TY - GEN

T1 - Counting colors in boxes

AU - Kaplan, Haim

AU - Rubin, Natan

AU - Sharir, Micha

AU - Verbin, Elad

N1 - Funding Information:
Work by Micha Sharir was partially supported by NSF Grant CCF-05-14079, by a grant from the U.S.-Israeli Binational Science Foundation, by grant 155/05 from the Israel Science Fund, Israeli Academy of Sciences, and by the Hermann Minkowski-MINERVA Center for Geometry at Tel Aviv University. This work is part of the second author's M.Sc. Dissertation, prepared under the supervision of the first and third authors.
Publisher Copyright:
Copyright © 2007 by the Association for Computing Machinery, Inc. and the Society for Industrial and Applied Mathematics.

PY - 2007/1/1

Y1 - 2007/1/1

N2 - Let P be a set of n points in ℝd, so that each point is colored by one of C given colors. We present algorithms for preprocessing P into a data structure that efficiently supports queries of the form: Given an axis-parallel box Q, count the number of distinct colors of the points of P ∩ Q. We present a general and relatively simple solution that has polylogarithmic query time and worst-case storage about O(nd). It is based on several interesting structural properties of the problem that we derive. We also show that for random inputs, the data structure requires almost linear expected storage. We then present several techniques for achieving space-time tradeoff. In ℝ2, the most efficient solution uses fast matrix multiplication in the preprocessing stage. In higher dimensions we obtain a tradeoff using simpler mechanisms. We give a reduction from matrix multiplication to the offline version of problem, which shows that in ℝ2 our time-space tradeoffs are close to optimal in the sense that improving them substantially would improve the best exponent of matrix multiplication. Finally, we present a generalized matrix multiplication problem and show its intimate relation to counting colors in boxes in any dimension.

AB - Let P be a set of n points in ℝd, so that each point is colored by one of C given colors. We present algorithms for preprocessing P into a data structure that efficiently supports queries of the form: Given an axis-parallel box Q, count the number of distinct colors of the points of P ∩ Q. We present a general and relatively simple solution that has polylogarithmic query time and worst-case storage about O(nd). It is based on several interesting structural properties of the problem that we derive. We also show that for random inputs, the data structure requires almost linear expected storage. We then present several techniques for achieving space-time tradeoff. In ℝ2, the most efficient solution uses fast matrix multiplication in the preprocessing stage. In higher dimensions we obtain a tradeoff using simpler mechanisms. We give a reduction from matrix multiplication to the offline version of problem, which shows that in ℝ2 our time-space tradeoffs are close to optimal in the sense that improving them substantially would improve the best exponent of matrix multiplication. Finally, we present a generalized matrix multiplication problem and show its intimate relation to counting colors in boxes in any dimension.

UR - http://www.scopus.com/inward/record.url?scp=84969135590&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84969135590

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 785

EP - 794

BT - Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007

PB - Association for Computing Machinery

T2 - 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007

Y2 - 7 January 2007 through 9 January 2007

ER -