TY - JOUR

T1 - Range Minima Queries with Respect to a Random Permutation, and Approximate Range Counting

AU - Kaplan, Haim

AU - Ramos, Edgar

AU - Sharir, Micha

N1 - Funding Information:
The work by Haim Kaplan was partially supported by Grant 2006/204 from the U.S.—Israel Binational Science Foundation, and by Grant 975/06 from the Israel Science Fund (ISF). The work by Micha Sharir was partially supported by NSF Grants CCR-05-14079 and CCR-08-30272, by Grant 2006/194 from the U.S.—Israel Binational Science Foundation, by Grants 155/05 and 338/09 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.

PY - 2011/1

Y1 - 2011/1

N2 - In approximate halfspace range counting, one is given a set P of n points in ℝd, and an ε>0, and the goal is to preprocess P into a data structure which can answer efficiently queries of the form: Given a halfspace h, compute an estimate N such that (1-ε){pipe}P∩h{pipe}≤N≤(1+ε){pipe}P∩h{pipe}. Several recent papers have addressed this problem, including a study by Kaplan and Sharir (Proc. 17th Annu. ACM-SIAM Sympos. Discrete Algo., pp. 484-493, 2006), which is based, as is the present paper, on Cohen's technique for approximate range counting (Cohen in J. Comput. Syst. Sci. 55:441-453, 1997). In this approach, one chooses a small number of random permutations of P, and then constructs, for each permutation π, a data structure that answers efficiently minimum range queries: Given a query halfspace h, find the minimum-rank element (according to π) in P∩h. By repeating this process for all chosen permutations, the approximate count can be obtained, with high probability, using a certain averaging process over the minimum-rank outputs. In the previous study, the authors have constructed such a data structure in ℝ3, using a combinatorial result about the overlay of minimization diagrams in a randomized incremental construction of lower envelopes. In the present work, we propose an alternative approach to the range-minimum problem, based on cuttings, which achieves better performance. Specifically, it uses, for each permutation, O(n⌊d/2⌋(log log n)c/log ⌊d/2⌋n) expected storage and preprocessing time, for some constant c, and answers a range-minimum query in O(log n) expected time. We also present a different approach, based on "antennas," which is simple to implement, although the bounds on its expected storage, preprocessing, and query costs are worse by polylogarithmic factors.

AB - In approximate halfspace range counting, one is given a set P of n points in ℝd, and an ε>0, and the goal is to preprocess P into a data structure which can answer efficiently queries of the form: Given a halfspace h, compute an estimate N such that (1-ε){pipe}P∩h{pipe}≤N≤(1+ε){pipe}P∩h{pipe}. Several recent papers have addressed this problem, including a study by Kaplan and Sharir (Proc. 17th Annu. ACM-SIAM Sympos. Discrete Algo., pp. 484-493, 2006), which is based, as is the present paper, on Cohen's technique for approximate range counting (Cohen in J. Comput. Syst. Sci. 55:441-453, 1997). In this approach, one chooses a small number of random permutations of P, and then constructs, for each permutation π, a data structure that answers efficiently minimum range queries: Given a query halfspace h, find the minimum-rank element (according to π) in P∩h. By repeating this process for all chosen permutations, the approximate count can be obtained, with high probability, using a certain averaging process over the minimum-rank outputs. In the previous study, the authors have constructed such a data structure in ℝ3, using a combinatorial result about the overlay of minimization diagrams in a randomized incremental construction of lower envelopes. In the present work, we propose an alternative approach to the range-minimum problem, based on cuttings, which achieves better performance. Specifically, it uses, for each permutation, O(n⌊d/2⌋(log log n)c/log ⌊d/2⌋n) expected storage and preprocessing time, for some constant c, and answers a range-minimum query in O(log n) expected time. We also present a different approach, based on "antennas," which is simple to implement, although the bounds on its expected storage, preprocessing, and query costs are worse by polylogarithmic factors.

KW - Approximate range counting

KW - Random sampling

KW - Randomized incremental constructions

KW - Range minima queries

KW - Range searching

UR - http://www.scopus.com/inward/record.url?scp=78751625624&partnerID=8YFLogxK

U2 - 10.1007/s00454-010-9308-6

DO - 10.1007/s00454-010-9308-6

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AN - SCOPUS:78751625624

SN - 0179-5376

VL - 45

SP - 3

EP - 33

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 1

ER -