Press."},{"key":"e_1_2_1_9_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0196-6774(02)00216-X"},{"key":"e_1_2_1_10_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-001-0015-1"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-004-2822-7"},{"volume-title":"Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 158--167","author":"Hershberger J.","key":"e_1_2_1_12_1","unstructured":"Hershberger , J. and Suri , S . 2001. Simplified kinetic connectivity for rectangles and hypercubes . In Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 158--167 . Hershberger, J. and Suri, S. 2001. Simplified kinetic connectivity for rectangles and hypercubes. In Proceedings of the 12th ACM-SIAM Symposium on Discrete Algorithms (SODA). SIAM, 158--167."},{"volume-title":"Fundamentals of Average Case Analysis of Particular Algorithms","author":"Kemp R.","key":"e_1_2_1_13_1","unstructured":"Kemp , R. 1984. Fundamentals of Average Case Analysis of Particular Algorithms . John Wiley & amp; Sons and B. G. Teubner. Kemp, R. 1984. Fundamentals of Average Case Analysis of Particular Algorithms. John Wiley &amp; Sons and B. G. Teubner."},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195902000724"},{"key":"e_1_2_1_15_1","volume-title":"Sorting and Searching","author":"Knuth D. E.","unstructured":"Knuth , D. E. 1998. Sorting and Searching , 2 nd Ed. The Art of Computer Programming, vol. 3 . Addison--Wesley Longman Publishing Co. , Redwood City, CA. Knuth, D. E. 1998. Sorting and Searching, 2nd Ed. The Art of Computer Programming, vol. 3. Addison--Wesley Longman Publishing Co., Redwood City, CA.","edition":"2"},{"key":"e_1_2_1_16_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.tcs.2007.02.035"},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-85238-4_38"},{"key":"e_1_2_1_18_1","doi-asserted-by":"crossref","unstructured":"Motwani R. and Raghavan P. 1995. Randomized Algorithms. Cambridge University Press.   Motwani R. and Raghavan P. 1995. Randomized Algorithms. Cambridge University Press.","DOI":"10.1017\/CBO9780511814075"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1145\/765568.765571"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00535300"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/990308.990310"},{"key":"e_1_2_1_22_1","volume-title":"-H","author":"Spielman D. A.","year":"2006","unstructured":"Spielman , D. A. and Teng , S . -H . 2006 . Smoothed analysis of algorithms and heuristics: Progress and open questions. In Foundations of Computational Mathematics, Santander 2005, L. M. Pardo, A. Pinkus, E. S\u00fcli, and M. J. Todd, Eds., Cambridge University Press , 274--342. Spielman, D. A. and Teng, S.-H. 2006. Smoothed analysis of algorithms and heuristics: Progress and open questions. In Foundations of Computational Mathematics, Santander 2005, L. M. Pardo, A. Pinkus, E. S\u00fcli, and M. J. Todd, Eds., Cambridge University Press, 274--342."},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/262839.263075"}],"container-title":["ACM Transactions on Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2229163.2229174","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2229163.2229174","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T08:48:52Z","timestamp":1750236532000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2229163.2229174"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,7]]},"references-count":23,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2012,7]]}},"alternative-id":["10.1145\/2229163.2229174"],"URL":"https:\/\/doi.org\/10.1145\/2229163.2229174","relation":{},"ISSN":["1549-6325","1549-6333"],"issn-type":[{"type":"print","value":"1549-6325"},{"type":"electronic","value":"1549-6333"}],"subject":[],"published":{"date-parts":[[2012,7]]},"assertion":[{"value":"2009-09-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2010-06-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-07-24","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}