# Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case +
## 6 Stage 1-Interior Depletion  
The goal of this stage is to move the corner, side, and interior robots of \(\mathbf{P}\) to the corners and sides of \(\mathbf{P}^{\prime\prime}\). At the start of Stage 1, each robot is colored start or ready (Lemma 8). A robot with color ready is located at a corner of \(\mathbf{P}\). A robot with color start is located at a corner or side or in the interior of \(\mathbf{P}\). From Stage 0, we have that \(\mathbf{P}\) is not a line segment. Let \(\mathcal{Q}_{c}\), \(\mathcal{Q}_{s}\), \(\mathcal{Q}_{i}\) be the sets of robots at corners, sides, and the interior of \(\mathbf{P}\), respectively. For a robot \(r_{i}\), if all other visible robots are within an angle of \(<\)180\({}^{\circ}\) (=180\({}^{\circ}\), \(>\)180\({}^{\circ}\), respectively), then \(r_{i}\) is a corner (side, interior, respectively) robot. Stage 1 moves robots in all \(\mathcal{Q}_{c}\), \(\mathcal{Q}_{s}\), \(\mathcal{Q}_{i}\) to corners and sides of \(\mathbf{P}^{\prime\prime}\) and colors them as corner and side. Figure 9 illustrates Stage 1. This stage needs four rounds in the \(\mathcal{FSYNC}\) setting [9]. In Round 1.1, all corners of \(\mathbf{P}\) become corners of \(\mathbf{P}^{\prime}\) with color corner, and the side robots of \(\mathbf{P}\) change their color to side1 without moving. In Round 1.2, all interior robots of \(\mathbf{P}\) (also interior in \(\mathbf{P}^{\prime}\)) assume color transit moving closer to their closest corners in \(\mathbf{P}^{\prime}\), and the robots with color side1 move to the closest sides of \(\mathbf{P}^{\prime}\) assuming color side. In Round 1.3, some transit colored  
Figure 9: The five sub-stages, Stages 1.1–1.5, of Stage 1: (**a**) starting configuration for Stage 1, and (**b–f**) after Stages 1.1–1.5, respectively. robots become side robots of \(\mathbf{P}^{\prime}\), and, by the end of Round 1.4, all transit colored robots become side robots of \(\mathbf{P}^{\prime}\). Stage 1 organizes into five sub-stages in the \(\mathcal{ASYNC}\) setting: Stage 1.1 translates Round 1.1, Stages 1.2, 1.3, and 1.5 translate Round 1.2, and Stage 1.4 translates Rounds 1.3 and 1.4 of the \(\mathcal{FSYNC}\) algorithm. Table 3 explains how robots explicitly synchronize stages (and sub-stages) so that a next stage begins only after the current stage finishes. * **Stage 1.1** is to move the corner robots of \(\mathbf{P}\) (Figure 9a) to the corners of \(\mathbf{P}^{\prime}\) (Figure 9b) so that all interior robots of \(\mathbf{P}\) see them (and vice versa). Each corner robot of \(\mathbf{P}\) first moves to some point inside the eligible area in the interior of \(\mathbf{P}\) and colors itself corner1. After that, each corner that has a robot in its corner triangle changes color to corner2. Otherwise, if an interior robot in present in \(\mathbf{P}^{\prime}\), it changes color to corner3, while if no interior robot is present, it changes color to corner. The side robots of \(\mathbf{P}\) first color themselves side1, and some of them later change color to special. * **Stage 1.2** is to move all the robots that are inside corner triangle \(T(c_{i})\) of each corner \(c_{i}\) of \(\mathbf{P}^{\prime}\) to the points on the corner line segment \(L_{i}\) and color them transit. After that, the corners of \(\mathbf{P}^{\prime}\) with color corner2 change color to corner3. Figure 9c shows how the robots inside \(T(c_{3})\) in Figure 9b move to \(L_{3}\). * **Stage 1.3** is to position all the robots that are inside \(\mathbf{S}(\mathbf{P}^{\prime})\) to the corner and triangle line segments of the corners of \(\mathbf{P}^{\prime}\). The robots change color to transit after they reach their respective (corner or triangle) line segments. Figure 9d shows how the robots inside \(\mathbf{S}(\mathbf{P}^{\prime})\) in Figure 9c move to triangle and corner line segments. * **Stage 1.4** is to move the robots in the corner line segments and the triangle line segments to the sides of \(\mathbf{P}^{\prime}\).