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

At any time a user may wish to change the spatial conditions that constrain the region of a continuous $k$ CNN query. For example a user may change the angle range that bounds the pie-region which may expand or shrink the constrained region. Consider the running example of Fig.  where $p_2$ is the CNN of $q$ . The user changes (expands) the constrained region and the new pie-region is now constrained by angle range $\langle \theta_1,\theta_3 \rangle$ . Below we describe how we update the result of such queries by visiting the minimum possible number of cells.

If the angle range is decreased, the old $k$ CNNs that lie outside the new angle range are deleted and the algorithm continues with the ArcTrip of new angle range and radius $r=q.dist_k$ until $k$ CNNs are found. If the angle range increases, we visit only the cells that lie in the angle range that is not covered by the old angle range. The example below illustrates how the update works when the constrained region is expanded.

Figure: A Varying Pie-Region Continuous $1$ -CNN Query

[change in constrained region and location of $p_2$ ] [$p_4$ is new CNN]