{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:24:02Z","timestamp":1787340242545,"version":"build-2736575974"},"reference-count":29,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2007,1]]},"abstract":"<jats:p>Point location is the problem of preprocessing a planar polygonal subdivision S of size n into a data structure in order to determine efficiently the cell of the subdivision that contains a given query point. We consider this problem from the perspective of expected query time. We are given the probabilities $p_z$ that the query point lies within each cell $z \\in S$. The entropy H of the resulting discrete probability distribution is the dominant term in the lower bound on the expected-case query time. We show that it is possible to achieve query time $H + O(\\sqrt{H}+1)$ with space $O(n)$, which is optimal up to lower order terms in the query time. We extend this result to subdivisions with convex cells, assuming a uniform query distribution within each cell. In order to achieve space efficiency, we introduce the concept of entropy-preserving cuttings.<\/jats:p>","DOI":"10.1137\/s0097539704446724","type":"journal-article","created":{"date-parts":[[2007,6,14]],"date-time":"2007-06-14T18:01:41Z","timestamp":1181844101000},"page":"584-610","source":"Crossref","is-referenced-by-count":23,"title":["Optimal Expected-Case Planar Point Location"],"prefix":"10.1137","volume":"37","author":[{"given":"Sunil","family":"Arya","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Theocharis","family":"Malamatos","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"David M.","family":"Mount","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ka Chun","family":"Wong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,6,12]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"S. Arya, S.W. Cheng, D. M. Mount, and H. Ramesh,\n                      Efficient expected-case algorithms for planar point location\n                      , in Proceedings of the 7th Annual Scandinavian Workshop on Algorithm Theory (Bergen, Norway, 2000), Lecture Notes in Comput. Sci. 1851, Springer-Verlag, Berlin, 2000, pp. 353\u2013366.","DOI":"10.1007\/3-540-44985-X_31"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539799366340"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"S. Arya, T. Malamatos, and D. M. Mount,\n                      Nearly optimal expected-case planar point location\n                      , in Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, 2000, pp. 208\u2013218.","DOI":"10.1109\/SFCS.2000.892108"},{"key":"R4","unstructured":"S. Arya, T. Malamatos, and D. M. Mount,\n                      Entropy-preserving cuttings and space-efficient planar point location\n                      , in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, Washington, DC, 2001, SIAM, Philadelphia, 2001, pp. 256\u2013261."},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1145\/1240233.1240240"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1109\/TC.1979.1675432"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02122778"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(86)90004-0"},{"key":"R9","unstructured":"T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein,\n                      Introduction to Algorithms\n                      , 2nd ed., MIT Press, Cambridge, MA, 2001."},{"key":"R10","unstructured":"T. M. Cover and J. A. Thomas,\n                      Elements of Information Theory\n                      , Wiley-Interscience, New York, 1991."},{"key":"R11","doi-asserted-by":"crossref","unstructured":"M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf,\n                      Computational Geometry: Algorithms and Applications\n                      , 2nd ed., Springer-Verlag, Berlin, 2000.","DOI":"10.1007\/978-3-662-04245-8"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"D. P. Dobkin and L. Snyder,\n                      On a general method for maximizing and minimizing among certain geometric problems\n                      , in Proceedings of the 20th Annual IEEE Symposium on Foundations of Computer Science, 1979, pp. 9\u201317.","DOI":"10.1109\/SFCS.1979.28"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1137\/0215023"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1145\/77635.77639"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"L. Fejes T\u00f3th,\n                      Lagerungen in der Ebene, auf der Kugel und im Raum\n                      , 1st ed., Springer-Verlag, Berlin, 1953.","DOI":"10.1007\/978-3-662-01206-2_1"},{"key":"R16","unstructured":"M. T. Goodrich, M. Orletsky, and K. Ramaiyer,\n                      Methods for achieving fast query times in point location data structures\n                      , in Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, LA, 1997, SIAM, Philadelphia, 1997, pp. 757\u2013766."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/0121057"},{"key":"R18","unstructured":"J. Iacono,\n                      Optimal planar point location\n                      , in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, Washington, DC, 2001, SIAM, Philadelphia, 2001, pp. 340\u2013341."},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/0212002"},{"key":"R20","unstructured":"D. E. Knuth,\n                      Sorting and Searching\n                      , The Art of Computer Programming, Vol. 3, 2nd ed., Addison-Wesley, Reading, MA, 1998."},{"key":"R21","unstructured":"T. Malamatos,\n                      Expected-case Planar Point Location\n                      , Ph.D. thesis, Department of Computer Science, Hong Kong University of Science and Technology, Hong Kong, 2002."},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/0206017"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(08)80064-8"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/0210035"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1145\/6138.6151"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/0925-7721(91)90012-4"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.2000.1101"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1002\/j.1538-7305.1948.tb01338.x"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1145\/3828.3835"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0097539704446724","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:26:34Z","timestamp":1787336794000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0097539704446724"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,1]]},"references-count":29,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2007,1]]}},"alternative-id":["10.1137\/S0097539704446724"],"URL":"https:\/\/doi.org\/10.1137\/s0097539704446724","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,1]]}}}