Small-size ε-nets for axis-parallel rectangles and boxes
Proceedings of the forty-first annual ACM symposium on Theory of computing
Near-linear approximation algorithms for geometric hitting sets
Proceedings of the twenty-fifth annual symposium on Computational geometry
A note about weak ε-nets for axis-parallel boxes in d-space
Information Processing Letters
More Algorithms for All-Pairs Shortest Paths in Weighted Graphs
SIAM Journal on Computing
Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes
SIAM Journal on Computing
Tight lower bounds for the size of epsilon-nets
Proceedings of the twenty-seventh annual symposium on Computational geometry
Counting colours in compressed strings
CPM'11 Proceedings of the 22nd annual conference on Combinatorial pattern matching
Colored range queries and document retrieval
Theoretical Computer Science
Colored top-K range-aggregate queries
Information Processing Letters
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Let $P$ be a set of $n$ points in $\mathbb{R}^d$, 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 following form: Given an axis-parallel box $Q$, count the number of distinct colors of the points of $P\cap Q$. We present a general and relatively simple solution that has a polylogarithmic query time and worst-case storage about $O(n^d)$. It is based on several interesting structural properties of the problem, which we establish here. 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 $\mathbb{R}^2$, the most efficient solution uses fast matrix multiplication in the preprocessing stage. In higher dimensions we use simpler tradeoff mechanisms, which behave just as well. We give a reduction from matrix multiplication to the off-line version of problem, which shows that in $\mathbb{R}^2$ our time-space tradeoffs are reasonably sharp, 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 higher dimension.