{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T09:21:09Z","timestamp":1787390469124,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-0832787"],"award-info":[{"award-number":["CCF-0832787"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1218711"],"award-info":[{"award-number":["CCF-1218711"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["332622"],"award-info":[{"award-number":["332622"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>Finding the length of the longest increasing subsequence (LIS) is a classic algorithmic problem. Let $n$ denote the size of the array. Simple $O(n\\log n)$ algorithms are known for this problem. We develop a polylogarithmic time randomized algorithm that for any constant $\\delta &gt; 0$, estimates the length of the LIS of an array to within an additive error of $\\delta n$. More precisely, the running time of the algorithm is $(\\log n)^c (1\/\\delta)^{O(1\/\\delta)}$, where the exponent $c$ is independent of $\\delta$. Previously, the best known polylogarithmic time algorithms could only achieve an additive $n\/2$ approximation. With a suitable choice of parameters, our algorithm also gives, for any fixed $\\tau&gt;0$, a multiplicative $(1+\\tau)$-approximation to the distance to monotonicity $\\varepsilon_f$ (the fraction of entries not in the LIS), whose running time is polynomial in $\\log(n)$ and $1\/\\varepsilon_f$. The best previously known algorithm could only guarantee an approximation within a factor (arbitrarily close to) 2.<\/jats:p>","DOI":"10.1137\/130942152","type":"journal-article","created":{"date-parts":[[2017,4,20]],"date-time":"2017-04-20T12:04:13Z","timestamp":1492689853000},"page":"774-823","source":"Crossref","is-referenced-by-count":10,"title":["Estimating the Longest Increasing Sequence in Polylogarithmic Time"],"prefix":"10.1137","volume":"46","author":[{"given":"M.","family":"Saks","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C.","family":"Seshadhri","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,4,20]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20167"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-99-00796-X"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"A. Andoni, P. Indyk, and R. Krauthgamer,\n                      Overcoming the $\\ell_1$ non-embeddability barrier: Algorithms for product matrices\n                      , in Proceedings of the 20th Symposium on Discrete Algorithms, 2009, pp. 865-874.","DOI":"10.1137\/1.9781611973068.94"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/080716530"},{"key":"atypb5","first-page":"8","author":"Andoni A.","year":"2012","journal-title":"ACM Trans. Algorithms"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"A. Andoni and H. L. Nguyen,\n                      Near-optimal sublinear time algorithms for Ulam distance\n                      , in Proceedings of the 21st Symposium on Discrete Algorithms, 2010.","DOI":"10.1137\/1.9781611973075.8"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"A. Bhattacharyya, E. Grigorescu, K. Jung, S. Raskhodnikova, and D. Woodruff,\n                      Transitive-closure spanners\n                      , in Proceedings of the 18th Annual Symposium on Discrete Algorithms, 2009, pp. 531-540.","DOI":"10.1137\/1.9781611973068.101"},{"key":"atypb8","unstructured":"T. Cormen, C. Leiserson, R. Rivest, and C. Stein,\n                      Introduction to Algorithms\n                      , MIT Press, Cambridge, MA, 2000."},{"key":"atypb9","first-page":"97","author":"Dodis Y.","year":"1999","journal-title":"Computer Science"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"D. Dubhashi and A. Panconesi,\n                      Concentration of Measure for the Analysis of Randomized Algorithms\n                      , Cambridge University Press, Cambridge, UK, 2009.","DOI":"10.1017\/CBO9780511581274"},{"key":"atypb11","unstructured":"F. Ergun and H. Jowhari,\n                      On distance to monotonicity and longest increasing subsequence of a data stream\n                      , in Proceedings of the 19th Symposium on Discrete Algorithms, 2008, pp. 730-736."},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1006\/jcss.1999.1692"},{"key":"atypb13","first-page":"97","volume":"75","author":"Fischer E.","year":"2001","journal-title":"Bull. EATCS"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"E. Fischer, E. Lehman, I. Newman, S. Raskhodnikova, R. Rubinfeld, and A. Samorodnitsky,\n                      Monotonicity testing over general poset domains\n                      , in Proceedings of the 34th Annual Symposium on Theory of Computing, 2002, pp. 474-483.","DOI":"10.1145\/509907.509977"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(75)90103-X"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"A. Gal and P. Gopalan,\n                      Lower bounds on streaming algorithms for approximating the length of the longest increasing subsequence\n                      , in Proceedings of the 48th Symposium on Foundations of Computer Science, 2007, pp. 294-304.","DOI":"10.1109\/FOCS.2007.54"},{"key":"atypb17","first-page":"45","author":"Goldreich O.","year":"1999","journal-title":"RI"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s004930070011"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1145\/285055.285060"},{"key":"atypb20","unstructured":"P. Gopalan, T. S. Jayram, R. Krauthgamer, and R. Kumar,\n                      Estimating the sortedness of a data stream\n                      , in Proceedings of the 18th Symposium on Discrete Algorithms, 2007, pp. 318-327."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/050645804"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1080\/01621459.1963.10500830"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcss.2006.03.002"},{"key":"atypb24","doi-asserted-by":"crossref","unstructured":"D. Ron,\n                      Property testing\n                      , in Handbook on Randomization, vol. II, Kluwer, Dordrecht, the Netherlands, 2001, pp. 597-649.","DOI":"10.1007\/978-1-4615-0013-1_15"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539793255151"},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"M. Saks and C. Seshadhri,\n                      Estimating the longest increasing sequence in polylogarithmic time\n                      , in Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, 2010, pp. 458-467.","DOI":"10.1109\/FOCS.2010.51"},{"key":"atypb27","doi-asserted-by":"crossref","unstructured":"M. Saks and C. Seshadhri,\n                      Space efficient streaming algorithms for the distance to monotonicity and asymmetric edit distance\n                      , in Proceedings of the 24th Symposium on Discrete Algorithms, 2013.","DOI":"10.1137\/1.9781611973105.122"},{"key":"atypb28","unstructured":"X. Sun and D. Woodruff,\n                      The communication and streaming complexity of computing the longest common and increasing subsequences\n                      , in Proceedings of the 18th Symposium on Discrete Algorithms, 2007, pp. 336-345."}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/130942152","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:21:16Z","timestamp":1787340076000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/130942152"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":28,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/130942152"],"URL":"https:\/\/doi.org\/10.1137\/130942152","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}