{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:34:29Z","timestamp":1787340869043,"version":"build-2736575974"},"reference-count":48,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100004586","name":"Funda\u00e7\u00e3o Carlos Chagas Filho de Amparo \u00e0 Pesquisa do Estado do Rio de Janeiro","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004586","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1618866"],"award-info":[{"award-number":["CCF-1618866"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002920","name":"Research Grants Council, University Grants Committee","doi-asserted-by":"publisher","award":["610108"],"award-info":[{"award-number":["610108"]}],"id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002920","name":"Research Grants Council, University Grants Committee","doi-asserted-by":"publisher","award":["16200014"],"award-info":[{"award-number":["16200014"]}],"id":[{"id":"10.13039\/501100002920","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>In the polytope membership problem, a convex polytope $K$ in $\\mathbb{R}^d$ is given, and the objective is to preprocess $K$ into a data structure so that, given any query point $q \\in \\mathbb{R}^d$, it is possible to determine efficiently whether $q \\in K$. We consider this problem in an approximate setting. Given an approximation parameter $\\varepsilon$, the query can be answered either way if the distance from $q$ to $K$'s boundary is at most $\\varepsilon$ times $K$'s diameter. We assume that the dimension $d$ is fixed, and $K$ is presented as the intersection of $n$ halfspaces. Previous solutions to approximate polytope membership were based on straightforward applications of classic polytope approximation techniques by Dudley [ Approx. Theory, 10 (1974), pp. 227--236] and Bentley, Faust, and Preparata [ Commun. ACM, 25 (1982), pp. 64--68]. The former is optimal in the worst case with respect to space, and the latter is optimal with respect to query time. We present four main results. First, we show how to combine the two above techniques to obtain a simple space-time trade-off. Second, we present an algorithm that dramatically improves this trade-off. In particular, for any constant $\\alpha \\ge 4$, this data structure achieves query time roughly $O(1\/\\varepsilon^{(d-1)\/\\alpha})$ and space roughly $O(1\/\\varepsilon^{(d-1)(1 - \\Omega(\\log \\alpha)\/\\alpha)})$. We do not know whether this space bound is tight, but our third result shows that there is a convex body such that our algorithm achieves a space of at least $\\Omega( 1\/\\varepsilon^{(d-1)(1-O(\\sqrt{\\alpha})\/\\alpha} )$. Our fourth result shows that it is possible to reduce approximate Euclidean nearest neighbor searching to approximate polytope membership queries. Combined with the above results, this provides significant improvements to the best known space-time trade-offs for approximate nearest neighbor searching in $\\mathbb{R}^d$. For example, we show that it is possible to achieve a query time of roughly $O(\\log n + 1\/\\varepsilon^{d\/4})$ with space roughly $O(n\/\\varepsilon^{d\/4})$, thus reducing by half the exponent in the space bound.<\/jats:p>","DOI":"10.1137\/16m1061096","type":"journal-article","created":{"date-parts":[[2018,1,2]],"date-time":"2018-01-02T14:32:51Z","timestamp":1514903571000},"page":"1-51","source":"Crossref","is-referenced-by-count":2,"title":["Approximate Polytope Membership Queries"],"prefix":"10.1137","volume":"47","author":[{"given":"Sunil","family":"Arya","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guilherme D.","family":"da Fonseca","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David M.","family":"Mount","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,1,2]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1145\/1008731.1008736"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"P. K. Agarwal, S. Har-Peled, and K. R. Varadarajan,\n                      Geometric approximation via coresets\n                      , in Combinatorial and Computational Geometry, J. E. Goodman, J. Pach, and E. Welzl, eds., MSRI Publications, Cambridge, UK, 2005.","DOI":"10.1017\/9781009701259.002"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/0222051"},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"N. Alon and J. H. Spencer,\n                      The Probabilistic Method\n                      , Wiley-Interscience, New York, 2000.","DOI":"10.1002\/0471722154"},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"S. Arya and T. M. Chan,\n                      Better $\\eps$-dependencies for offline approximate nearest neighbor search, Euclidean minimum spanning trees, and $\\eps$-kernels\n                      , in Proceedings of the 30th Annual Symposium on Computational Geometry, 2014, pp. 416-425.","DOI":"10.1145\/2582112.2582161"},{"key":"atypb6","doi-asserted-by":"crossref","unstructured":"S. Arya, G. D. da Fonseca, and D. M. Mount,\n                      A unified approach to approximate proximity searching\n                      , in Proceedings of the 18th Annual European Symposium on Algorithms, 2010, pp. 374-385.","DOI":"10.1007\/978-3-642-15775-2_32"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"S. Arya, G. D. da Fonseca, and D. M. Mount,\n                      Optimal area-sensitive bounds for polytope approximation\n                      , in Proceedings of the 28th Annual Symposium on Computational Geometry, 2012, pp. 363-372,https:\/\/doi.acm.org\/10.1145\/2261250.2261305.","DOI":"10.1145\/2261250.2261305"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"S. Arya, G. D. da Fonseca, and D. M. Mount,\n                      Approximate Polytope Membership Queries\n                      ,arXiv:1604.01183[cs.CG], 2016.","DOI":"10.1137\/1.9781611974782.18"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"S. Arya, G. D. da Fonseca, and D. M. 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Chan,\n                      Fixed-dimensional linear programming queries made easy\n                      , in Proceedings of the 12th Annual Symposium on Computational Geometry, 1996, pp. 284-290,https:\/\/doi.acm.org\/10.1145\/237218.237397.","DOI":"10.1145\/237218.237397"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/BF02712874"},{"key":"atypb22","unstructured":"T. M. Chan,\n                      Closest-point problems simplified on the RAM\n                      , in Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms, 2002, pp. 472-473."},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.comgeo.2005.10.002"},{"key":"atypb24","doi-asserted-by":"crossref","unstructured":"T. M. 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