Spatial range joins have many applications, including geographic information systems, location-based social networking services, neuroscience, and visualization. However, joins incur not only expensive computational costs but also too large result sets. A practical and reasonable approach to alleviating these issues is to return random samples of the join results. Although this is promising and sufficient for many applications involving spatial range joins, efficiently computing random samples is not trivial. This is because we must obtain random join samples without running spatial range joins. We address this challenging problem for the first time and aim at designing time- and space-efficient algorithms. First, we design baseline algorithms that employ existing random sampling techniques and show that they are not efficient. We then propose a new data structure that can deal with our problem in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\tilde{O}(n + m + t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>O</mi> <mo stretchy="false">~</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> expected time and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$O(n+m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> space, where n and m are the sizes of two point sets and t is the required number of samples. Furthermore, we devise a new streaming algorithm that obtains t random samples in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\tilde{O}(n + m + t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>O</mi> <mo stretchy="false">~</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> expected time and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$O(m+t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> space for the case where one dataset consists of n streaming points. We conduct extensive experiments using four real spatial datasets, and the results demonstrate that our algorithms are significantly faster than the baselines in most tests.

Static and streaming algorithms for random sampling over spatial range joins

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2026年8月1日

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Daichi Amagata,
Duhyun Kim

摘要

Spatial range joins have many applications, including geographic information systems, location-based social networking services, neuroscience, and visualization. However, joins incur not only expensive computational costs but also too large result sets. A practical and reasonable approach to alleviating these issues is to return random samples of the join results. Although this is promising and sufficient for many applications involving spatial range joins, efficiently computing random samples is not trivial. This is because we must obtain random join samples without running spatial range joins. We address this challenging problem for the first time and aim at designing time- and space-efficient algorithms. First, we design baseline algorithms that employ existing random sampling techniques and show that they are not efficient. We then propose a new data structure that can deal with our problem in $\tilde{O}(n + m + t)$ $\tilde{O} (n + m + t)$ expected time and $O(n+m)$ $O (n + m)$ space, where n and m are the sizes of two point sets and t is the required number of samples. Furthermore, we devise a new streaming algorithm that obtains t random samples in $\tilde{O}(n + m + t)$ $\tilde{O} (n + m + t)$ expected time and $O(m+t)$ $O (m + t)$ space for the case where one dataset consists of n streaming points. We conduct extensive experiments using four real spatial datasets, and the results demonstrate that our algorithms are significantly faster than the baselines in most tests.