Reverse $k$ Nearest Neighbor Search over Trajectories

GPS enables mobile devices to continuously provide new opportunities to improve our daily lives. For example, the data collected in applications created by Uber or Public Transport Authorities can be used to plan transportation routes, estimate capacities, and proactively identify low coverage areas. In this paper, we study a new kind of query—<italic>Reverse <inline-formula><tex-math notation="LaTeX">$k$</tex-math><alternatives> <inline-graphic xlink:href="wang-ieq2-2776268.gif"/></alternatives></inline-formula> Nearest Neighbor Search over Trajectories</italic> (<inline-formula><tex-math notation="LaTeX">$\mathbf{R}{k}\mathbf{NNT}$</tex-math><alternatives> <inline-graphic xlink:href="wang-ieq3-2776268.gif"/></alternatives></inline-formula>), which can be used for route planning and capacity estimation. Given a set of existing routes <inline-formula><tex-math notation="LaTeX"> $\mathcal{D}_{\mathcal{R}}$</tex-math><alternatives><inline-graphic xlink:href="wang-ieq4-2776268.gif"/></alternatives> </inline-formula>, a set of passenger transitions <inline-formula><tex-math notation="LaTeX"> $\mathcal{D}_{\mathcal{T}}$</tex-math><alternatives><inline-graphic xlink:href="wang-ieq5-2776268.gif"/></alternatives> </inline-formula>, and a query route <inline-formula><tex-math notation="LaTeX">$Q$</tex-math><alternatives> <inline-graphic xlink:href="wang-ieq6-2776268.gif"/></alternatives></inline-formula>, an <inline-formula> <tex-math notation="LaTeX">$\mathbf{R}{k}\mathbf{NNT}$</tex-math><alternatives> <inline-graphic xlink:href="wang-ieq7-2776268.gif"/></alternatives></inline-formula> query returns all transitions that take <inline-formula><tex-math notation="LaTeX">$Q$</tex-math><alternatives> <inline-graphic xlink:href="wang-ieq8-2776268.gif"/></alternatives></inline-formula> as one of its <inline-formula> <tex-math notation="LaTeX">$k$</tex-math><alternatives><inline-graphic xlink:href="wang-ieq9-2776268.gif"/> </alternatives></inline-formula> nearest travel routes. To solve the problem, we first develop an index to handle dynamic trajectory updates, so that the most up-to-date transition data are available for answering an <inline-formula> <tex-math notation="LaTeX">$\mathbf{R}{k}\mathbf{NNT}$</tex-math><alternatives> <inline-graphic xlink:href="wang-ieq10-2776268.gif"/></alternatives></inline-formula> query. Then we introduce a filter refinement framework for processing <inline-formula><tex-math notation="LaTeX">$\mathbf{R}{k}\mathbf{NNT}$</tex-math> <alternatives><inline-graphic xlink:href="wang-ieq11-2776268.gif"/></alternatives></inline-formula> queries using the proposed indexes. Next, we show how to use <inline-formula><tex-math notation="LaTeX">$\mathbf{R}{k}\mathbf{NNT}$ </tex-math><alternatives><inline-graphic xlink:href="wang-ieq12-2776268.gif"/></alternatives></inline-formula> to solve the optimal route planning problem <inline-formula><tex-math notation="LaTeX">$\mathbf{MaxR}{k}\mathbf{NNT}$</tex-math> <alternatives><inline-graphic xlink:href="wang-ieq13-2776268.gif"/></alternatives></inline-formula> (<inline-formula> <tex-math notation="LaTeX">$\mathbf{MinR}{k}\mathbf{NNT}$</tex-math><alternatives> <inline-graphic xlink:href="wang-ieq14-2776268.gif"/></alternatives></inline-formula>), which is to search for the optimal route from a start location to an end location that could attract the maximum (or minimum) number of passengers based on a predefined travel distance threshold. Experiments on real datasets demonstrate the efficiency and scalability of our approaches. To the best of our knowledge, this is the first work to study the <inline-formula> <tex-math notation="LaTeX">$\mathbf{R}{k}\mathbf{NNT}$</tex-math><alternatives> <inline-graphic xlink:href="wang-ieq15-2776268.gif"/></alternatives></inline-formula> problem for route planning.