On Efficiently Monitoring Continuous Aggregate k Nearest Neighbors in Road Networks

Given a set <inline-formula><tex-math notation="LaTeX">$O$</tex-math><alternatives><mml:math><mml:mi>O</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq1-2911950.gif"/></alternatives></inline-formula> of data objects, a set <inline-formula><tex-math notation="LaTeX">$Q$</tex-math><alternatives><mml:math><mml:mi>Q</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq2-2911950.gif"/></alternatives></inline-formula> of query points, a positive integer <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq3-2911950.gif"/></alternatives></inline-formula>, and an aggregate function <inline-formula><tex-math notation="LaTeX">$f$</tex-math><alternatives><mml:math><mml:mi>f</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq4-2911950.gif"/></alternatives></inline-formula> (e.g., <italic>sum, max</italic>, and <italic>min</italic>), an aggregate <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq5-2911950.gif"/></alternatives></inline-formula> nearest neighbor query finds the <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq6-2911950.gif"/></alternatives></inline-formula> data objects from <inline-formula><tex-math notation="LaTeX">$O$</tex-math><alternatives><mml:math><mml:mi>O</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq7-2911950.gif"/></alternatives></inline-formula> that have the smallest aggregate distances with respect to the query points in <inline-formula><tex-math notation="LaTeX">$Q$</tex-math><alternatives><mml:math><mml:mi>Q</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq8-2911950.gif"/></alternatives></inline-formula>. This query has a large application base such as location-based services, transportation scheduling, traffic monitoring, emergency management, etc. With the rapid development of positioning technologies, many real-life applications appeal to the continuous aggregate <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq9-2911950.gif"/></alternatives></inline-formula> nearest neighbor monitoring in road networks, where both the data objects and query points move along the networks, and the edge weights (e.g., the driving time) fluctuate over time. In this paper, we study the problem of <italic>continuous aggregate <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq10-2911950.gif"/></alternatives></inline-formula> nearest neighbor monitoring</italic> (CA<inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq11-2911950.gif"/></alternatives></inline-formula>NN monitoring for short) <italic>in road networks</italic>. We propose an efficient generic CA<inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq12-2911950.gif"/></alternatives></inline-formula>NN monitoring framework, termed as <inline-formula><tex-math notation="LaTeX">$\mathsf{GMF}$</tex-math><alternatives><mml:math><mml:mi mathvariant="sans-serif">GMF</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq13-2911950.gif"/></alternatives></inline-formula>, which is capable of processing three types of update, including data object update, query point update, and edge weight update. We introduce an essential concept, i.e., <italic>safe distance</italic>, into this framework, which helps to boost the update efficiency for CA<inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq14-2911950.gif"/></alternatives></inline-formula>NN monitoring problem. Using an effective structure, termed as <italic>partial distance matrix</italic>, we identify the safe distance and form the candidate object set for CA<inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq15-2911950.gif"/></alternatives></inline-formula>NN monitoring efficiently. Extensive experimental evaluation on real road networks demonstrates that, our proposed CA<inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives><mml:math><mml:mi>k</mml:mi></mml:math><inline-graphic xlink:href="gao-ieq16-2911950.gif"/></alternatives></inline-formula>NN monitoring framework is superior to the state-of-the-art method.