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Improved Streaming Algorithm for the Klee's Measure Problem and Generalizations .
Mridul Nandi ⓡ N. V. Vinodchandran ⓡ Arijit Ghosh ⓡ Kuldeep S. Meel ⓡ Soumit Pal ⓡ Chakraborty,Sourav .
In Proceedings of the International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), August 2024.
Estimating the size of the union of a stream of sets $S_1, S_2, łdots S_M$ where each set is a subset of a known universe $\Omega$ is a fundamental problem in data streaming. This problem naturally generalizes the well-studied $\FO$ estimation problem in the streaming literature, where each set contains a single element from the universe. We consider the general case when the sets $S_i$ can be succinctly represented and allow efficient membership, cardinality, and sampling queries (called a Delphic family of sets). A notable example in this framework is the Klee's Measure Problem (KMP), where every set $S_i$ is an axis-parallel rectangle in $d$-dimensional spaces ($\Omega = [\Delta]^d$ where $[\Delta] := łeft\1, \dots, \Delta \right\$ and $\Delta \in \mathbbN$). Recently, Meel, Chakraborty, and Vinodchandran (PODS-21, PODS-22) designed a streaming algorithm for $(\epsilon,\delta)$-estimation of the size of the union of set streams over Delphic family with space and update time complexity $Ołeft(\fracłog^3 |\Omega|\varepsilon^2\cdot łog \frac1\delta\right)$ and $\widetildeOłeft(\fracłog^4 |\Omega|\varepsilon^2\cdot łog \frac1\delta\right)$, respectively. This work presents a new, sampling-based algorithm for estimating the size of the union of Delphic sets that has space and update time complexity $\widetildeOłeft(\fracłog^2 |\Omega|\varepsilon^2\cdot łog \frac1\delta\right)$. This improves the space complexity bound by a $łog|\Omega|$ factor and update time complexity bound by a $łog^2 |\Omega|$ factor. A critical question is whether quadratic dependence of $łog |\Omega|$ on space and update time complexities is necessary. Specifically, can we design a streaming algorithm for estimating the size of the union of sets over Delphic family with space and complexity linear in $łog |\Omega|$ and update time $\poly(łog |\Omega|)$? While this appears technically challenging, we show that establishing a lower bound of $ømega(łog |\Omega|)$ with $\poly(łog |\Omega|)$ update time is beyond the reach of current techniques. Specifically, we show that under certain hard-to-prove computational complexity hypothesis, there is a streaming algorithm for the problem with optimal space complexity $O(łog |\Omega|)$ update time $\poly(łog(|\Omega|))$. Thus, establishing a space lower bound of $ømega(łog |\Omega|)$ will lead to break-through complexity class separation results.
@inproceedings{MVGMPC24,
title={
Improved Streaming Algorithm for the Klee's Measure Problem and
Generalizations
},
author={
Nandi, Mridul and Vinodchandran, N. V.
and Ghosh, Arijit
and Meel, Kuldeep S.
and Pal, Soumit
and Chakraborty,Sourav
},
booktitle=APPROX,
abstract={
Estimating the size of the union of a stream of sets $S_1, S_2, \ldots S_M$
where each set is a subset of a known universe $\Omega$ is a fundamental
problem in data streaming. This problem naturally generalizes the
well-studied $\FO$ estimation problem in the streaming literature, where
each set contains a single element from the universe. We consider the
general case when the sets $S_i$ can be succinctly represented and allow
efficient membership, cardinality, and sampling queries (called a Delphic
family of sets). A notable example in this framework is the Klee's Measure
Problem (KMP), where every set $S_i$ is an axis-parallel rectangle in
$d$-dimensional spaces ($\Omega = [\Delta]^d$ where $[\Delta] := \left\{1,
\dots, \Delta \right\}$ and $\Delta \in \mathbb{N}$). Recently, Meel,
Chakraborty, and Vinodchandran (PODS-21, PODS-22) designed a streaming
algorithm for $(\epsilon,\delta)$-estimation of the size of the union of set
streams over Delphic family
with space and update time complexity $O\left(\frac{\log^3
|\Omega|}{\varepsilon^2}\cdot \log \frac{1}{\delta}\right)$ and
$\widetilde{O}\left(\frac{\log^{4} |\Omega|}{\varepsilon^2}\cdot \log
\frac{1}{\delta}\right)$, respectively.
This work presents a new, sampling-based algorithm for estimating the size
of the union of Delphic sets that has space and update time complexity
$\widetilde{O}\left(\frac{\log^2 |\Omega|}{\varepsilon^2}\cdot \log
\frac{1}{\delta}\right)$. This improves the space complexity bound by a
$\log|\Omega|$ factor and update time complexity bound by a $\log^2
|\Omega|$ factor.
A critical question is whether quadratic dependence of $\log |\Omega|$ on
space and update time complexities is necessary. Specifically, can we design
a streaming algorithm for estimating the size of the union of sets over
Delphic family with space and complexity linear in $\log |\Omega|$ and
update time $\poly(\log |\Omega|)$? While this appears technically
challenging, we show that establishing a lower bound of $\omega(\log
|\Omega|)$ with $\poly(\log |\Omega|)$ update time is beyond the reach of
current techniques. Specifically, we show that under certain hard-to-prove
computational complexity hypothesis, there is a streaming algorithm for the
problem with optimal space complexity $O(\log |\Omega|)$ update time
$\poly(\log(|\Omega|))$. Thus, establishing a space lower bound of
$\omega(\log |\Omega|)$ will lead to break-through complexity class
separation results.
},
nameorder={random},
year={2024},
month=aug,
bib2html_rescat={Data Streams},
bib2html_pubtype={Refereed Conference},
bib2html_dl_pdf={},
}
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