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- W4301927926 abstract "This paper studies the emph{$varepsilon$-approximate range emptiness} problem, where the task is to represent a set $S$ of $n$ points from ${0,ldots,U-1}$ and answer emptiness queries of the form $[a ; b]cap S neq emptyset$ ? with a probability of emph{false positives} allowed. This generalizes the functionality of emph{Bloom filters} from single point queries to any interval length $L$. Setting the false positive rate to $varepsilon/L$ and performing $L$ queries, Bloom filters yield a solution to this problem with space $O(n lg(L/varepsilon))$ bits, false positive probability bounded by $varepsilon$ for intervals of length up to $L$, using query time $O(L lg(L/varepsilon))$. Our first contribution is to show that the space/error trade-off cannot be improved asymptotically: Any data structure for answering approximate range emptiness queries on intervals of length up to $L$ with false positive probability $varepsilon$, must use space $Omega(n lg(L/varepsilon)) - O(n)$ bits. On the positive side we show that the query time can be improved greatly, to constant time, while matching our space lower bound up to a lower order additive term. This result is achieved through a succinct data structure for (non-approximate 1d) range emptiness/reporting queries, which may be of independent interest." @default.
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- W4301927926 date "2014-07-10" @default.
- W4301927926 modified "2023-10-16" @default.
- W4301927926 title "Approximate Range Emptiness in Constant Time and Optimal Space" @default.
- W4301927926 doi "https://doi.org/10.48550/arxiv.1407.2907" @default.
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