Extension to Varying Donut-Pie Region Continuous $k$ CNN Queries

In this section we will extend our approach to answer the queries where the user can change the constraints at any time. Consider the running example of Fig.  where $p_2$ is the CNN of $q$ . The user changes the constrained region by changing the angle range and distance bounds. The angle range is changed from $\langle \theta_1,\theta_2 \rangle$ to $\langle \theta_1,\theta_3 \rangle$ and the lower distance bound has been changed from $d'_L$ to $d_L$ . Now the algorithm needs to monitor the nearest neighbors in a bigger constrained region (the closed shape shown with thick boundary). The idea behind updating the result of such queries is simply look in the area that was not looked before. ArcTrip with suitable angle range and radius can be used for this purpose. Below we give an extensive example that will illustrate the monitoring of such queries.

Figure: Computation of a Varying Donut-Pie Region $1$ -CNN Query

[constrained region and location of $p_2$ changes]

[$p_4$ is the new CNN]