On Minimal Perimeter Polyminoes

Altshuler, Yaniv; Yanovsky, Vladimir; Vainsencher, Daniel; Wagner, Israel A.; Bruckstein, Alfred M.

doi:10.1007/11907350_2

Cited by 16 publications

(19 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We are interested in the minimal number of tiles which can become contaminated at this stage. The minimal number of 4Neighbors of any number of tiles is achieved when the tiles are organized in the shape of a "digital sphere" (see a complete proof in Altshuler et al (2006b) or Appendix 9.8, and a full discussion of isoperimetric inequalities for discrete grids and optimally packed shapes in Vainsencher and Bruckstein (2008)), as demonstrated in Figure 1. Therefore, for a region of any given area S the minimal number of its 4Neighbors is at least 2 √ 2 • S t − 1.…”

Section: Discussionmentioning

confidence: 99%

Multi-agent Cooperative Cleaning of Expanding Domains

Altshuler

Yanovski

Wagner

et al. 2010

The International Journal of Robotics Research

Self Cite

View full text Add to dashboard Cite

Several recent works considered multi-a(ge)nt robotics in static environments. In this work we examine ways of operating in dynamic environments, where changes take place independently of the agents' activity. The work focuses on a dynamic variant of the cooperative cleaners problem, a problem that requires several simple agents to clean a connected region of "dirty" pixels in Z 2 . A number of simple agents move in this dirty region, each having the ability to "clean" the place it is located in. Their goal is to jointly clean the given dirty region. The dynamic variant of the problem involves a deterministic expansion of dirt in the environment, simulating spreading of contamination or fire. Theoretical lower bounds for the problem are presented, as well as various impossibility results. A cleaning protocol for the problem is presented, and a wealth of experimental results testing its performance in comparison to the lower bounds. Several analytic upper bounds for the proposed protocol are also presented, accompanied with appropriate experimental results.

show abstract

Section: Discussionmentioning

confidence: 99%

Multi-agent Cooperative Cleaning of Expanding Domains

Altshuler

Yanovski

Wagner

et al. 2010

The International Journal of Robotics Research

Self Cite

View full text Add to dashboard Cite

show abstract

“…The present notion of roundness is distinct from the one given in [6] where they consider minimizing the site perimeter of lattice sets, that is the number of points with Manhattan distance 1 from the sets. For a given N, Eq.…”

Section: Fig 2 a Polyomino (A) V-convex (B) H-convex (C) Hv-convex mentioning

confidence: 98%

Discrete sets with minimal moment of inertia

Brlek

Labelle

Lacasse

2008

Theoretical Computer Science

View full text Add to dashboard Cite

a b s t r a c tWe analyze the moment of inertia I(S), relative to the center of gravity, of finite plane lattice sets S. We classify these sets according to their roundness: a set S is rounder than a set T if I(S) < I(T ). We introduce the notion of quasi-discs and show that roundest sets are strongly-convex quasi-discs in the discrete sense. We use weakly unimodal partitions and an inequality for the radius to make a table of roundest discrete sets up to size 40. Surprisingly, it turns out that the radius of the smallest disc containing a roundest discrete set S is not necessarily the radius of S as a quasi-disc.Crown

show abstract

“…In this paper we are using these two types of neighbourhood relations. In this paper we recall some results of [1,29] (regarding isoperimetric inequality in the square grid), moreover we give an alternative (and in some sense simpler) proof of the result based on combinatorics ( [25]). Our main results are extensions of these results to the triangular grid with the two types of Jordan neighbourhood.…”

Section: Introductionmentioning

confidence: 99%