Fast Reachability Queries Answering Based on <inline-formula><tex-math notation="LaTeX">$\mathsf{RCN}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">RCN</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq1-3108433.gif"/></alternatives></inline-formula> Reduction

Answering reachability queries is one of the fundamental graph operations. One way to make acceleration is directly building index on the input graph. Considering that the size of the input graph has a great impact on the query performance, the other way is reducing the size of the input graph, such that the given queries can be answered over a smaller graph. Although the input graph can be reduced significantly in some cases by existing approaches, a smaller reduced graph does not always mean a positive effect on the query performance. In this paper, we study graph reduction to accelerate reachability queries answering. We propose a novel graph reduction approach, namely <inline-formula><tex-math notation="LaTeX">$\mathsf{RCN}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">RCN</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq2-3108433.gif"/></alternatives></inline-formula> reduction, to reduce the input graph <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq3-3108433.gif"/></alternatives></inline-formula> with <inline-formula><tex-math notation="LaTeX">$|V|$</tex-math><alternatives><mml:math><mml:mrow><mml:mo>|</mml:mo><mml:mi>V</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="zhou-ieq4-3108433.gif"/></alternatives></inline-formula> nodes into a smaller one <inline-formula><tex-math notation="LaTeX">$G^{r}$</tex-math><alternatives><mml:math><mml:msup><mml:mi>G</mml:mi><mml:mi>r</mml:mi></mml:msup></mml:math><inline-graphic xlink:href="zhou-ieq5-3108433.gif"/></alternatives></inline-formula> with <inline-formula><tex-math notation="LaTeX">$|V^r|$</tex-math><alternatives><mml:math><mml:mrow><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:msup><mml:mi>V</mml:mi><mml:mi>r</mml:mi></mml:msup><mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:math><inline-graphic xlink:href="zhou-ieq6-3108433.gif"/></alternatives></inline-formula> nodes. Assume that the probability of a node of <inline-formula><tex-math notation="LaTeX">$G$</tex-math><alternatives><mml:math><mml:mi>G</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq7-3108433.gif"/></alternatives></inline-formula> to be a query node is <inline-formula><tex-math notation="LaTeX">$1/|V|$</tex-math><alternatives><mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mo>|</mml:mo><mml:mi>V</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="zhou-ieq8-3108433.gif"/></alternatives></inline-formula>, we show that based on our approach, the lower bound probability that a query <inline-formula><tex-math notation="LaTeX">$q$</tex-math><alternatives><mml:math><mml:mi>q</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq9-3108433.gif"/></alternatives></inline-formula> can be answered in constant time is <inline-formula><tex-math notation="LaTeX">$1-(\frac{|V^r|}{|V|})^2$</tex-math><alternatives><mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:msup><mml:mi>V</mml:mi><mml:mi>r</mml:mi></mml:msup><mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>V</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math><inline-graphic xlink:href="zhou-ieq10-3108433.gif"/></alternatives></inline-formula>, denoting that the probability that <inline-formula><tex-math notation="LaTeX">$q$</tex-math><alternatives><mml:math><mml:mi>q</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq11-3108433.gif"/></alternatives></inline-formula> needs to be answered over the reduced graph is <inline-formula><tex-math notation="LaTeX">$(\frac{|V^r|}{|V|})^2$</tex-math><alternatives><mml:math><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mfrac><mml:mrow><mml:mrow><mml:mo>|</mml:mo></mml:mrow><mml:msup><mml:mi>V</mml:mi><mml:mi>r</mml:mi></mml:msup><mml:mrow><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>V</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mfrac><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math><inline-graphic xlink:href="zhou-ieq12-3108433.gif"/></alternatives></inline-formula>, which means the smaller the reduced graph, the larger the probability that <inline-formula><tex-math notation="LaTeX">$q$</tex-math><alternatives><mml:math><mml:mi>q</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq13-3108433.gif"/></alternatives></inline-formula> can be answered in constant time. We show the difficulties of <inline-formula><tex-math notation="LaTeX">$\mathsf{RCN}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">RCN</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq14-3108433.gif"/></alternatives></inline-formula> reduction and propose efficient algorithms to improve the reduction ratio. Based on the result of <inline-formula><tex-math notation="LaTeX">$\mathsf{RCN}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">RCN</mml:mi></mml:math><inline-graphic xlink:href="zhou-ieq15-3108433.gif"/></alternatives></inline-formula> reduction, we further propose a novel labeling scheme to accelerate queries answering. We confirm the efficiency of our approach by extensive experimental results for graph reduction and reachability queries processing using 20 real datasets.