We consider the problem of exploring an anonymous unoriented ring by a team of k identical, oblivious, asynchronous mobile robots that can view the environment but cannot communicate. This weak scenario is standard when the spatial universe in which the robots operate is the two-dimensional plane, but (with one exception) has not been investigated before for networks. Our results imply that, although these weak capabilities of robots render the problem considerably more difficult, ring exploration by a small team of robots is still possible.

We first show that, when k and n are not co-prime, the problem is not solvable in general, e.g., if k divides n there are initial placements of the robots for which gathering is impossible. We then prove that the problem is always solvable provided that n and k are co-prime, for kź17, by giving an exploration algorithm that always terminates, starting from arbitrary initial configurations. Finally, we consider the minimum number ź(n) of robots that can explore a ring of size n. As a consequence of our positive result we show that ź(n) is O(logn). We additionally prove that Ω(logn) robots are necessary for infinitely many n.

/pdf/computing-without-communicating-ring-exploration-by-4brne4w5vo.pdf

Computing Without Communicating: Ring Exploration by Asynchronous Oblivious Robots

We consider the problem of exploring an anonymous unoriented ring by a team of k identical, oblivious, asynchronous mobile robots that can view the environment but cannot communicate. This weak scenario is standard when the spatial universe in which the robots operate is the two-dimentional plane, but (with one exception) has not been investigated before. We indeed show that, although the lack of these capabilities renders the problems considerably more difficult, ring exploration is still possible.

We show that the minimum number ρ(n) of robots that can explore a ring of size n is O(log n) and that ρ(n) = Ω(log n) for arbitrarily large n. On one hand we give an algorithm that explores the ring starting from any initial configuration, provided that n and k are co-prime, and we show that there always exist such k in O(log n). On the other hand we show that Ω(log n) agents are necessary for arbitrarily large n. Notice that, when k and n are not co-prime, the problem is sometimes unsolvable (i.e., there are initial configurations for which the exploration cannot be done). This is the case, e.g., when k divides n.

/pdf/computing-without-communicating-ring-exploration-by-3771u7engk.pdf

Computing without communicating: ring exploration by asynchronous oblivious robots

We give the first efficient (1−e)-approximation algorithm for the following problem: Given an axis-parallel d-dimensional box R in ℝ
 d
 containing n points, compute a maximum-volume empty axis-parallel
 d-dimensional box contained in R. The minimum of this quantity over all such point sets is of the order $\Theta (\frac {1}{n} )$
 . Our algorithm finds an empty axis-aligned box whose volume is at least (1−e) of the maximum in O((8ede
 −2)
 d
 ⋅nlog
 d
 n) time. No previous efficient exact or approximation algorithms were known for this problem for d≥4. As the problem has been recently shown to be NP-hard in arbitrarily high dimensions (i.e., when d is part of the input), the existence of an efficient exact algorithm is unlikely. We also present a (1−e)-approximation algorithm that, given an axis-parallel d-dimensional cube R in ℝ
 d
 containing n points, computes a maximum-volume empty axis-parallel hypercube contained in R. The minimum of this quantity over all such point sets is also shown to be of the order $\Theta (\frac{1}{n} )$
 . A faster (1−e)-approximation algorithm, with a milder dependence on d in the running time, is obtained in this case.

On the Largest Empty Axis-Parallel Box Amidst n Points

This is a short introduction to the theory of Stirling numbers of the second kind S(m, k) from the point of view of analysis. It is written as an historical survey centered on the representation of...

https://www.jstor.org/stable/10.4169/math.mag.85.4.252

Close Encounters with the Stirling Numbers of the Second Kind

This is a historical introduction to the theory of Stirling numbers of the second kind S(n,k) from the point of view of analysis. We tell the story of their birth in the book of James Stirling (1730) and show how they mature in the works of Johann Grunert (1843). We demonstrate their usefulness in several differentiation formulas. The reader can also see the connection to Bernoulli numbers, to Euler polynomials and to power sums.

Sebastian Martin Ruiz

Papers

80.52 An algebraic identity leading to Wilson’s theorem *

81.27 A result on prime numbers