{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T04:24:58Z","timestamp":1750220698322,"version":"3.41.0"},"reference-count":17,"publisher":"Association for Computing Machinery (ACM)","license":[{"start":{"date-parts":[[2020,5,3]],"date-time":"2020-05-03T00:00:00Z","timestamp":1588464000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["ACM J. Exp. Algorithmics"],"published-print":{"date-parts":[[2020,12,6]]},"abstract":"<jats:p>In recent decades, a number of profound theorems concerning approximation of hard counting problems have appeared. These include estimation of the permanent, estimating the volume of a convex polyhedron, and counting (approximately) the number of linear extensions of a partially ordered set. All of these results have been achieved using probabilistic sampling methods, specifically Monte Carlo Markov Chain (MCMC) techniques. In each case, a rapidly mixing Markov chain is defined that is guaranteed to produce, with high probability, an accurate result after only a polynomial number of operations.<\/jats:p>\n          <jats:p>Although of polynomial complexity, none of these results lead to a practical computational technique, nor do they claim to. The polynomials are of high degree and a non-trivial amount of computing is required to get even a single sample. Our aim in this article is to present practical Monte Carlo methods for one of these problems, counting linear extensions. Like related work on estimating the coefficients of the reliability polynomial, our technique is based on improving the so-called Knuth counting algorithm by incorporating an importance function into the node selection technique giving a sequential importance sampling (SIS) method. We define and report performance on two importance functions.<\/jats:p>","DOI":"10.1145\/3385650","type":"journal-article","created":{"date-parts":[[2020,5,4]],"date-time":"2020-05-04T18:55:08Z","timestamp":1588618508000},"page":"1-14","source":"Crossref","is-referenced-by-count":1,"title":["A Sequential Importance Sampling Algorithm for Counting Linear Extensions"],"prefix":"10.1145","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2399-0815","authenticated-orcid":false,"given":"Alathea","family":"Jensen","sequence":"first","affiliation":[{"name":"Susquehanna University, Selinsgrove, PA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Isabel","family":"Beichl","sequence":"additional","affiliation":[{"name":"National Institute of Standards and Technology, Gaithersburg, MD"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2020,5,3]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1137\/0201008"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1177005202"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jda.2018.04.001"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.6149"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00383444"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(98)00333-1"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2017.2699226"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.dsp.2015.05.014"},{"key":"#cr-split#-e_1_2_1_9_1.1","doi-asserted-by":"crossref","unstructured":"Fan Chung Persi Diaconis and Ron Graham. 2019. Permanental generating functions and sequential importance sampling. Adv. Appl. Math. (2019). DOI:https:\/\/doi.org\/10.1016\/j.aam.2019.05.004 10.1016\/j.aam.2019.05.004","DOI":"10.1016\/j.aam.2019.05.004"},{"key":"#cr-split#-e_1_2_1_9_1.2","doi-asserted-by":"crossref","unstructured":"Fan Chung Persi Diaconis and Ron Graham. 2019. Permanental generating functions and sequential importance sampling. Adv. Appl. Math. (2019). DOI:https:\/\/doi.org\/10.1016\/j.aam.2019.05.004","DOI":"10.1016\/j.aam.2019.05.004"},{"key":"e_1_2_1_10_1","volume-title":"Bugallo","author":"Elvira V\u00edctor","year":"2019","unstructured":"V\u00edctor Elvira , Luca Martino , David Luengo , and M\u00f3nica F . Bugallo . 2019 . Generalized multiple importance sampling. Statist. Sci . 34, 1 (2 2019), 129--155. DOI:https:\/\/doi.org\/10.1214\/18-STS668 10.1214\/18-STS668 V\u00edctor Elvira, Luca Martino, David Luengo, and M\u00f3nica F. Bugallo. 2019. Generalized multiple importance sampling. Statist. Sci. 34, 1 (2 2019), 129--155. DOI:https:\/\/doi.org\/10.1214\/18-STS668"},{"key":"e_1_2_1_11_1","doi-asserted-by":"publisher","DOI":"10.1007\/s00453-012-9703-x"},{"key":"e_1_2_1_12_1","doi-asserted-by":"publisher","DOI":"10.1145\/368996.369025"},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.2307\/2005469"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.strusafe.2016.06.002"},{"volume-title":"Identifying Clusters in Spatial Data Via Sequential Importance Sampling","author":"Raveendran Nishanthi","key":"e_1_2_1_15_1","unstructured":"Nishanthi Raveendran and Georgy Sofronov . 2019. Identifying Clusters in Spatial Data Via Sequential Importance Sampling . Springer International Publishing , 175--189. DOI:https:\/\/doi.org\/10.1007\/978-3-319-99648-6_10 10.1007\/978-3-319-99648-6_10 Nishanthi Raveendran and Georgy Sofronov. 2019. Identifying Clusters in Spatial Data Via Sequential Importance Sampling. Springer International Publishing, 175--189. 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