Parallel Thinning algorithm using KxK mask

International audienc

A Parallel Thinning Algorithm Using KxK Mask

International audienc

Meta-Heuristic Search and Square Erickson Matrices

Abstract — A Ramsey theory problem, that can be seen as a 2 dimensional extension of the Van der Waerden theorem, was posed by Martin J. Erickson in his book [1]: “find the minimum n such that if the lattice points of [n]×[n] are two-colored, there exist four points of one color lying on the vertices of a square with sides parallel to the axes”. This was solved recently by Bacher and Eliahou in 2009 [2], who showed thatn = 15. In this paper we tackle a derived version of this problem, searching for the minimumnthat forces the existence of a monochromatic [3]×[3] subgrid of [n]×[n] of the form{i,i +t,i + 2t}× {j,j+t,j+2t}. We use meta-heuristics on this open problem to find instances of 2-colorations without monochromatic [3]×[3] subgrid of the above form, setting a lower bound onn. In particular we found such a binary square grid of size 662, implying thatn>662. I