TY - GEN
T1 - Approximate nearest neighbor search amid higher-dimensional flats
AU - Agarwal, Pankaj K.
AU - Rubin, Natan
AU - Sharir, Micha
N1 - Funding Information:
∗ Work by P. A. and M. S. was supported by Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Work by P. A. was also supported by NSF under grants CCF-11-61359, IIS-14-08846, and CCF-15-13816, and by an ARO grant W911NF-15-1-0408. Work by N. R. was supported by grant 1452/15 from Israel Science Foundation by grant 2014384 from the U.S.-Israeli Binational Science Foundation, and by Ralph Selig Career Development Chair in Information Theory; the project leading to this application has received funding from European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 678765. Work by M. S. has also been supported by Grant 892/13 from the Israel Science Foundation, by the Blavatnik Research Fund in Computer Science at Tel Aviv University, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University.
Publisher Copyright:
© Pankaj K. Agarwal, Natan Rubin, and Micha Sharir.
PY - 2017/9/1
Y1 - 2017/9/1
N2 - We consider the approximate nearest neighbor (ANN) problem where the input set consists of n k-flats in the Euclidean Rd, for any fixed parameters 0 ≤ k < d, and where, for each query point q, we want to return an input flat whose distance from q is at most (1 + ϵ) times the shortest such distance, where ϵ > 0 is another prespecified parameter. We present an algorithm that achieves this task with nk+1(log(n)/ ϵ)O(1) storage and preprocessing (where the constant of proportionality in the big-O notation depends on d), and can answer a query in O(polylog(n)) time (where the power of the logarithm depends on d and k). In particular, we need only nearquadratic storage to answer ANN queries amid a set of n lines in any fixed-dimensional Euclidean space. As a by-product, our approach also yields an algorithm, with similar performance bounds, for answering exact nearest neighbor queries amid k-flats with respect to any polyhedral distance function. Our results are more general, in that they also provide a tradeoff between storage and query time.
AB - We consider the approximate nearest neighbor (ANN) problem where the input set consists of n k-flats in the Euclidean Rd, for any fixed parameters 0 ≤ k < d, and where, for each query point q, we want to return an input flat whose distance from q is at most (1 + ϵ) times the shortest such distance, where ϵ > 0 is another prespecified parameter. We present an algorithm that achieves this task with nk+1(log(n)/ ϵ)O(1) storage and preprocessing (where the constant of proportionality in the big-O notation depends on d), and can answer a query in O(polylog(n)) time (where the power of the logarithm depends on d and k). In particular, we need only nearquadratic storage to answer ANN queries amid a set of n lines in any fixed-dimensional Euclidean space. As a by-product, our approach also yields an algorithm, with similar performance bounds, for answering exact nearest neighbor queries amid k-flats with respect to any polyhedral distance function. Our results are more general, in that they also provide a tradeoff between storage and query time.
KW - Approximate nearest neighbor search
KW - K-flats
KW - Linear programming queries
KW - Polyhedral distance functions
UR - http://www.scopus.com/inward/record.url?scp=85030543725&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ESA.2017.4
DO - 10.4230/LIPIcs.ESA.2017.4
M3 - Conference contribution
AN - SCOPUS:85030543725
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 25th European Symposium on Algorithms, ESA 2017
A2 - Sohler, Christian
A2 - Sohler, Christian
A2 - Pruhs, Kirk
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 25th European Symposium on Algorithms, ESA 2017
Y2 - 4 September 2017 through 6 September 2017
ER -