Technique

Let $\langle \theta_1,\theta_2 \rangle$ be the angle range and $d_L$ and $d_U$ be the lower and upper distance bounds, respectively. The algorithm starts calling ArcTrip with angle range $\langle \theta_1,\theta_2 \rangle$ and radius set as $d_L$ . ArcTrip is iteratively called with radius incremented by $\delta$ unless $k$ NNs are found. Fig.  shows an example where $p_2$ is reported as CNN. The algorithm visits the shaded cells.

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

Continuous monitoring of such $k$ CNN queries is similar to continuous monitoring of $k$ NN queries with the difference that ArcTrip is called instead of CircularTrip and the distance of any object $p$ that lies outside the constrained region is considered infinity.