Continuous $k$ CNN Queries Over Irregular Region

In this section we show that our technique can easily be extended to continuously monitor the $k$ CNN queries over irregular region. Consider the irregular region $R$ shown in Fig. . We can easily monitor the nearest neighbors to $q$ in this irregular region $R$ . Let $d_L$ be the minimum distance of $R$ from $q$ and $d_U$ be the maximum distance of $R$ from $q$ . Similarly, let $\langle \theta_1,\theta_2 \rangle$ be the minimum angle range that fully covers $R$ as shown in Fig. .

Figure: Continuous $k$ CNN Query Over Irregular Region $R$

[The irregular region]

[forming a donut-pie region $k$ CNN query]

Continuous monitoring of $k$ CNN query over this irregular region $R$ is exactly same as continuous monitoring of a donut-pie region bounded by angle range $\langle \theta_1,\theta_2 \rangle$ and distance bounds $d_L$ and $d_U$ as presented in section . The only difference is that the cells and points that lie outside $R$ are ignored.