# Constant-Time Complete Visibility for Robots with Lights: The Asynchronous Case +
## 6 Stage 1-Interior Depletion
### Stage 1.1-Making Corners of \(\mathbf{P}\) the Corners of \(\mathbf{P}^{\prime}\)
#### 6.2.1 Stage 1.2.1: Moving Four Interior Robots in \(\mathbf{P^{\prime}}\) to \(L_{i}\)
Moreover, \(c_{i}\) sees all the robots on \(EL_{i}\) since there is no other robot in the interior of \(\mathbf{P}^{\prime}\) divided by \(EL_{i}\) towards \(c_{i}\). Observe that it is possible for some corner \(c_{j}\) of \(\mathbf{P}^{\prime}\) to not have (at least) four eligible robots in \(EL_{j}\) even after all the robots inside \(CP(c_{j})\) moved to \(EL_{j}\). In this case, \(c_{j}\) assumes color corner5 (directly from corner3). Observe also that there can be, at most, eight robots on \(EL_{i}\) before any corner \(c_{i}\) changes its color to corner4 since all four robots \(t_{1},t_{2},t_{3},t_{4}\) that are positioned on \(EL_{i}\) in Stage 1.2.1.2 are such that they are positioned within \(QC(c_{i})\) and four new robots inside \(CP(c_{i})\) need to be moved to \(EL_{i}\) so that there will be two robots to act as left beacons and two robots to act as right beacons on \(EL_{i}\). This configuration is needed for an admissible configuration for Stage 1.3. **Lemma 18**. : _During Stage 1.2.2, (at least) four internal robots of \(\mathbf{P}^{\prime}\) are positioned on the eligible lines and colored eligible. Stage 1.2.2 runs for \(\mathcal{O}(1)\) epochs avoiding collisions and Stage 1.2.2 starts only after Stage 1.2.1 finishes._  
Proof. : We have from Lemma 16 that, when Stage 1.2.1 finishes, any robot with color internal in the interior of \(\mathbf{P}^{\prime}\) sees \(c_{i}\) and all four eligible colored robots on \(EL_{i}\). Robot \(r_{i}\) also sees \(c_{i-1}\) and \(c_{i+1}\) if no robot is currently in transit to \(EL_{i}\) (irrespective of the interior robots in transit to \(EL_{i-1}\) and \(EL_{i+1}\)). Note that only the robots inside \(CP(c_{i})\) move to \(c_{i}\) since, otherwise, they will be closest to some other corner of \(\mathbf{P}^{\prime}\) than \(c_{i}\). Therefore, any robot \(r_{i}\) inside \(CP(c_{i})\) can see all \(c_{i},c_{i-1},c_{i+1}\) if no robot inside \(CP(c_{i})\) is currently in transit to \(EL_{i}\) (Lemma 17). Therefore, \(r_{i}\) can compute the cone areas \(QC^{\prime}(r_{i})\) and \(QC^{\prime\prime}(r_{i})\). Moreover, since \(r_{i}\) sees all \(c_{i},c_{i-1},c_{i+1}\), it can find whether there is some robot inside cone \(QC^{\prime}(r_{i})\) and/or \(QC^{\prime\prime}(r_{i})\). Therefore, \(r_{i}\) can correctly move to \(EL_{i}\). We now show that Stage 1.2.2 runs for \(\mathcal{O}(1)\) epochs. Note that since four robots are already on \(EL_{i}\) due to the execution in Stage 1.2.1, at most, four new robots move to \(EL_{i}\) at Stage 1.2.2. A robot can move to \(EL_{i}\) in two epochs; hence, it needs eight rounds to put four new robots on \(EL_{i}\). After that, the corner \(c_{i}\) can color itself corner4 in one epoch. The execution is collision-free since the robots moving to different eligible lines do not collide. Moreover, the robots moving to the same eligible line also do not collide since the straight lines (joining them with \(c_{i}\)) they follow to reach \(EL_{i}\) do not intersect before \(c_{i}\). Finally, Stage 1.2.2 starts only after Stage 1.2.1 finishes because the robots in the interior of \(\mathbf{P}^{\prime}\) with color internal see at least a corner with color \(\
otin\) {corner3, corner5, corner} during Stage 1.2.1. Stage 1.3-Positioning the Remaining Internal Robots of \(\mathbf{P}^{\prime}\) on the Eligible Lines  
In the beginning of Stage 1.3, all corners of \(\mathbf{P}^{\prime}\) have color \(\in\) {corner4, corner5, corner} with at least a corner colored corner4 (otherwise, there is no interior robot with color internal in \(\mathbf{P}^{\prime}\)). Using the \(\mathcal{O}(\log N)\)-epoch approach developed in IPDPS'17 [13], all interior robots of \(\mathbf{P}^{\prime}\) that are on the corner line segments have color eligible and the rest have color internal. The goal is to move the internal colored robots to position themselves on the corner line segments. This is done as follows: Each robot colored internal find the closest corner of \(\mathbf{P}^{\prime}\).