Tomographic Equivalence and Switching Operations

Abstract

A binary picture on an arbitrary grid is a mapping f from the set of all grid points to {0,1} such that f (x) = 1 for only finitely many grid points x. If two binary pictures f 1 and f 2 on the same grid have the property that for every grid line P the sets {p ∈ l| f 1(p)= 1} and{ p ∈ l|f 2(p)= 1} contain exactly the same number of grid points, then we say that f 1 and f 2 are tomographically equivalent. Given a binary picture f on the usual 2-dimensional square grid, there may exist an upright rectangle R (of any size) whose sides are grid lines, such that f = 1 at two diagonally opposite corner points of R and f= 0 at the other two corner points. If so, then we call the process of changing the value of the picture f from 1 to 0 and 0 to 1 at the four corner points of R (without changing the value of f at any other grid point) a rectangular 4-switch. Ryser showed in the 1950s that two binary pictures on the square grid are tomographically equivalent if and only if one picture can be transformed to the other by a finite sequence of rectangular 4-switches. We present a few different versions of this theorem, describe an application, and also give a proof of the result. We then show that the result has no analog on grids that have grid lines in three or more directions (such as the 3-dimensional cubic grid),because on such grids one can find for every integer L two tomographically equivalent binary pictures that differ at more than L grid points and are not tomographically equivalent to any other binary picture.KeywordsGrid PointInduction HypothesisCorner PointGrid LineSwitching OperationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.