Minimizing Query Frequency to Bound Congestion Potential for Moving Entities at a Fixed Target Time^†

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper studies the problem of avoidance collision with many moving robots, through querying the exact locations of some robots (those unqueried entities would grow into a large uncertain region). The problem is to minimize the query cost to minimize the uncertainty max-degree, ply and thickness of the uncertainty regions.

The paper is well-written and since I reviewed the conf version, I would pay more attention to the difference between the 2 versions. Technically, Section 3 is very much in the conf version, but Section 4 (for the continuous case) is new compared with the conf version. However, the proof of Lemma 1 was completely omitted; instead, only a ref to an arXiv paper was given. I am not sure what is the best way to handle this. ArXiv is unrefereed, but completely copying the proof from it doesn't seem to be the best solution, so maybe you can have a condensed version of proof? Apparently, Lemma 1 is crucial for the continuous case.

Other than that, the paper is well-written and could be accepted.

Author Response

As the reviewer has noted, Section 4 addresses issues that arise when the objective is to minimize congestion continuously, rather than at some fixed time. We make no claim to address this issue in a comprehensive way in this paper.  The “companion paper” that we refer to is in preparation for journal submission; a combined ArXiv version is available for the interested reader. The proof of Lemma 1 (renamed as Proposition 1, to help clarify its auxiliary role in the current paper) is contained in the ArXiv version but is not essential for determining the validity of the results in the current paper. Proposition 1 is stated only to provide motivation for Lemma 2 (now renamed Lemma 1), that describes a generalization of our earlier fixed-target-time results.  To quote from the start of Section 4:
    "In the following subsection we outline how, once suitably initialized, accurate perception of x-separation can be sustained using just-in-time queries              based on perceived x-separation, leaving the details to the companion paper. This serves to motivate the second subsection in which we describe how     the   accurate perception of x-separation can be initialized at some time t_0 > 0, from a state of unbounded uncertainty of entity locations at time zero,     using query granularity that is competitive with any other initialization scheme. This is achieved by a modified version of the FTT-degree[x+∆] scheme     of Section 3, using higher query frequency and a more restrictive criterion than (x+∆)-degree-safety. "

Reviewer 2 Report

Comments and Suggestions for Authors

In the the introduction you give an example of robots moving with bounded speed along unpredictable trajectories. For me this is somehow a very constructed and unnatural scenario. When the trajectories can be arbitrary - why should the speed be constant. Are there more (realistic) examples where your result could be applied ?

Author Response

Having an upper bound on speed, which of course need not be constantly, or even ever, realized, is completely realistic.  Arbitrary trajectories simply model the absence of information available to the observer.  To assume otherwise, while potentially more accurate in reflecting some situations, such as when queries reveal instantaneous location and speed or direction, would limit the broader applicability of our model.