{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T11:30:44Z","timestamp":1753875044052,"version":"3.41.2"},"reference-count":16,"publisher":"Proceedings of the National Academy of Sciences","issue":"14","license":[{"start":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T00:00:00Z","timestamp":1743638400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"MEC | Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/00326\/2025"],"award-info":[{"award-number":["UIDB\/00326\/2025"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"MEC | Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDP\/00326\/2025"],"award-info":[{"award-number":["UIDP\/00326\/2025"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000781","name":"EC | European Research Council","doi-asserted-by":"publisher","award":["101125586"],"award-info":[{"award-number":["101125586"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2025,4,8]]},"abstract":"<jats:p>\n            Uniformly distributed point sets of low discrepancy are heavily used in experimental design and across a very wide range of applications such as numerical integration, computer graphics, and finance. Recent methods based on Graph Neural Networks [T. K. Rusch, N. Kirk, M. M. Bronstein, C. Lemieux, D. Rus,\n            <jats:italic>Proc. Natl. Acad. Sci. U.S.A.<\/jats:italic>\n            121, e2409913121 (2024).] and solver-based optimization identified point sets having much lower discrepancy than previously known constructions. We show in this note that further substantial improvements are possible by separating the construction of low-discrepancy point sets into i) the relative position of the points, and ii) the optimal placement respecting these relationships. Using tailored permutations, we construct point sets that are of 20% smaller discrepancy on average than those proposed by Rusch et al. In terms of inverse discrepancy, our sets reduce the number of points in dimension 2 needed to obtain a discrepancy of 0.005 from more than 500 points to less than 350. For applications where the sets are used to query time-consuming models, this is a significant reduction.\n          <\/jats:p>","DOI":"10.1073\/pnas.2424464122","type":"journal-article","created":{"date-parts":[[2025,4,4]],"date-time":"2025-04-04T03:43:22Z","timestamp":1743738202000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":["Searching permutations for constructing uniformly distributed point sets"],"prefix":"10.1073","volume":"122","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9206-3698","authenticated-orcid":false,"given":"Fran\u00e7ois","family":"Cl\u00e9ment","sequence":"first","affiliation":[{"name":"Department of Mathematics","place":["Seattle"]},{"name":"University of Washington","place":["Seattle"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4981-3227","authenticated-orcid":false,"given":"Carola","family":"Doerr","sequence":"additional","affiliation":[{"name":"Sorbonne Universit\u00e9","place":["Paris, France"]},{"name":"CNRS, LIP6","place":["Paris, France"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9119-0732","authenticated-orcid":false,"given":"Kathrin","family":"Klamroth","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science","place":["Wuppertal, Germany"]},{"name":"School of Mathematics and Natural Sciences","place":["Wuppertal, Germany"]},{"name":"University of Wuppertal","place":["Wuppertal, Germany"]}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7525-8901","authenticated-orcid":false,"given":"Lu\u00eds","family":"Paquete","sequence":"additional","affiliation":[{"name":"Department of Informatics Engineering","place":["Coimbra, Portugal"]},{"name":"University of Coimbra","place":["Coimbra, Portugal"]},{"name":"Centre for Informatics and Systems of the University of Coimbra\/Laborat\u00d3rio Associado de Sistemas Inteligentes (CISUC\/LASI)","place":["Coimbra, Portugal"]},{"name":"Departamento de Engenharia Inform\u00e1tica (DEI)","place":["Coimbra, Portugal"]}]}],"member":"341","published-online":{"date-parts":[[2025,4,3]]},"reference":[{"key":"e_1_3_3_1_2","first-page":"7","article-title":"A general theorem from the theory of the uniform distribution modulo 1","volume":"1","author":"Koksma J.","year":"1942","unstructured":"J. Koksma, A general theorem from the theory of the uniform distribution modulo 1. Math. B (Zutphen) 1, 7\u201311 (1942\/1943).","journal-title":"Math. B (Zutphen)"},{"key":"e_1_3_3_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02415361"},{"key":"e_1_3_3_3_2","unstructured":"M. Cauwet et al. \u201cFully parallel hyperparameter search: Reshaped space-filling\u201d in Proceedings International Conference on Machine Learning H. Daum\u00e9 III A. Singh Eds. (PMLR 2020) vol. 119 pp. 1338\u20131348."},{"key":"e_1_3_3_4_2","doi-asserted-by":"publisher","DOI":"10.1145\/3528223.3530063"},{"key":"e_1_3_3_5_2","doi-asserted-by":"crossref","unstructured":"S. Galanti A. Jung \u201cLow-discrepancy sequences: Monte-Carlo simulation of option prices\u201d in Journal of Derivatives J. M. Pimbley Ed. (Euromoney Publications PLC 1997) pp. 63\u201383.","DOI":"10.3905\/jod.1997.407985"},{"key":"e_1_3_3_6_2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511761188"},{"key":"e_1_3_3_7_2","doi-asserted-by":"crossref","unstructured":"T. J. Santner B. Williams W. I. Notz The Design and Analysis of Computer Experiments Springer Series in Statistics P. l\u2019\u00c9cuyer A. B. Owen Eds. (Springer 2003).","DOI":"10.1007\/978-1-4757-3799-8"},{"key":"e_1_3_3_8_2","doi-asserted-by":"crossref","unstructured":"H. Niederreiter Random Number Generation and Quasi-Monte Carlo Methods SIAM CBMS-NSF Regional Conference Series in Applied Mathematics (Society for Industrial and Applied Mathematics 1992) vol. 63.","DOI":"10.1137\/1.9781611970081"},{"key":"e_1_3_3_9_2","doi-asserted-by":"publisher","DOI":"10.1145\/3687764"},{"key":"e_1_3_3_10_2","doi-asserted-by":"crossref","unstructured":"C. Doerr F. M. de Rainville \u201cConstructing low star discrepancy point sets with genetic algorithms\u201d in Proceedings of Genetic and Evolutionary Computation Conference C. Blum E. Alba Eds. (ACM 2013) pp. 789\u2013796.","DOI":"10.1145\/2463372.2463469"},{"key":"e_1_3_3_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jco.2024.101852"},{"key":"e_1_3_3_12_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.2409913121"},{"key":"e_1_3_3_13_2","unstructured":"F. Cl\u00e9ment C. Doerr K. Klamroth L. Paquete Constructing optimal L\u221e star discrepancy sets. arXiv [Preprint] (2024). https:\/\/doi.org\/10.48550\/arXiv.2311.17463 (Accessed 01 June 2024)."},{"key":"e_1_3_3_14_2","doi-asserted-by":"crossref","unstructured":"V. Ostromoukhov \u201cRecent progress in improvement of extreme discrepancy and star discrepancy of one-dimensional sequences 2008\u201d in Monte Carlo and Quasi-Monte Carlo Methods (2009) pp. 561\u2013572.","DOI":"10.1007\/978-3-642-04107-5_36"},{"key":"e_1_3_3_15_2","doi-asserted-by":"publisher","DOI":"10.2140\/moscow.2022.11.215"},{"key":"e_1_3_3_16_2","unstructured":"F. Cl\u00e9ment C. Doerr K. Klamroth L. Paquete Low dIscrepancy permutations. GitHub. https:\/\/github.com\/frclement\/LowDiscPermu\/. Deposited 4 February 2025."}],"container-title":["Proceedings of the National Academy of Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pnas.org\/doi\/pdf\/10.1073\/pnas.2424464122","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,4]],"date-time":"2025-04-04T03:47:34Z","timestamp":1743738454000},"score":1,"resource":{"primary":{"URL":"https:\/\/pnas.org\/doi\/10.1073\/pnas.2424464122"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,3]]},"references-count":16,"journal-issue":{"issue":"14","published-print":{"date-parts":[[2025,4,8]]}},"alternative-id":["10.1073\/pnas.2424464122"],"URL":"https:\/\/doi.org\/10.1073\/pnas.2424464122","relation":{},"ISSN":["0027-8424","1091-6490"],"issn-type":[{"type":"print","value":"0027-8424"},{"type":"electronic","value":"1091-6490"}],"subject":[],"published":{"date-parts":[[2025,4,3]]},"assertion":[{"value":"2024-12-03","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-02-28","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-04-03","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e2424464122"}}