# Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case +
## 6 Stage 1-Interior Depletion
### Stage 2-Edge Depletion
Therefore, in Stage 2.5, all remaining side points, \(s_{4},s_{5},\cdots,s_{m-3}\) move to \(\mathsf{Circle}(S)\) (Figure 16(5)). When no points remain in the interior of the polygon with color \(\mathsf{side}\), the beacons color themselves as corners in Stage 2.6 and the algorithm terminates (Figure 16(6)). During Stage 2, each robot can unambiguously determine with which side \(S\) it is associated. This allows us to consider each side in isolation. We show below that this stage satisfies all the conditions required to apply the Beacon-Directed Curve Positioning framework, so it runs in \(\mathcal{O}(1)\) epochs. During Stage 2, all the side robots of \(\mathbf{P}^{\prime\prime}\) become corners of \(\mathbf{P}^{\prime\prime\prime}\) and take color \(\mathsf{corner}\). Stage 2 starts only after Stage 1.5.2 finishes. Moreover, Stage 2 runs for \(\mathcal{O}(1)\) epochs avoiding collisions and then the algorithm terminates. Proof. : Stages 2.1-2.4 run in \(\mathcal{O}(1)\) epochs since only, at most, two robots are moving in these stages. Stage 2.6 also runs in \(\mathcal{O}(1)\) epochs as no robots move in this stage and only color computation is needed. Therefore, it remains to show that Stage 2.5 runs in \(\mathcal{O}(1)\) epochs, even by \(\mathcal{ASYNC}\) robots, using the Beacon-Directed Curve Positioning framework. Observe that side \(S\) of \(\mathsf{Circle}(S)\) has \(1\leq m<N-8\) side robots. Moreover, due to the moves of the side robots of \(S\) in Stages 2.1-2.4, six beacons are already on the safe circle, \(\mathsf{Circle}(S)\). We now argue that Stage 2.5 satisfies all the conditions required to apply the Beacon-Directed Curve Positioning technique. We first show that the waiting robots on \(S\) that are yet to move to \(\mathsf{Circle}(S)\) satisfy all the conditions for an admissible configuration (Definition 2). Condition (a) is satisfied as the robots in \(S\) always move perpendicularly to \(S\) to positions on \(\mathsf{Circle}(S)\). Condition (b) is also satisfied due to their perpendicular moves. Condition (c) is satisfied as the path of any robot on \(S\) perpendicular to \(S\) intersects \(\mathsf{Circle}(S)\) at exactly one point, as \(\mathsf{Circle}(S)\) is a circle that passes through the beacons \(b_{1}\) and \(b_{m}\) and the corners \(c_{i}\) and \(c_{i+1}\). Condition (d) is also satisfied as all the waiting robots are on a line \(S\) and \(\mathsf{Circle}(S)\) passes through the corners \(c_{i}\) and \(c_{i+1}\) in the corridor of \(S\). Furthermore, all the waiting robots on \(S\) are between the three left beacons and three right beacons with color \(\mathsf{beacon}\) and the waiting robots on \(S\) have color \(\mathsf{side}\). Therefore, from Theorem 2, Stage 2.5 finishes in \(\mathcal{O}(1)\) epochs as \(k=3\) due to our use of \(\mathsf{Circle}(S)\) as a curve for the final positions of the robots on \(S\). Thus, Stage 2 runs for \(\mathcal{O}(1)\) epochs. The collision-freedom in the execution of Stage 2 is immediate as the robots move perpendicularly to \(S\). We also observe that side robots can distinguish the interior of the polygon from the outside; hence, each side \(S\) can proceed independently. Stage 2 starts only after Stage 1.5.2 finishes as, otherwise, the robots on the sides of \(\mathbf{P}^{\prime\prime}\) with color \(\mathsf{side}\) see at least a robot with color that is not \(\mathsf{side}\) or \(\mathsf{corner}\). The algorithm terminates after Stage 2 as all the robots have color \(\mathsf{corner}\) after Stage 2.5 finishes and that signifies that all the robots in \(\mathcal{Q}\) are on the corners of a hull \(\mathbf{P}\), solving Complete Visibility. Stage 2 executes in \(\mathcal{O}(1)\) epochs avoiding collisions and uses 6 colors (2 colors are of Stage 1). Proof of Theorem 1.: We have the runtime of Theorem 1 in the \(\mathcal{ASYNC}\) setting combining the results of Theorems 3-5. We have the number of colors bound of Theorem 1 counting the colors listed in Table 2, which totals 47 different colors.