Time-Color Tradeoff on Uniform Circle Formation by Asynchronous Robots
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Bibliographic record
Abstract
We consider the distributed setting of n autonomous mobile robots operating in Look-Compute-Move (LCM) cycles on a plane. Robots are equipped with lights (i.e., the robots with lights model) that can assume a color at a time from a fixed color set. We consider obstructed visibility in which a robot cannot see another robot if a third robot is positioned between them on the straight line connecting them. Robots are said to collide if they share positions or their paths intersect within concurrent LCM cycles. In this paper, we consider the problem of Uniform Circle Formation, where starting from distinct initial locations in the plane, the robots relocate autonomously to occupy positions on the vertices of a regular n-gon not fixed in advance. The objective is to simultaneously minimize (or provide tradeoff between) two fundamental performance metrics: (i) time to solve Uniform Circle Formation and (ii) size of the color set used by each robot light. There exists an O(1)-time O(1)-color algorithm for this problem in the fully synchronous and semi-synchronous settings and O(log n)-time O(1)-color algorithm in the asynchronous setting, avoiding collisions. In this paper, we consider the asynchronous setting and develop a deterministic generic algorithmic framework that provides time-color tradeoff on solving Uniform Circle Formation avoiding collisions. Specifically, our framework achieves a solution with time O(x) using $O\left( {{n^{1/{2^x}}}} \right)$ colors in the asynchronous setting. Setting x some constant, we achieve the first asynchronous, asymptotically time-optimal, algorithm with O(1) time using $O(\sqrt n )$ colors, whereas setting x = O(log log n), we) achieve the second asynchronous, asymptotically color-optimal, algorithm with O(log log n time using O(1) colors. In sum, our framework shows the size of the color set provides a tradeoff on time for Uniform Circle Formation in the asynchronous setting.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.003 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it