Math @ Duke

Publications [#235361] of Pankaj K. Agarwal
Papers Published
 Agarwal, PK; Matoušek, J; Sharir, M, On range searching with semialgebraic sets. II,
SIAM Journal on Computing, vol. 42 no. 6
(December, 2013),
pp. 20392062, ISSN 00975397 [doi]
(last updated on 2018/06/22)
Abstract: Let P be a set of n points in ℝd. We present a linearsize data structure for answering range queries on P with constantcomplexity semialgebraic sets as ranges, in time close to O(n11/d). It essentially matches the performance of similar structures for simplex range searching, and, for d ≥ 5, significantly improves earlier solutions by the first two authors obtained in 1994. This almost settles a longstanding open problem in range searching. The data structure is based on a partitioning technique of Guth and Katz [On the Erdos distinct distances problem in the plane, arXiv:1011.4105, 2010], which shows that for a parameter r, 1 < r ≤ n, there exists a dvariate polynomial f of degree O(r1/d) such that each connected component of ℝd \ Z(f) contains at most n/r points of P, where Z(f) is the zero set of f. We present an efficient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications. © 2013 Society for Industrial and Applied Mathematics.


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